Extended Least Squares Making Evident Nonlinear Relationships between Variables: Portfolios of Financial Assets

Abstract

This research work extends the least squares criterion. The regression models which have been treated so far in the literature do not study multilinear relationships between variables. Such relationships are of a nonlinear nature. They take place whenever two or more than two univariate variables are the components of a multiple variable of order 2 or an order greater than 2. A multiple variable of order 2 is not a bivariate variable, and a multiple variable of an order greater than 2 is not a multivariate variable. A multiple variable allows for the construction of a tensor. The $\alpha$-norm of this tensor gives rise to an aggregate measure of a multilinear nature. In particular, given a multiple variable of order 2, four regression lines can be estimated in the same subset of a two-dimensional linear space over $\mathbb{R}$. How these four regression lines give rise to an aggregate measure of a multilinear nature is shown by this paper. In this research work, such a measure is an estimate concerning the expected return on a portfolio of financial assets. The metric notion of $\alpha$-product is used to summarize the sampling units which are observed.

1. Introduction

Relationships between variables can be of a linear or nonlinear nature. In this paper, particular nonlinear relationships are made evident. Their intrinsic nature is multilinear. Multilinear relationships between variables take place whenever two or more than two univariate variables are the components of a multiple variable of order 2 or greater than 2. It is possible to write $X_{12}=\{{ }_{1}X,{ }_{2}X\}$, where ${ }_{1}X$ and ${ }_{2}X$ are the components of $X_{12}$. If one writes $X_{123}={\{}_{1}X,{ }_{2}X,{ }_{3}X\}$, then ${ }_{1}X$, ${ }_{2}X$, and ${ }_{3}X$ are the components of $X_{123}$. Similarly, if one writes $X_{12\dots m}=\{{ }_{1}X,{ }_{2}X,{ }_{3}X,\dots ,{ }_{m}X\}$, then ${ }_{1}X$, ${ }_{2}X$, ${ }_{3}X$, …, and ${ }_{m}X$ are the components of $X_{12\dots m}$. Within this context, $X_{12}$, $X_{123}$, and $X_{12\dots m}$ are multiple variables. Their components are univariate variables which influence each other. A multiple variable of order 2 is not a bivariate variable, and a multiple variable of order greater than 2 is not a multivariate variable. This paper will clarify why multilinear indices associated with a multiple variable of order 2 are based on $2^2=4$ bivariate distributions of probability. Similarly, it will also be clarified why multilinear indices associated with a multiple variable of order m, where $m>2$ is an integer, are based on $m^2$ bivariate distributions of probability. Each bivariate distribution of probability underlying a multilinear index is of a two-dimensional nature. Hence, it is possible to decompose it into two marginal probability distributions. This paper focuses on a multilinear approach of a Bayesian nature according to which more than two univariate variables are always studied in pairs. Since a multilinear and quadratic metric is used to study multilinear relationships between variables, various indices of a bivariate nature are firstly taken into account. This means that a disaggregation process first takes place. An aggregation process takes place second whenever a multilinear measure is shown. For these reasons, in this paper, given a multiple variable of order 2, four regression lines are estimated in the same subset of a two-dimensional linear space over $\mathbb{R}$. The multilinear regression model introduced in this research work is different from those regression models which have been treated so far in the literature (see Aneiros et al. 2022; Ewald and Schneider 2020). The multilinear regression model handled in this paper allows us to make fair estimates based on the notion of prevision (see Cassese et al. 2020). Prevision consists of considering all the observed values drawn by a given population in order to distribute among them those nonnegative and finitely additive masses such that only absolutely inadmissible evaluations have to be excluded (see Angelini 2024b). Two dynamical phases are then involved with respect to the notion of prevision. They depend on a specific state of information and knowledge associated with a given individual. This state is intrinsically variable. The first phase gives rise to a closed convex set. This is because possible opinions about the evaluations of probability are infinite in number. Strictly speaking, if the observed values are put on a straight line, then a closed line segment appears in the first phase. Every point belonging to this closed line segment is a fair estimate of the parameter under consideration. The second phase gives rise to the choice of a point belonging to the closed line segment into account. This choice avoids that an extrapolation process takes place outside the closed line segment under consideration (see Berti et al. 2001). The notion of prevision is different from the one of prediction. Within this context, to make a prediction does not mean to handle barycenters of masses in the first phase and choosing one of them in the second one. To make a prediction means to make a prophecy. Hence, one out of the possible and observed values belonging to the set of possible alternatives can arbitrarily be chosen. It is then a prophecy. For this reason, one says that a prediction transforms the uncertainty into an artificial certainty (see Angelini 2024a). The distinction between prevision and prediction is not clear-cut in a non-specific language. Nevertheless, if one wants to identify an existing dichotomy, then this distinction is appropriate and necessary. There is therefore a fundamental difference: after knowing the specific outcome, a prediction can be said to be right or wrong in an objective way; conversely, whatever happens, nothing similar in any sense can be said about a prevision. The intrinsic nature of a prevision is then subjective.

The Objectives of This Paper

Given two variables ${ }_{1}X$ and ${ }_{2}X$, if one writes ${ }_{2}X=f\left({ }_{1}X\right)$, then ${ }_{2}X$ is the dependent or explained variable and ${ }_{1}X$ is the independent or explanatory variable. Whenever a specific value of ${ }_{1}X$ is predicted, the corresponding level associated with the other univariate variable ${ }_{2}X$ can be one out of the observed values, or it can be found beyond them. If this specific value is found beyond observed values, then it can be found either to the left or right of them. The further one extrapolates beyond the observed values, the greater the risk one runs. Whenever the model under consideration is extrapolated far beyond the given sampling data, the result can become meaningless. In this paper, after studying four linear relationships between the two components, ${ }_{1}X$ and ${ }_{2}X$, of a multiple variable of order 2 denoted by $X_{12}$, one is always inclined to avoid the fact that aggregate estimates of a specific parameter take place by choosing, as their constituent elements, points outside the closed line segments belonging to the four regression lines, respectively. Such lines capture the given sampling units via the least squares criterion. For this reason, one refers oneself to the notion of prevision. Such a notion takes all given sampling units into account. It is possible to summarize them using the concept of $\alpha$-product. Such a concept is of a metric nature. In Section 2, the two-variable multilinear regression model treated in this paper is conceptually and formally shown. In Section 3, a specific regression line is estimated. In Section 4, another regression line is estimated using the same observations in a different way. In Section 5, two regression lines, whose least squares residuals are intrinsically equal to zero, are handled. Section 6 shows that the notion of $\alpha$-product is an essential tool of prevision. In Section 7, the bivariate correlation coefficient is obtained using an aggregate measure of a multilinear nature. This means that the bivariate correlation coefficient can intrinsically be referred to a multiple variable of order 2. Section 8 contains how the multilinear regression model treated in this paper can be used to make rational estimates related to the expected return on portfolios of financial assets. Section 9 shows that a tensor identifies a multiple variable. A fundamental difference between a multiple variable and a multivariate one is made evident. Moreover, the empirical validation of the proposed multilinear regression model is handled. Finally, Section 10 contains conclusions and some future perspectives.

2. The Two-Variable Multilinear Regression Model: Conceptual and Formal Properties

The two-variable multilinear regression model treated in this paper studies multilinear relationships between variables. $X_{12}$ is a multiple variable of order 2. Its components are denoted by ${ }_{1}X$ and ${ }_{2}X$, respectively. One writes

$$X_{12}=\{{ }_{1}X,{ }_{2}X\},$$

(1)

where ${ }_{1}X$ and ${ }_{2}X$ are two univariate variables. The two-variable multilinear regression model handled in this research work is composed of four regression lines. They are depicted in the same subset of a two-dimensional linear space over $\mathbb{R}$. They are studied in the same subset of the Cartesian plane. With respect to these four regression lines, two regression lines are estimated. The other two regression lines exactly coincide with the same 45-degree line. Such a model is formally given by

$$\mathrm{two-variable}\mathrm{multilinear}\mathrm{regression}=\left\{\begin{array}{c}{ }_{2}X=\alpha +\beta {(}_{1}X),\hfill \\ { }_{1}X={\alpha }^{*}+{\beta }^{*}{(}_{2}X),\hfill \\ { }_{2}X={ }_{1}X,\hfill \\ { }_{1}X={ }_{2}X.\hfill \end{array}\right.$$

(2)

Since ${ }_{1}X$ and ${ }_{2}X$ are the two components of $X_{12}$, it is possible to observe that ${ }_{1}X$ influences ${ }_{2}X$, and ${ }_{2}X$ influences ${ }_{1}X$. There is no one-way causation. Hence, ${ }_{1}X$ influences ${ }_{2}X$. Since there is feedback in the opposite direction, ${ }_{2}X$ influences ${ }_{1}X$ as well. Four different pairs of marginal variables are studied. They are $({ }_{1}X,{ }_{1}X)$, $({ }_{1}X,{ }_{2}X)$, $({ }_{2}X,{ }_{1}X)$, and $({ }_{2}X,{ }_{2}X)$. For this reason, a multilinear relationship between ${ }_{1}X$ and ${ }_{2}X$ is dealt with.

3. An Estimated Regression Line

Let

$$({ }_1{X}_i,{ }_2{X}_i),$$

(3)

with $i=1,\dots ,n$, be a sample of n ordered pairs of observations associated with ${ }_{1}X$ and ${ }_{2}X$, respectively. Given this sample, whose size is equal to n, the linear relationship between the two components of $X_{12}$ expressed by

$${ }_{2}X=\alpha +\beta \left({ }_{1}X\right)$$

(4)

is first studied. It is possible to enlarge (4) by considering

$${ }_{2}X=\alpha +\beta \left({ }_{1}X\right)+u,$$

(5)

where u is a stochastic variable which is normally distributed. One writes

$${ }_2{X}_i=\alpha +\beta \left({ }_1{X}_i\right)+u_i,$$

(6)

with $i=1,\dots ,n$, where it turns out to be

$$\left\{\begin{array}{c}\mathbb{E}\left(u_i\right)=0,\mathrm{for}\mathrm{all}i,\hfill \\ \mathrm{var}\left(u_i\right)=\mathbb{E}\left(u_i^2\right)={\sigma }^2,\mathrm{for}\mathrm{all}i,\hfill \\ \mathrm{cov}\left(u_i{u}_j\right)=\mathbb{E}\left(u_i{u}_j\right)=0,\mathrm{for}i\ne j.\hfill \end{array}\right.$$

(7)

The disturbances denoted by $u_i$ are independently and identically distributed, with zero mean and variance expressed by ${\sigma }^2$.

The scatter of n observations is conventionally studied inside the first quadrant of a two-dimensional Cartesian coordinate system. An estimation of the linear relationship coinciding with (5) is given by the straight line expressed by

$${ }_2\widehat{X}=a+b\left({ }_{1}X\right),$$

(8)

where ${ }_2\widehat{X}$ indicates the height of the line at ${ }_{1}X$. If a and b are selected to minimize the residual sum of squares given by

$$\mathrm{RSS}=\sum _{i=1}^n\left({ }_2{e}_i\right){ }^2=\sum _{i=1}^n{\left({ }_2{X}_i-a-b\left({ }_1{X}_i\right)\right)}^2=f(a,b),$$

(9)

then the least squares principle takes place (see Watson 1967; Bartlett and Macdonald 1968). The necessary conditions for a stationary value of RSS are such that it is possible to write

$$a={ }_2\overline{X}-b\left({ }_1\overline{X}\right),$$

(10)

and

$$b={\displaystyle \frac{{\sum }_{i=1}^n{ }_{1}x{ }_i{ }_{2}x{ }_i}{{\sum }_{i=1}^n\left({ }_{1}x{ }_i\right){ }^2}},$$

(11)

where one has

$${ }_1{x}_i={ }_1{X}_i-{ }_1\overline{X},$$

(12)

with $i=1,\dots ,n$, and

$${ }_2{x}_i={ }_2{X}_i-{ }_2\overline{X},$$

(13)

with $i=1,\dots ,n$. Once the sample means have been calculated, the data characterizing (11) are expressed using deviations from them. ${ }_1\overline{X}$ and ${ }_2\overline{X}$ denote the sample means of ${ }_{1}X$ and ${ }_{2}X$, respectively.

The least squares line passes through the mean point given by $({ }_1\overline{X},{ }_2\overline{X})$. The least squares residuals have zero correlation in the sample with the values of ${ }_{1}X$. They have also zero correlation with the values denoted by ${ }_2\widehat{X}$. The decomposition of the sum of squares associated with the linear regression of ${ }_{2}X$ on ${ }_{1}X$ is written as

$$\mathrm{TSS}=\mathrm{ESS}+\mathrm{RSS},$$

(14)

where one has

$$\left\{\begin{array}{c}\mathrm{TSS}=\mathrm{total}\mathrm{sum}\mathrm{of}\mathrm{squared}\mathrm{deviations}\mathrm{in}\mathrm{the}{ }_{2}X\mathrm{variable},\hfill \\ \mathrm{RSS}=\mathrm{residual}\mathrm{sum}\mathrm{of}\mathrm{squares}\mathrm{from}\mathrm{the}\mathrm{regression}\mathrm{of}{ }_{2}X\mathrm{on}{ }_{1}X,\hfill \\ \mathrm{ESS}=\mathrm{explained}\mathrm{sum}\mathrm{of}\mathrm{squares}\mathrm{from}\mathrm{the}\mathrm{regression}\mathrm{of}{ }_{2}X\mathrm{on}{ }_{1}X.\hfill \end{array}\right.$$

(15)

Since it is possible to write

$$r^2={\displaystyle \frac{\mathrm{ESS}}{\mathrm{TSS}}},$$

(16)

where the coefficient of determination given by $r^2$ is the square of the sample correlation coefficient expressed by r, (16) is the proportion of the ${ }_{2}X$ variation attributable to the linear regression of ${ }_{2}X$ on ${ }_{1}X$ (see Fisher 1922). The limits of r are $\pm 1$. In the limiting case, the sample points all lie on a single straight line. The disturbance variance denoted by ${\sigma }^2$ cannot be estimated from a sample of u values denoted by $u_i$, $i=1,\dots ,n$ (see Liu et al. 2020). These values are unobservable. If one knows $\alpha$ and $\beta$, then they can be measured. Nevertheless, $\alpha$ and $\beta$ are unknown. It follows that the disturbance variance can be estimated from the calculated residuals. It is accordingly possible to use the following estimation expressed by

$${ }_2{s}^2={\displaystyle \frac{{\sum }_{i=1}^n\left({ }_2{e}_i\right){ }^2}{n-2}}$$

(17)

to assess the disturbance variance.

Linear estimators of $\alpha$ and $\beta$ are unbiased (see Puntanen and Styan 1989). One writes

$$\mathbb{E}\left(a\right)=\alpha$$

(18)

and

$$\mathbb{E}\left(b\right)=\beta .$$

(19)

Their sampling variances are the smallest that can be achieved by any linear unbiased estimator. One writes

$$\mathrm{var}\left(a\right)={\sigma }^2\left[{\displaystyle \frac{1}{n}}+{\displaystyle \frac{{{(}_1\overline{X})}^2}{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}}\right]$$

(20)

and

$$\mathrm{var}\left(b\right)={\displaystyle \frac{{\sigma }^2}{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}}.$$

(21)

The sampling distribution of the intercept term denoted by a is given by

$$a\sim N\left(\alpha ,{\sigma }^2\left[{\displaystyle \frac{1}{n}}+{\displaystyle \frac{{{(}_1\overline{X})}^2}{{\sum }_{i=1}^n{\left({ }_1{x}_i\right)}^2}}\right]\right),$$

(22)

whereas the sampling distribution of the least squares slope denoted by b is written as

$$b\sim N\left(\beta ,{\displaystyle \frac{{\sigma }^2}{{\sum }_{i=1}^n{\left({ }_1{x}_i\right)}^2}}\right).$$

(23)

Since ${\sigma }^2$ is unknown, it is necessary to use (17). It is an unbiased estimator because one observes

$$\mathbb{E}\left({\displaystyle \frac{{\sum }_{i=1}^n\left({ }_2{e}_i\right){ }^2}{n-2}}\right)={\sigma }^2.$$

(24)

One observes

$${\displaystyle \frac{{\sum }_{i=1}^n\left({ }_2{e}_i\right){ }^2}{{\sigma }^2}}\sim {\chi }^2(n-2).$$

(25)

Also, another fundamental result associated with inference procedures is that ${\sum }_{i=1}^n\left({ }_2{e}_i\right){ }^2$ is distributed independently of $f(a,b)$. Tests on the intercept are based on the t distribution, so one writes

$${\displaystyle \frac{a-\alpha }{{ }_{2}s\sqrt{\left(\frac{1}{n}+\frac{{{(}_1\overline{X})}^2}{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}\right)}}}\sim t(n-2).$$

(26)

Accordingly, a $100(1-ϵ)$ percent confidence interval for $\alpha$ is expressed by

$$a\pm t_{ϵ/2}{ }_{2}s\sqrt{\left({\displaystyle \frac{1}{n}}+{\displaystyle \frac{{{(}_1\overline{X})}^2}{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}}\right)},$$

(27)

whereas the null hypothesis given by $H_0: \alpha ={\alpha }_0$ is rejected at the $100ϵ$ percent level of significance if one observes

$$\left|{\displaystyle \frac{a-{\alpha }_0}{{ }_{2}s\sqrt{\left(\frac{1}{n}+\frac{{{(}_1\overline{X})}^2}{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}\right)}}}\right|>t_{ϵ/2}.$$

(28)

Tests on b are based on the t distribution as well, so one writes

$${\displaystyle \frac{b-\beta }{\frac{{ }_{2}s}{\sqrt{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}}}}\sim t(n-2).$$

(29)

A $100(1-ϵ)$ percent confidence interval for $\beta$ is expressed by

$$b\pm t_{ϵ/2}{\displaystyle \frac{{ }_{2}s}{\sqrt{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}}},$$

(30)

whereas the null hypothesis given by $H_0: \beta ={\beta }_0$ is rejected at the $100ϵ$ percent level of significance if one observes

$$\left|{\displaystyle \frac{b-{\beta }_0}{\frac{{ }_{2}s}{\sqrt{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}}}}\right|>t_{ϵ/2}.$$

(31)

Tests on ${\sigma }^2$ can be derived as well. It is possible to use a ${\chi }^2$ variable to derive them.

The test for the significance of ${ }_{1}X$ is given by $H_0: \beta =0$. It is known that there are three versions connected with it. First, this test can be set out in an analysis of variance framework using the F statistic expressed by

$$F={\displaystyle \frac{\mathrm{ESS}/1}{\mathrm{RSS}/(n-2)}}.$$

(32)

On the other hand, it is also possible to base it on the sample correlation coefficient or on the regression slope using the t distribution in both cases.

Given an estimated regression line with intercept denoted by a and slope expressed by b, it is possible to focus own attention on a specific value given by ${ }_1{X}_0$ of the regressor variable. For example, if one is required to predict a 95 percent confidence interval for ${ }_2{X}_0$, then one writes

$$\left(a+b\left({ }_1{X}_0\right)\right)\pm t_{0.025}{ }_{2}s\sqrt{\left(1+{\displaystyle \frac{1}{n}}+{\displaystyle \frac{{({ }_1{X}_0-{ }_1\overline{X})}^2}{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}}\right)}.$$

(33)

To test whether a new observation given by $({ }_1{X}_0,{ }_2{X}_0)$ comes from the structure generating the sample data, it is possible to compare this new observation with the confidence interval for ${ }_2{X}_0$. If one is required to predict a 95 percent confidence interval for $\mathbb{E}\left({ }_2{X}_0\right)$, then it is possible to write

$$\left(a+b\left({ }_1{X}_0\right)\right)\pm t_{0.025}{ }_{2}s\sqrt{\left({\displaystyle \frac{1}{n}}+{\displaystyle \frac{{({ }_1{X}_0-{ }_1\overline{X})}^2}{{\sum }_{i=1}^n{\left({ }_{1}x{ }_i\right)}^2}}\right)}.$$

(34)

In general, a sampling distribution shows the behavior of the estimators in repeated applications of the estimating formulae. A given sample provides a particular numerical estimation. Another sample from the same population will provide another numerical estimation. A sampling distribution shows the results that will be obtained for the estimators over the virtually infinite set of samples which could be drawn from the population into account. If one refers oneself to $\mathbb{E}\left({ }_{2}X\right|{ }_{1}X)=\alpha +\beta \left({ }_{1}X\right)$ or $\mathbb{E}\left({ }_{2}X\right|{ }_1{X}_i)=\alpha +\beta \left({ }_1{X}_i\right)$, $i=1,\dots ,n$, then the parameters of interest are $\alpha$, $\beta$, and ${\sigma }^2$. They are parameters of interest of a conditional distribution. In this conditional distribution, the only source of variation from one hypothetical sample to another one is a variation in the stochastic disturbance denoted by u. This stochastic disturbance, together with the given values denoted by ${ }_1{X}_1,\dots ,{ }_1{X}_n$, will determine the values denoted by ${ }_2{X}_1,\dots ,{ }_2{X}_n$. It follows that the sample values of a, b, and ${ }_2{s}^2$ take place. Accordingly, analyzing ${ }_{2}X$ conditional on ${ }_{1}X$ deals with the ${ }_1{X}_1,\dots ,{ }_1{X}_n$ values viewed as fixed in repeated sampling. This is based on the implicit assumption that the marginal distribution for ${ }_{1}X$ does not involve the parameters of interest. In other words, the marginal distribution for ${ }_{1}X$ contains no information on $\alpha$, $\beta$, and ${\sigma }^2$.

4. Another Estimated Regression Line

This paper focuses on a regression model which is not only interested in viewing ${ }_{1}X$ as the fixed regressor. This paper focuses on a model which is also interested in viewing ${ }_{2}X$ as the fixed regressor. Accordingly, n ordered pairs of observations of the form given by

$$({ }_2{X}_i,{ }_1{X}_i),$$

(35)

with $i=1,\dots ,n$, are now dealt with. One takes care to interchange ${ }_1{X}_i$ and ${ }_2{X}_i$. One presently studies n ordered pairs of the same previous observations. Nevertheless, the first element of the previous n pairs of observations becomes the second one and vice versa. At present, one refers oneself to $\mathbb{E}\left({ }_{1}X\right|{ }_{2}X)={\alpha }^{*}+{\beta }^{*}\left({ }_{2}X\right)$ or $\mathbb{E}\left({ }_{1}X\right|{ }_2{X}_i)={\alpha }^{*}+{\beta }^{*}\left({ }_2{X}_i\right)$, $i=1,\dots ,n$. If one refers oneself to $\mathbb{E}\left({ }_{1}X\right|{ }_{2}X)={\alpha }^{*}+{\beta }^{*}\left({ }_{2}X\right)$ or $\mathbb{E}\left({ }_{1}X\right|{ }_2{X}_i)={\alpha }^{*}+{\beta }^{*}\left({ }_2{X}_i\right)$, $i=1,\dots ,n$, then the parameters of interest are ${\alpha }^{*}$, ${\beta }^{*}$, and ${\sigma }^2$. They are parameters of interest of another conditional distribution (see Gilio and Sanfilippo 2014). In this conditional distribution, the only source of variation from one hypothetical sample to another one is a variation in the stochastic disturbance. It is again denoted by u. This stochastic disturbance, together with the given values denoted by ${ }_2{X}_1,\dots ,{ }_2{X}_n$, will determine the values denoted by ${ }_1{X}_1,\dots ,{ }_1{X}_n$. It follows that the sample values of $a^{*}$, $b^{*}$, and ${ }_1{s}^2$ take place. Accordingly, analyzing ${ }_{1}X$ conditional on ${ }_{2}X$ deals with the ${ }_2{X}_1,\dots ,{ }_2{X}_n$ values viewed as fixed in repeated sampling. This is based on the implicit assumption that the marginal distribution for ${ }_{2}X$ does not involve the parameters of interest. In other words, the marginal distribution for ${ }_{2}X$ contains no information on ${\alpha }^{*}$, ${\beta }^{*}$, and ${\sigma }^2$. The linear relationship between the two components of $X_{12}$ expressed by

$${ }_{1}X={\alpha }^{*}+{\beta }^{*}\left({ }_{2}X\right)$$

(36)

is studied second. It is possible to enlarge (36) by considering

$${ }_{1}X={\alpha }^{*}+{\beta }^{*}\left({ }_{2}X\right)+u,$$

(37)

where u is a stochastic variable which is normally distributed. One writes

$${ }_1{X}_i={\alpha }^{*}+{\beta }^{*}\left({ }_2{X}_i\right)+u_i,$$

(38)

with $i=1,\dots ,n$, where one observes

$$\left\{\begin{array}{c}\mathbb{E}\left(u_i\right)=0,\mathrm{for}\mathrm{all}i,\hfill \\ \mathrm{var}\left(u_i\right)=\mathbb{E}\left(u_i^2\right)={\sigma }^2,\mathrm{for}\mathrm{all}i,\hfill \\ \mathrm{cov}\left(u_i{u}_j\right)=\mathbb{E}\left(u_i{u}_j\right)=0,\mathrm{for}i\ne j.\hfill \end{array}\right.$$

The disturbances denoted by $u_i$ are independently and identically distributed, with zero mean and variance expressed by ${\sigma }^2$.

An estimation of the linear relationship coinciding with (37) is given by the straight line expressed by

$${ }_1\widehat{X}=a^{*}+b^{*}\left({ }_{2}X\right),$$

(39)

where ${ }_1\widehat{X}$ indicates the value of the straight line which is observed on the horizontal axis at ${ }_{2}X$. If $a^{*}$ and $b^{*}$ are selected to minimize the residual sum of squares given by

$$\mathrm{RSS}=\sum _{i=1}^n\left({ }_1{e}_i\right){ }^2=\sum _{i=1}^n{\left({ }_1{X}_i-a^{*}-b^{*}\left({ }_2{X}_i\right)\right)}^2=f(a^{*},b^{*}),$$

(40)

then the least squares principle takes place. The necessary conditions for a stationary value of RSS are such that it is possible to write

$$a^{*}={ }_1\overline{X}-b^{*}\left({ }_2\overline{X}\right)$$

(41)

and

$$b^{*}={\displaystyle \frac{{\sum }_{i=1}^n{ }_{2}x{ }_i{ }_{1}x{ }_i}{{\sum }_{i=1}^n\left({ }_{2}x{ }_i\right){ }^2}},$$

(42)

where one has

$${ }_1{x}_i={ }_1{X}_i-{ }_1\overline{X},i=1,\dots ,n$$

and

$${ }_2{x}_i={ }_2{X}_i-{ }_2\overline{X},i=1,\dots ,n.$$

Once the sample means have been calculated, the data characterizing (42) are expressed using deviations from them.

The least squares line passes through the mean point given by $({ }_1\overline{X},{ }_2\overline{X})$. The least squares residuals have zero correlation in the sample with the values of ${ }_{2}X$. They have also zero correlation with the values denoted by ${ }_1\widehat{X}$. The decomposition of the sum of squares associated with the linear regression of ${ }_{1}X$ on ${ }_{2}X$ is written as

$$\mathrm{TSS}=\mathrm{ESS}+\mathrm{RSS},$$

where one has

$$\left\{\begin{array}{c}\mathrm{TSS}=\mathrm{total}\mathrm{sum}\mathrm{of}\mathrm{squared}\mathrm{deviations}\mathrm{in}\mathrm{the}{ }_{1}X\mathrm{variable},\hfill \\ \mathrm{RSS}=\mathrm{residual}\mathrm{sum}\mathrm{of}\mathrm{squares}\mathrm{from}\mathrm{the}\mathrm{regression}\mathrm{of}{ }_{1}X\mathrm{on}{ }_{2}X,\hfill \\ \mathrm{ESS}=\mathrm{explained}\mathrm{sum}\mathrm{of}\mathrm{squares}\mathrm{from}\mathrm{the}\mathrm{regression}\mathrm{of}{ }_{1}X\mathrm{on}{ }_{2}X.\hfill \end{array}\right.$$

(43)

Since it is possible to write

$$r^2={\displaystyle \frac{\mathrm{ESS}}{\mathrm{TSS}}},$$

$r^2$ is the proportion of the ${ }_{1}X$ variation attributable to the linear regression of ${ }_{1}X$ on ${ }_{2}X$. The disturbance variance denoted by ${\sigma }^2$ cannot be estimated from a sample of u values denoted by $u_i$, $i=1,\dots ,n$. These values are unobservable. If one knows ${\alpha }^{*}$ and ${\beta }^{*}$, then they can be measured. Unfortunately, ${\alpha }^{*}$ and ${\beta }^{*}$ are unknown. It follows that the disturbance variance can be estimated from the calculated residuals. It is accordingly possible to use the following estimation expressed by

$${ }_1{s}^2={\displaystyle \frac{{\sum }_{i=1}^n\left({ }_1{e}_i\right){ }^2}{n-2}}$$

(44)

to assess the disturbance variance.

Linear estimators of ${\alpha }^{*}$ and ${\beta }^{*}$ are unbiased. One writes

$$\mathbb{E}\left(a^{*}\right)={\alpha }^{*}$$

(45)

and

$$\mathbb{E}\left(b^{*}\right)={\beta }^{*}.$$

(46)

Their sampling variances are the smallest that can be achieved by any linear unbiased estimator. One writes

$$\mathrm{var}\left(a^{*}\right)={\sigma }^2\left[{\displaystyle \frac{1}{n}}+{\displaystyle \frac{{\left({ }_2\overline{X}\right)}^2}{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}}\right]$$

(47)

and

$$\mathrm{var}\left(b^{*}\right)={\displaystyle \frac{{\sigma }^2}{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}}.$$

(48)

The sampling distribution of the intercept term denoted by $a^{*}$ is given by

$$a^{*}\sim N\left({\alpha }^{*},{\sigma }^2\left[{\displaystyle \frac{1}{n}}+{\displaystyle \frac{{\left({ }_2\overline{X}\right)}^2}{{\sum }_{i=1}^n{\left({ }_2{x}_i\right)}^2}}\right]\right),$$

(49)

whereas the sampling distribution of the least squares slope denoted by $b^{*}$ is written as

$$b^{*}\sim N\left({\beta }^{*},{\displaystyle \frac{{\sigma }^2}{{\sum }_{i=1}^n{\left({ }_2{x}_i\right)}^2}}\right).$$

(50)

Since ${\sigma }^2$ is unknown, it is necessary to use (44). It is an unbiased estimator because one observes

$$\mathbb{E}\left({\displaystyle \frac{{\sum }_{i=1}^n\left({ }_1{e}_i\right){ }^2}{n-2}}\right)={\sigma }^2.$$

(51)

One observes

$${\displaystyle \frac{{\sum }_{i=1}^n\left({ }_1{e}_i\right){ }^2}{{\sigma }^2}}\sim {\chi }^2(n-2).$$

(52)

Also, another fundamental result associated with inference procedures is that ${\sum }_{i=1}^n\left({ }_1{e}_i\right){ }^2$ is distributed independently of $f(a^{*},b^{*})$. Tests on the intercept are based on the t distribution, so one writes

$${\displaystyle \frac{a^{*}-{\alpha }^{*}}{{ }_{1}s\sqrt{\left(\frac{1}{n}+\frac{{\left({ }_2\overline{X}\right)}^2}{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}\right)}}}\sim t(n-2).$$

(53)

Accordingly, a $100(1-ϵ)$ percent confidence interval for ${\alpha }^{*}$ is expressed by

$$a^{*}\pm t_{ϵ/2}{ }_{1}s\sqrt{\left({\displaystyle \frac{1}{n}}+{\displaystyle \frac{{\left({ }_2\overline{X}\right)}^2}{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}}\right)},$$

(54)

whereas the null hypothesis given by $H_0: {\alpha }^{*}={\alpha }_0^{*}$ is rejected at the $100ϵ$ percent level of significance if one observes

$$\left|{\displaystyle \frac{a^{*}-{\alpha }_0^{*}}{{ }_{1}s\sqrt{\left(\frac{1}{n}+\frac{{\left({ }_2\overline{X}\right)}^2}{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}\right)}}}\right|>t_{ϵ/2}.$$

(55)

Tests on $b^{*}$ are based on the t distribution as well, so one writes

$${\displaystyle \frac{b^{*}-{\beta }^{*}}{\frac{{ }_{1}s}{\sqrt{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}}}}\sim t(n-2).$$

(56)

A $100(1-ϵ)$ percent confidence interval for ${\beta }^{*}$ is expressed by

$$b^{*}\pm t_{ϵ/2}{\displaystyle \frac{{ }_{1}s}{\sqrt{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}}},$$

(57)

whereas the null hypothesis given by $H_0: {\beta }^{*}={\beta }_0^{*}$ is rejected at the $100ϵ$ percent level of significance if one observes

$$\left|{\displaystyle \frac{b^{*}-{\beta }_0^{*}}{\frac{{ }_{1}s}{\sqrt{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}}}}\right|>t_{ϵ/2}.$$

(58)

Tests on ${\sigma }^2$ can be derived as well. A ${\chi }^2$ variable is used to derive them.

The test for the significance of ${ }_{2}X$ is given by $H_0: {\beta }^{*}=0$. There are three versions connected with it.

Given an estimated regression line with intercept denoted by $a^{*}$ and slope expressed by $b^{*}$, it is possible to focus own attention on a specific value given by ${ }_2{X}_0$ of the regressor variable. For example, if one is required to predict a 95 percent confidence interval for ${ }_1{X}_0$, then one writes

$$\left(a^{*}+b^{*}\left({ }_2{X}_0\right)\right)\pm t_{0.025}{ }_{1}s\sqrt{\left(1+{\displaystyle \frac{1}{n}}+{\displaystyle \frac{{({ }_2{X}_0-{ }_2\overline{X})}^2}{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}}\right)}.$$

(59)

To test whether a new observation given by $({ }_1{X}_0,{ }_2{X}_0)$ comes from the structure generating the sample data, it is possible to compare this new observation with the confidence interval for ${ }_1{X}_0$. If one is required to predict a 95 percent confidence interval for $\mathbb{E}\left({ }_1{X}_0\right)$, then it is possible to write

$$\left(a^{*}+b^{*}\left({ }_2{X}_0\right)\right)\pm t_{0.025}{ }_{1}s\sqrt{\left({\displaystyle \frac{1}{n}}+{\displaystyle \frac{{({ }_2{X}_0-{ }_2\overline{X})}^2}{{\sum }_{i=1}^n{\left({ }_{2}x{ }_i\right)}^2}}\right)}.$$

(60)

5. Two Regression Lines Whose Least Squares Residuals Are Equal to Zero

In addition to (3) and (35), this paper focuses on a regression model for which it is possible to consider n ordered pairs of observations of the form given by

$$({ }_1{X}_i,{ }_1{X}_i),$$

(61)

with $i=1,\dots ,n$. This means that n ordered pairs of observations all lie on the 45-degree line denoted by

$${ }_{2}X={ }_{1}X.$$

(62)

Its intercept is exactly equal to 0, whereas its slope is exactly equal to 1. The least squares residuals are intrinsically equal to 0 (see del Pino 1989). Also, in this limiting case, the least squares estimation coincides with the maximum likelihood one (see McCullagh 1983; Nelder and Pregibon 1987).

In addition to (3), (35), and (61), this paper shows a regression model for which it is also possible to study n ordered pairs of observations of the form expressed by

$$({ }_2{X}_i,{ }_2{X}_i),$$

(63)

with $i=1,\dots ,n$. This means that n ordered pairs of observations all lie on the 45-degree line denoted by

$${ }_{1}X={ }_{2}X.$$

(64)

Its intercept is exactly equal to 0, whereas its slope is exactly equal to 1. The least squares residuals are intrinsically equal to 0. In this limiting case, the least squares estimation coincides with the maximum likelihood one. From (62) and (64), it follows that it is true that the 45-degree line is the same graph depicted in the same subset of the Cartesian plane.

6. The Notion of α-Product as an Essential Tool of Prevision

With respect to the two estimated regression lines, one observes

$${ }_{2}X={ }_2\widehat{X}+{ }_{2}e,\mathrm{with}\sum { }_{2}e=0,$$

(65)

and

$${ }_{1}X={ }_1\widehat{X}+{ }_{1}e,\mathrm{with}\sum { }_{1}e=0.$$

(66)

This means that two n-dimensional vectors denoted by

$${ }_{\mathbf{2}}\mathbf{X}=\left[\begin{array}{c}{ }_2{X}_1\\ { }_2{X}_2\\ ⋮\\ { }_2{X}_n\end{array}\right]=\left[\begin{array}{c}{ }_2{X}_1{ }_2{X}_2\dots { }_2{X}_n\end{array}\right]$$

(67)

and

$${ }_{\mathbf{1}}\mathbf{X}=\left[\begin{array}{c}{ }_1{X}_1\\ { }_1{X}_2\\ ⋮\\ { }_1{X}_n\end{array}\right]=\left[\begin{array}{c}{ }_1{X}_1{ }_1{X}_2\dots { }_1{X}_n\end{array}\right]$$

(68)

are considered together with two estimated regression lines. One also writes

$${ }_{\mathbf{2}}\hat{\mathbf{X}}=\left[\begin{array}{c}{ }_2\widehat{X_1}\\ { }_2\widehat{X_2}\\ ⋮\\ { }_2\widehat{X_n}\end{array}\right]=\left[\begin{array}{c}{ }_2\widehat{X_1}{ }_2\widehat{X_2}\dots { }_2\widehat{X_n}\end{array}\right]$$

(69)

and

$${ }_{\mathbf{1}}\widehat{\mathbf{X}}=\left[\begin{array}{c}{ }_1\widehat{X_1}\\ { }_1\widehat{X_2}\\ ⋮\\ { }_1\widehat{X_n}\end{array}\right]=\left[\begin{array}{c}{ }_1\widehat{X_1}{ }_1\widehat{X_2}\dots { }_1\widehat{X_n}\end{array}\right].$$

(70)

These vectors belong to an n-dimensional linear space over $\mathbb{R}$ of a Euclidean nature. The last two vectors expressed by (69) and (70) are two $n\times 1$ or $1\times n$ matrices involving regression residuals given by ${ }_{2}e$ and ${ }_{1}e$, respectively. In fact, one observes

$${ }_2\widehat{X}={ }_{2}X-{ }_{2}e$$

(71)

and

$${ }_1\widehat{X}={ }_{1}X-{ }_{1}e.$$

(72)

In general, the scalar or inner product of two n-dimensional vectors is commutative, associative, and distributive. It may also be orthogonal, so the zero-product property does not hold. Since one can always consider the transpose of an $n\times 1$ or $1\times n$ matrix, one writes

$${\left({ }_{\mathbf{2}}\widehat{\mathbf{X}}\right)}^T{ }_{\mathbf{1}}\mathbf{X}=\left[\begin{array}{c}{ }_2\widehat{X_1}{ }_2\widehat{X_2}\dots { }_2\widehat{X_n}\end{array}\right]\left[\begin{array}{c}{ }_1{X}_1\\ { }_1{X}_2\\ ⋮\\ { }_1{X}_n\end{array}\right],$$

(73)

where ${ }_{\mathbf{2}}\widehat{\mathbf{X}}$ and ${ }_{\mathbf{1}}\mathbf{X}$ are two $n\times 1$ matrices, or

$${ }_{\mathbf{2}}\widehat{\mathbf{X}}{\left({ }_{\mathbf{1}}\mathbf{X}\right)}^T=\left[\begin{array}{c}{ }_2\widehat{X_1}{ }_2\widehat{X_2}\dots { }_2\widehat{X_n}\end{array}\right]\left[\begin{array}{c}{ }_1{X}_1\\ { }_1{X}_2\\ ⋮\\ { }_1{X}_n\end{array}\right],$$

(74)

where ${ }_{\mathbf{2}}\widehat{\mathbf{X}}$ and ${ }_{\mathbf{1}}\mathbf{X}$ are two $1\times n$ matrices. Anyway, one has

$${\left({ }_{\mathbf{2}}\widehat{\mathbf{X}}\right)}^T{ }_{\mathbf{1}}\mathbf{X}={ }_{\mathbf{2}}\widehat{\mathbf{X}}{\left({ }_{\mathbf{1}}\mathbf{X}\right)}^T.$$

(75)

The two-variable multilinear regression model treated in this research work uses the metric notion of $\alpha$-product (see Angelini 2023). For example, the sampling quantities of the following system

$$\left\{\begin{array}{cc}\sum { }_{2}X\hfill & =na+b\sum { }_{1}X\hfill \\ \sum { }_{1}X{ }_{2}X\hfill & =a\sum { }_{1}X+b\sum {\left({ }_{1}X\right)}^2\hfill \end{array}\right.$$

(76)

are five. They are n, $\sum { }_{1}X$, $\sum { }_{2}X$, $\sum { }_{1}X{ }_{2}X$, and $\sum {\left({ }_{1}X\right)}^2$. They characterize the normal equations for the linear regression of ${ }_{2}X$ on ${ }_{1}X$. In this paper, such quantities are $\alpha$-products. For example, the following

Table 1. How to obtain $\sum { }_{1}X$ using the notion of $\alpha$-product.

	Vector 2	1	…	1	Sum
Vector 1
${ }_1{X}_1$		1	…	0	1
⋮		⋮	⋱	⋮	⋮
${ }_1{X}_n$		0	…	1	1
Sum		1	…	1	n

shows that the notion of $\alpha$-product of two n-dimensional vectors is used to obtain $\sum { }_{1}X$. The following

Table 2. How to obtain $\sum { }_{1}X{ }_{2}X$ using the notion of $\alpha$-product.

	Vector 2	${ }_2{\mathit{X}}_1$	…	${ }_2{\mathit{X}}_{\mathit{n}}$	Sum
Vector 1
${ }_1{X}_1$		1	…	0	1
⋮		⋮	⋱	⋮	⋮
${ }_1{X}_n$		0	…	1	1
Sum		1	…	1	n

shows that the notion of $\alpha$-product of two n-dimensional vectors is used to obtain $\sum { }_{1}X{ }_{2}X$. Both two-way tables with n rows and n columns which have just been written use masses outside their main diagonal which are all equal to zero. In the first table, an n-dimensional vector, whose elements are all equal to 1, is used. With respect to the associative property of the notion of $\alpha$-product, one can always use the constant given by $1/n$, where $1/n\in \mathbb{R}$. This constant mathematically satisfies the conditions of coherence associated with the notion of prevision of a random entity (see von Rosen 1989; Coletti et al. 2016). From Table 1, it is possible to obtain

$${\displaystyle \frac{{\sum }_{i=1}^n{ }_1{X}_i}{n}}={ }_1\overline{X},$$

(77)

where the sum of positive and equiprobable masses associated with ${ }_1{X}_1$, …, ${ }_1{X}_n$ is coherently equal to 1. On the other hand, if all deviations from ${ }_1\overline{X}$ with respect to the first vector of Table 2 are considered together with all deviations from ${ }_2\overline{X}$ with respect to the second one of the same table, then it is possible to obtain the expression given by

$$\mathrm{cov}({ }_{1}X,{ }_{2}X)={\displaystyle \frac{{\sum }_{i=1}^n{ }_{1}x{ }_i{ }_{2}x{ }_i}{n}}.$$

(78)

Let $({ }_1\widehat{X},{ }_1\widehat{X})$, $({ }_{1}X,{ }_2\widehat{X})$, $({ }_{2}X,{ }_1\widehat{X})$, and $({ }_2\widehat{X},{ }_2\widehat{X})$ be four ordered pairs of variables. They identify four two-way tables, where each of them has n rows and n columns. The n nonnegative masses appearing on the main diagonal of each two-way table are such that their sum is always equal to 1. All observed values are possible. Since there are no degrees of possibility, one can attribute to the various observed values a greater or lesser degree of a specific factor of a psychological nature expressing a more or less strong propensity to expect that a given value rather than others will turn out to be true at the right time (see Chen and Escobar-Anel 2021; Flores et al. 2021). The two-variable multilinear regression model treated in this paper can make coherent previsions. After establishing how all the n nonnegative masses appearing on the main diagonal of each two-way table are, the $\alpha$-norm of a tensor of order 2 is handled (see Maturo and Angelini 2023). Such a norm coincides with the determinant of a square matrix of order 2, whose elements are all $\alpha$-products. One writes

$$\left[\begin{array}{cc}\mathbb{E}\left({ }_1\widehat{X}{ }_1\widehat{X}\right)& \mathbb{E}\left({ }_{1}X{ }_2\widehat{X}\right)\\ \mathbb{E}\left({ }_{2}X{ }_1\widehat{X}\right)& \mathbb{E}\left({ }_2\widehat{X}{ }_2\widehat{X}\right)\end{array}\right],$$

(79)

so it is possible to obtain

$$\mathbb{E}\left({\widehat{X}}_{12}\right)=\mathbf{P}\left({\widehat{X}}_{12}\right)=\left|\begin{array}{cc}\mathbb{E}\left({ }_1\widehat{X}{ }_1\widehat{X}\right)& \mathbb{E}\left({ }_{1}X{ }_2\widehat{X}\right)\\ \mathbb{E}\left({ }_{2}X{ }_1\widehat{X}\right)& \mathbb{E}\left({ }_2\widehat{X}{ }_2\widehat{X}\right)\end{array}\right|=\left|\begin{array}{cc}\mathbf{P}\left({ }_1\widehat{X}{ }_1\widehat{X}\right)& \mathbf{P}\left({ }_{1}X{ }_2\widehat{X}\right)\\ \mathbf{P}\left({ }_{2}X{ }_1\widehat{X}\right)& \mathbf{P}\left({ }_2\widehat{X}{ }_2\widehat{X}\right)\end{array}\right|,$$

(80)

where $\mathbb{E}\left({\widehat{X}}_{12}\right)=\mathbf{P}\left({\widehat{X}}_{12}\right)$ is a real number. The same symbol $\mathbf{P}$ signifies the unique notion which, in general, is called prevision or mathematical expectation. In the case of single events, it is also called probability. The $\alpha$-products characterizing the two elements of the main diagonal of (79) are associated with the 45-degree line, whereas the $\alpha$-products characterizing the two elements of the antidiagonal of the same square matrix are connected with the two estimated regression lines. $\mathbb{E}\left({ }_1\widehat{X}{ }_1\widehat{X}\right)$, $\mathbb{E}\left({ }_{1}X{ }_2\widehat{X}\right)$, $\mathbb{E}\left({ }_{2}X{ }_1\widehat{X}\right)$, and $\mathbb{E}\left({ }_2\widehat{X}{ }_2\widehat{X}\right)$ are all bilinear previsions obtained by considering n nonnegative and finitely additive masses associated with n points. Each bilinear prevision is always decomposed into two linear previsions. This means that points on a specific regression line are always studied in the Cartesian plane. In the first phase, each mass of n masses can take infinite values between 0 and 1, endpoints included, into account. Hence, a closed convex set is handled with respect to each regression line. It is a closed line segment that is bounded by the two extreme points which are studied on each regression line. Given a two-way table, all the masses outside its main diagonal are always equal to zero. Straightforward cases are the following:

Example 1. 

If $y=x$ is a regression line, then the following

Table 3. How two marginal previsions are on $y=x$.

	Vector 2	0	2	3	4	Sum
Vector 1
0		0	0	0	0	0
2		0	$0.2$	0	0	$0.2$
3		0	0	$0.3$	0	$0.3$
4		0	0	0	$0.5$	$0.5$
Sum		0	$0.2$	$0.3$	$0.5$	1

shows how a bivariate (two-dimensional) probability distribution can be decomposed. A point belonging to $y=x$ is given by $(3.3,3.3)$. A closed line segment appears on $y=x$ in the first phase. Its endpoints are $(0,0)$ and $(4,4)$. The second phase depends on the state of information and knowledge associated with a given individual. On the other hand, if $y=2x$ is an estimated regression line, then the following

Table 4. How two marginal previsions are on $y=2x$.

	Vector 2	0	2	4	6	Sum
Vector 1
0		0	0	0	0	0
1		0	$0.05$	0	0	$0.05$
2		0	0	$0.3$	0	$0.3$
3		0	0	0	$0.65$	$0.65$
Sum		0	$0.05$	$0.3$	$0.65$	1

shows that a point belonging to $y=2x$ is expressed by $(2.6,5.2)$. A closed line segment appears on $y=2x$ in the first phase. Its endpoints are $(0,0)$ and $(3,6)$. The choice of a point among infinite points on the estimated regression line depends on the state of information and knowledge associated with a given individual. This choice is absolutely rational.

7. The Bivariate Correlation Coefficient Obtained Using an Aggregate Measure of a Multilinear Nature

The two-variable multilinear regression model treated in this paper focuses on a multiple variable of order 2. This section shows that the sample correlation coefficient is intrinsically associated with a multiple variable of order 2. Such a coefficient is written in the following form given by

$$r=r_{12}={\displaystyle \frac{\sum { }_{1}x{ }_{2}x}{\sqrt{\sum {\left({ }_{1}x\right)}^2}\sqrt{\sum {\left({ }_{2}x\right)}^2}}},$$

(81)

where deviations from two mean values are considered (see Yule 1897). The population correlation coefficient is always associated with a multiple variable of order 2 in the same way as the sample correlation coefficient. If the notion of $\alpha$-product of two n-dimensional vectors expressed using deviations is taken into account, then it is possible to write the sample correlation coefficient in the following form given by

$$r=r_{12}={\displaystyle \frac{{ }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }\odot { }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }}{{\parallel { }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }\parallel }_{\alpha }{\parallel { }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }\parallel }_{\alpha }}},$$

(82)

where it turns out to be

$$\sqrt{{ }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }\odot { }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }}=\sqrt{\parallel { }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }{\parallel }_{\alpha }^2}={\parallel { }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }\parallel }_{\alpha }$$

(83)

and

$$\sqrt{{ }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }\odot { }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }}=\sqrt{\parallel { }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }{\parallel }_{\alpha }^2}={\parallel { }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }\parallel }_{\alpha }.$$

(84)

The symbol given by ⊙ identifies the notion of $\alpha$-product of two vectors. The sample correlation coefficient geometrically depends on the size of the angle between ${ }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }$ and ${ }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }$, where ${ }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }$ and ${ }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }$ are two n-dimensional vectors located at the origin of an n-dimensional linear space over $\mathbb{R}$ of a Euclidean nature. Since it is possible to write

$$-1\le {\left(1-{\displaystyle \frac{\parallel { }_{12}^{ }{d}_{ }^{ }{\parallel }_{\alpha }^2}{\parallel { }_{12}^{ }{\widehat{d}}_{ }^{ }{\parallel }_{\alpha }^2}}\right)}^{1/2}\le 1,$$

(85)

where one has

$$\begin{array}{c}\hfill {\parallel { }_{12}^{ }d\parallel }_{\alpha }^2=\left|\begin{array}{cc}\parallel { }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }{\parallel }_{\alpha }^2& { }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }\odot { }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }\\ { }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }\odot { }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }& \parallel { }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }{\parallel }_{\alpha }^2\end{array}\right|\end{array}$$

(86)

and

$$\begin{array}{c}\hfill {\parallel { }_{12}^{ }\widehat{d}\parallel }_{\alpha }^2=\left|\begin{array}{cc}\parallel { }_{\left(1\right)}^{ }{\mathbf{d}}_{ }^{ }{\parallel }_{\alpha }^2& 0\\ 0& \parallel { }_{\left(2\right)}^{ }{\mathbf{d}}_{ }^{ }{\parallel }_{\alpha }^2\end{array}\right|,\end{array}$$

(87)

the expression within the parentheses of (85) coincides with the sample correlation coefficient denoted by $r_{12}$. Such a coefficient is then proved to be intrinsically referred to $X_{12}=\{{ }_{1}X,{ }_{2}X\}$. The expression within the parentheses of (85) has been obtained using tensors of order 2 referred to a multiple variable of order 2. The sample correlation coefficient denoted by $r_{12}=r({ }_{1}X,{ }_{2}X)$ is invariant with respect to rotations and translations. One writes

$$r\left(a\left({ }_{1}X\right)+b,c\left({ }_{2}X\right)+d\right)=r({ }_{1}X,{ }_{2}X),$$

(88)

with $a,c\ne 0$. It is possible to verify (88) using a multilinear and quadratic metric. Moreover, one observes

$$r({ }_{1}X,{ }_{2}X)=r({ }_{2}X,{ }_{1}X)$$

(89)

and

$$r({ }_{1}X,{ }_{1}X)=r({ }_{2}X,{ }_{2}X)=1.$$

(90)

Whenever a correlation between ${ }_{1}X$ and ${ }_{2}X$ is studied, it is possible to express the underlying multilinear relationship between ${ }_{1}X$ and ${ }_{2}X$ through the following determinant

$$\begin{array}{c}\hfill \left|\begin{array}{cc}r({ }_{1}X,{ }_{1}X)& r({ }_{1}X,{ }_{2}X)\\ r({ }_{2}X,{ }_{1}X)& r({ }_{2}X,{ }_{2}X)\end{array}\right|.\end{array}$$

(91)

However, in this case, it does not make sense that the calculation of (91) takes place. On the other hand, it is known that one of the most meaningful relationships constructed from a finite set of elements is a linear combination of basis vectors of a given linear space. Nevertheless, given some vectors belonging to the same linear space, an actual linear combination of them does not always make sense in practice: this observation is therefore conceptually the same as the one above related to (91).

8. The Two-Variable Multilinear Regression Model Used to Study Portfolios of Financial Assets: Fair Estimations

An n-dimensional linear space over $\mathbb{R}$ of a Euclidean nature is denoted by ${\mathbb{R}}^n$. ${ }_{\mathbf{1}}\mathbf{X}$ and ${ }_{\mathbf{2}}\mathbf{X}$ are elements of ${\mathbb{R}}^n$, where ${\mathbb{R}}^n$ is isomorphic to ${\mathcal{R}}^n$. An n-dimensional affine space is expressed by ${\mathcal{R}}^n$. The two elements denoted by ${ }_{\mathbf{1}}\mathbf{X}$ and ${ }_{\mathbf{2}}\mathbf{X}$ are supposed to be linearly independent. They identify a linear subspace of ${\mathbb{R}}^n$. Its dimension is equal to 2. One observes a reduction of dimension in this way (see Yuan et al. 2007). With respect to ${ }_{\mathbf{1}}\mathbf{X}$, all collinear vectors referred to it are considered. Their number is infinite, so a straight line of ${\mathcal{R}}^n$ is established. This line is a linear subspace of ${\mathcal{R}}^n$. Its dimension is equal to 1. It is also a linear subspace of ${\mathbb{R}}^n$. It is possible to prove in the form of a theorem that there is a one-to-one correspondence between the elements of a sheaf of parallel hyperplanes embedded in ${\mathcal{R}}^n$ and the points of intersection of them with a straight line of ${\mathcal{R}}^n$. This line is orthogonal to all parallel hyperplanes taken into account. The points of intersection of all these parallel hyperplanes embedded in ${\mathcal{R}}^n$ with a straight line of ${\mathcal{R}}^n$ are real numbers transferred on a one-dimensional straight line, on which an origin, a unit of length, and an orientation are established. They coincide with all the elements of ${ }_{\mathbf{1}}\mathbf{X}$. The same is true with respect to all the elements of ${ }_{\mathbf{2}}\mathbf{X}$. Two mutually orthogonal one-dimensional straight lines, on which an origin, a same unit of length, and an orientation are established, identify a two-dimensional Cartesian coordinate system. Ordered pairs of sampling units referring to two univariate variables denoted by ${ }_{1}X$ and ${ }_{2}X$ can be studied inside the first quadrant of a two-dimensional Cartesian coordinate system. This is because it is always possible to write

$${ }_{1}X={ }_{1}X{ }_1-{ }_{1}X{ }_2,$$

(92)

with

$${ }_{1}X{ }_1=0\vee { }_{1}X,$$

(93)

and

$${ }_{1}X{ }_2=|0\wedge { }_{1}X|.$$

(94)

Moreover, it is always possible to write

$${ }_{2}X={ }_{2}X{ }_1-{ }_{2}X{ }_2,$$

(95)

with

$${ }_{2}X{ }_1=0\vee { }_{2}X,$$

(96)

and

$${ }_{2}X{ }_2=|0\wedge { }_{2}X|.$$

(97)

In general, one has

$$x\vee y=max(x,y),$$

(98)

$$x\wedge y=min(x,y),$$

(99)

and

$$\tilde{x}=1-x,$$

(100)

for any x and y that are real numbers.

The two-variable multilinear regression model treated in this paper shows that the real world cannot be analyzed in terms of a collection of bivariate or multivariate relationships only. The real world can also be analyzed in terms of multilinear relationships between variables (see Angelini and Maturo 2022). Surveys and technical discussions of the relationships between variables, which are studied in pairs, steadily take place. Various relationships between variables studied in pairs can be treated. Different pairs of variables are analyzed whenever a multilinear approach is used. This approach uses a multilinear and quadratic metric. For this reason, different pairs of variables have to be considered. Even though multilinear relationships between variables can be present whenever real problems are studied, one does not leave the two-dimensional diagrams behind. Bivariate relationships between variables are significant in themselves. The mathematical and statistical tools developed for them are fundamental building blocks for the analysis of multilinear situations characterizing real problems (see Bonat and Jørgensen 2016). $X_{12}={\{}_{1}X,{ }_{2}X\}$ can identify a two-financial asset portfolio. Its components can be two normally distributed variables. In other words, a bivariate normal population is considered. This means, among other things, that the ${ }_{1}X$ values identifying observed returns on ${ }_{1}X$ are normally distributed and so are the ${ }_{2}X$ values identifying observed returns on ${ }_{2}X$. A multiple variable of order 2 is associated with ${\widehat{X}}_{12}$, where ${\widehat{X}}_{12}$ identifies $2^2=4$ regression lines which are studied inside the same subset of a two-dimensional linear space over $\mathbb{R}$. In this paper, a two-financial asset portfolio is studied as a multiple random good of order 2. This matter can be extended, so

$$X_{123}={\{}_{1}X,{ }_{2}X,{ }_{3}X\}$$

(101)

can identify a three-financial asset portfolio. Its components are three normally distributed variables, so a multivariate normal population is considered. This means that the ${ }_{1}X$ values identifying observed returns on ${ }_{1}X$ are normally distributed. The ${ }_{2}X$ values identifying observed returns on ${ }_{2}X$ are normally distributed and so are the ${ }_{3}X$ values identifying observed returns on ${ }_{3}X$. A multiple variable of order 3 is associated with ${\widehat{X}}_{123}$, where ${\widehat{X}}_{123}$ identifies $3^2=9$ regression lines. They are studied inside the same subset of a two-dimensional linear space over $\mathbb{R}$. A quadratic metric is always used (see Markowitz 1952, 1956; Cont 2001). If $X_{12}={\{}_{1}X=X,{ }_{2}X=Y\}$ is a continuous and multiple variable of order 2, whose components are two continuous and univariate variables, then its expected return is given by the following determinant

$$\left|\begin{array}{cc}\mathbb{E}\left(X^2\right)={\int }_{-\infty }^{+\infty }{x}^{2}f\left(x\right)dx& \mathbb{E}\left(XY\right)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }xyf(x,y)dxdy\\ \mathbb{E}\left(YX\right)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }yxf(y,x)dydx& \mathbb{E}\left(Y^2\right)={\int }_{-\infty }^{+\infty }{y}^{2}f\left(y\right)dy\end{array}\right|,$$

(102)

where $f\left(x\right)$ and $f\left(y\right)$ are density functions of X and Y, whereas $f(x,y)$ is a bivariate probability density function like $f(y,x)$. This determinant has the same structure as the one characterizing the $\alpha$-norm of a tensor of order 2 connected with sampling quantities and their regression residuals. All elements of this determinant are integrals, so it is necessary to verify they exist. Similarly, if $X_{123}={\{}_{1}X=X,{ }_{2}X=Y,{ }_{3}X=Z\}$ is a continuous and multiple variable of order 3, whose components are three continuous and univariate variables, then its expected return is given by the following determinant

$$\left|\begin{array}{ccc}\mathbb{E}\left(X^2\right)={\int }_{-\infty }^{+\infty }{x}^{2}f\left(x\right)dx& \mathbb{E}\left(XY\right)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }xyf(x,y)dxdy& \mathbb{E}\left(XZ\right)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }xzf(x,z)dxdz\\ \mathbb{E}\left(YX\right)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }yxf(y,x)dydx& \mathbb{E}\left(Y^2\right)={\int }_{-\infty }^{+\infty }{y}^{2}f\left(y\right)dy& \mathbb{E}\left(YZ\right)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }yzf(y,z)dydz\\ \mathbb{E}\left(ZX\right)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }zxf(z,x)dzdx& \mathbb{E}\left(ZY\right)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }zyf(z,y)dzdy& \mathbb{E}\left(Z^2\right)={\int }_{-\infty }^{+\infty }{z}^{2}f\left(z\right)dz\end{array}\right|,$$

(103)

where $f\left(z\right)$ is a density function of Z, whereas $f(x,z)$ and $f(y,z)$ are bivariate probability density functions like $f(z,x)$ and $f(z,y)$. It is also possible to study an m-financial asset portfolio denoted by

$$X_{12\dots m}=\{{ }_{1}X,{ }_{2}X,{ }_{3}X,\dots ,{ }_{m}X\},$$

(104)

where $m>3$ is an integer. In particular, it is possible to treat its riskiness instead of its expected return (see Cepeda and Gamerman 2000; Angelini and Maturo 2023). Conceptually, if one thinks of the case of the barycenter of a solid body, then it is possible to understand why to set up a determinant to obtain a multilinear index related to a multiple variable is meaningful. The center of mass of a distribution of mass can always be established. Conversely, it is never possible in practice to know the exact distribution of mass related to a specific solid body. Similarly, to know the exact probability distribution related to a multiple variable is of little interest in many problems, but to know its mathematical expectation is essential.

The two-variable multilinear regression model handled in this research work extends other models widely studied in the literature (see Nelder and Wedderburn 1972). It is not in conflict with them (see Yuan and Lin 2006). Since the two-variable multilinear regression model is conceptually and mathematically associated with the bivariate correlation coefficient, the null hypothesis given by

$$H_0: \rho =0,$$

(105)

where $\rho$ is the population correlation coefficient, is rejected if the null hypothesis given by

$$H_0: \beta =0,$$

(106)

or

$$H_0: {\beta }^{*}=0$$

(107)

is rejected. To reject $\beta =0$ or ${\beta }^{*}=0$ implies that it is absolutely correct to reject $\rho =0$. It follows that there is a correlation between the two components of the two-financial asset portfolio under consideration.

9. Aggregate Measures: Multiple Random Variables and Their Identification through Tensors

If one studies a multiple random variable, then one obtains an aggregate measure. This is because one has the possibility of constructing a tensor, after which one has the possibility of calculating the $\alpha$-norm of this tensor by applying the techniques of multilinearity. In this way, the logical aspects of probability calculus are studied in deep. Also, they give a heuristic contribution. These aspects must be kept unconnected from the empirical ones. This approach is of a Bayesian nature. The probability distributions which are involved can first be of a nonparametric kind. In a multiple variable, the role played by the marginal variables characterized by their marginal probability distributions is essential. Within this context, the support of each marginal variable is a partition of events containing a finite number of them. An event is not a measurable set, but it is a possible alternative expressed by a real number to which a specific probability is associated. The notion of probability is not undefined, much the same as point and line are undefined in geometry. The notion of event is infinitely subdivisible. However, a subdivision stops when one studies the Cartesian product of two sets, where each set contains a finite number of possible alternatives. The Cartesian product of two sets is studied because otherwise things become unnecessarily complicated. The Cartesian product of two sets of possible alternatives identifies a bivariate variable. Nevertheless, the choice of the ordered pair of marginal variables identifying a bivariate variable is arbitrary. For this reason, it is appropriate to free oneself from the order in which the marginal variables are taken into account. Given a multiple random variable of order n containing n marginal variables written as

$$X_{12\dots n}=\{{ }_{1}X,{ }_{2}X,{ }_{3}X,\dots ,{ }_{n}X\},$$

(108)

it is possible to construct the following tensor expressed by

$$\left[\begin{array}{cccc}\mathbf{P}\left({ }_{1}X{ }_{1}X\right)& \mathbf{P}\left({ }_{1}X{ }_{2}X\right)& \dots & \mathbf{P}\left({ }_{1}X{ }_{n}X\right)\\ \mathbf{P}\left({ }_{2}X{ }_{1}X\right)& \mathbf{P}\left({ }_{2}X{ }_{2}X\right)& \dots & \mathbf{P}\left({ }_{2}X{ }_{n}X\right)\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{P}\left({ }_{n}X{ }_{1}X\right)& \mathbf{P}\left({ }_{n}X{ }_{2}X\right)& \dots & \mathbf{P}\left({ }_{n}X{ }_{n}X\right)\end{array}\right].$$

(109)

Each element of the square matrix of order n given by (109) is a component of a tensor of order n. The components of this tensor are symmetric. Each element of (109) is an $\alpha$-product obtained using an affine tensor of order 2. This tensor identifies the joint probabilities characterizing each bivariate probability distribution associated with a bivariate variable. Each bivariate variable is intrinsically composed of two marginal variables. The possible alternatives for each marginal variable are the components of a vector, so each bivariate probability distribution is summarized through an $\alpha$-product structurally referred to the tensor product of two vectors. The tensor product of two vectors is not a scalar or inner product. The tensor product of two vectors is not commutative. Hence, $n^2$ ordered pairs of marginal variables are handled. The $\alpha$-norm of the above-written tensor is given by

$$\mathbf{P}\left(X_{12\dots n}\right)=\left|\begin{array}{cccc}\mathbf{P}\left({ }_{1}X{ }_{1}X\right)& \mathbf{P}\left({ }_{1}X{ }_{2}X\right)& \dots & \mathbf{P}\left({ }_{1}X{ }_{n}X\right)\\ \mathbf{P}\left({ }_{2}X{ }_{1}X\right)& \mathbf{P}\left({ }_{2}X{ }_{2}X\right)& \dots & \mathbf{P}\left({ }_{2}X{ }_{n}X\right)\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{P}\left({ }_{n}X{ }_{1}X\right)& \mathbf{P}\left({ }_{n}X{ }_{2}X\right)& \dots & \mathbf{P}\left({ }_{n}X{ }_{n}X\right)\end{array}\right|.$$

(110)

It is the determinant of the square matrix of order n given by (109). There is no loss of information. On the contrary, there is a gain of information. This is because an aggregate measure compatible with objects of a homogeneous nature is obtained. On the other hand, an aggregate measure expressing the riskiness of a finite set of financial assets is given by

$$\begin{array}{c}\hfill \mathrm{var}\left(X_{12\dots n}\right)=\left|\begin{array}{cccc}{\displaystyle \frac{1}{2}}{ }_{ }^2{\Delta }^2{(}_{1}X)& \mathrm{cov}{(}_{1}X,{ }_{2}X)& \dots & \mathrm{cov}{(}_{1}X,{ }_{n}X)\\ \mathrm{cov}{(}_{2}X,{ }_{1}X)& {\displaystyle \frac{1}{2}}{ }_{ }^2{\Delta }^2{(}_{2}X)& \dots & \mathrm{cov}{(}_{2}X,{ }_{n}X)\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{cov}{(}_{n}X,{ }_{1}X)& \mathrm{cov}{(}_{n}X,{ }_{2}X)& \dots & {\displaystyle \frac{1}{2}}{ }_{ }^2{\Delta }^2{(}_{n}X)\end{array}\right|\end{array},$$

(111)

where the mean quadratic difference of ${ }_{i}X$ is denoted by ${ }_{ }^2\Delta {(}_{i}X)$, $i=1,\dots ,n$, the square of the mean quadratic difference of ${ }_{i}X$ is denoted by ${ }_{ }^2{\Delta }^2{(}_{i}X)$, $i=1,\dots ,n$, and one writes ${ }_{ }^2{\Delta }^2{(}_{i}X)=2{\sigma }_{{ }_{i}X}^2$, $i=1,\dots ,n$. It follows that it is possible to observe

$$\begin{array}{c}\hfill \mathrm{var}\left(X_{12\dots n}\right)=\left|\begin{array}{cccc}\mathrm{var}{(}_{1}X)& \mathrm{cov}{(}_{1}X,{ }_{2}X)& \dots & \mathrm{cov}{(}_{1}X,{ }_{n}X)\\ \mathrm{cov}{(}_{2}X,{ }_{1}X)& \mathrm{var}{(}_{2}X)& \dots & \mathrm{cov}{(}_{2}X,{ }_{n}X)\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{cov}{(}_{n}X,{ }_{1}X)& \mathrm{cov}{(}_{n}X,{ }_{2}X)& \dots & \mathrm{var}{(}_{n}X)\end{array}\right|\end{array}.$$

(112)

Aggregate measures are essential in many applications of an economic nature. For instance, the Sharpe ratio can be obtained using a multilinear approach. This approach gives rise to multilinear measures. If $r_f$ is the risk-free asset paying a fixed rate of return, then the Sharpe ratio is given by

$$\mathrm{SR}={\displaystyle \frac{\mathbf{P}\left(X_{12\dots n}\right)-r_f}{\sqrt{\mathrm{var}\left(X_{12\dots n}\right)}}}.$$

(113)

The Sharpe ratio measures how risk and return can be traded off in making portfolio choices. One assumes $\mathbf{P}\left(X_{12\dots n}\right)>r_f$. Furthermore, the beta of a given stock i is expressed by

$${\beta }_i={\displaystyle \frac{\mathrm{cov}(r_i,r_m)}{\mathrm{var}\left(r_m\right)}},$$

(114)

so aggregate measures obtained according to a multilinear approach can operationally be associated with the capital asset pricing model (CAPM). This model has many applications in the study of financial markets. The market expected return $r_m$ is expressed by (110). The mean-variance model based on mean return and standard deviation of return can use aggregate measures which are perfectly compatible with the satisfaction of the optimization principle by a given individual. The price of risk is then given by

$$p={\displaystyle \frac{r_m-r_f}{{\sigma }_m}},$$

(115)

where $r_m$ and ${\sigma }_m$ are always determinable within this context, and it represents the positive slope of the budget line. This is because one assumes that $r_m>r_f$. It is possible to identify a part of this budget line that is bounded by two distinct endpoints. This part is a closed line segment containing every point on the budget line that is between its endpoints. This paper focuses on a specific aspect: there are always infinitely many points for which the optimization principle can be satisfied by a given individual. They are points belonging to a closed convex set. It is known that the mean-variance model is a reasonable approximation to the expected utility model. Even the expected utility model can use aggregate measures obtained according to a multilinear approach. This means that Jensen’s inequality can be extended. Studying n marginal variables, which are the components of $X_{12\dots n}$, the certain gain, which is considered equivalent to $X_{12\dots n}$ by a given individual who is risk neutral, is expressed by $x_{12\dots n}=\mathbf{P}\left(X_{12\dots n}\right)$. His utility function is the 45-degree line. For a risk-loving individual, the random gain denoted by $X_{12\dots n}$ is preferred to a certain gain denoted by $x_{12\dots n}$, so $x_{12\dots n}$ is greater than $\mathbf{P}\left(X_{12\dots n}\right)$ on the x-axis. His utility function is therefore convex. For a risk-averse individual, the random gain $X_{12\dots n}$ is not preferred to a certain gain $x_{12\dots n}$, so $x_{12\dots n}$ is less than $\mathbf{P}\left(X_{12\dots n}\right)$ on the x-axis. His utility function is then concave.

9.1. A Multiple Random Variable: Emphasis on Marginal Variables

Check the following:

Example 2. 

Given $X_{123}=\{{ }_{1}X,{ }_{2}X,{ }_{3}X\}$, the corresponding tensor is a square matrix of order 3. It is then possible to consider $3^2=9$ ordered pairs of marginal variables. Whenever $3^2=9$ ordered pairs of marginal variables are studied, each marginal probability distribution is summarized in such a way that one obtains a marginal measure. After summarizing, each marginal measure never changes. The joint probabilities can freely be chosen by a given individual. The only constraint for them is that the mean value associated with each marginal probability distribution does not change. The following

Table 5. The mean values of ${ }_{1}X$ and ${ }_{2}X$ do not depend on the joint probabilities.

	Vector 2	0	22	41	63	Sum
Vector 1
0		$p_{11}$	$p_{12}$	$p_{13}$	$p_{14}$	0
10		$p_{21}$	$p_{22}$	$p_{23}$	$p_{24}$	$0.2$
20		$p_{31}$	$p_{32}$	$p_{33}$	$p_{34}$	$0.3$
30		$p_{41}$	$p_{42}$	$p_{43}$	$p_{44}$	$0.5$
Sum		0	$0.1$	$0.1$	$0.8$	1

shows how the mean value of ${ }_{1}X$ and ${ }_{2}X$ can be. The following

Table 6. The mean values of ${ }_{1}X$ and ${ }_{3}X$ do not depend on the joint probabilities.

	Vector 3	0	26	45	68	Sum
Vector 1
0		$p_{11}$	$p_{12}$	$p_{13}$	$p_{14}$	0
10		$p_{21}$	$p_{22}$	$p_{23}$	$p_{24}$	$0.2$
20		$p_{31}$	$p_{32}$	$p_{33}$	$p_{34}$	$0.3$
30		$p_{41}$	$p_{42}$	$p_{43}$	$p_{44}$	$0.5$
Sum		0	$0.3$	$0.3$	$0.4$	1

shows how the mean value of ${ }_{1}X$ and ${ }_{3}X$ can be. The following

Table 7. The mean values of ${ }_{2}X$ and ${ }_{3}X$ do not depend on the joint probabilities.

	Vector 3	0	26	45	68	Sum
Vector 2
0		$p_{11}$	$p_{12}$	$p_{13}$	$p_{14}$	0
22		$p_{21}$	$p_{22}$	$p_{23}$	$p_{24}$	$0.1$
41		$p_{31}$	$p_{32}$	$p_{33}$	$p_{34}$	$0.1$
63		$p_{41}$	$p_{42}$	$p_{43}$	$p_{44}$	$0.8$
Sum		0	$0.3$	$0.3$	$0.4$	1

shows how the mean value of ${ }_{2}X$ and ${ }_{3}X$ can be. Given the mean values of ${ }_{1}X$, ${ }_{2}X$, and ${ }_{3}X$, it is theoretically possible to consider all deviations from them, respectively. Thus, if the number of possible alternatives for ${ }_{1}X$, ${ }_{2}X$, and ${ }_{3}X$ is not small, then the joint probabilities can be chosen in such a way that the correlation coefficient is equal to $+1$, or 0, or $-1$. A value of $+1$ implies that a marginal variable increases as another one increases, whereas a value of $-1$ implies that a marginal variable increases while another one decreases. A value of 0 implies that there is no linear dependency between the corresponding marginal variables. If a marginal variable is studied together with itself, then only the marginal probabilities of its distribution appear as joint ones. Whenever the joint probabilities are freely chosen by a given individual, they do not change. The joint probabilities can be chosen based on the state of information and knowledge associated with a given individual.

It is necessary to handle errors associated with the notion of prevision. Natural or intrinsic errors connected with this notion appear whenever deviations from mean values take place. These errors are not unwelcome. They identify a parametric probability distribution. They identify a normal distribution characterized by specific parameters. Conversely, logical errors appear whenever objective conditions of coherence are not satisfied. Such conditions are not satisfied when and only when one chooses a point that does not belong to the corresponding closed convex set. These errors are absolutely unwelcome. It is possible to note the following:

Remark 1. 

The empirical validation of the multilinear regression model treated in this paper can easily be shown. This is because the α-norm of a tensor extends the properties of the barycenter of nonnegative and finitely additive masses. It is known that such properties are given by two distinct indices: the first index is associated with an equilibrium that is stable, the second one appears when the moment of inertia is minimum. Such properties are satisfied when and only when each component of the tensor into account is coherently expressed in a quantitative form obtained from a specific α-product. The α-norm of the tensor of order 3 related to Example 2 is given by

$$\mathbf{P}\left(X_{12\dots 3}\right)=\left|\begin{array}{ccc}\mathbf{P}\left({ }_{1}X{ }_{1}X\right)& \mathbf{P}\left({ }_{1}X{ }_{2}X\right)& \mathbf{P}\left({ }_{1}X{ }_{3}X\right)\\ \mathbf{P}\left({ }_{2}X{ }_{1}X\right)& \mathbf{P}\left({ }_{2}X{ }_{2}X\right)& \mathbf{P}\left({ }_{2}X{ }_{3}X\right)\\ \mathbf{P}\left({ }_{3}X{ }_{1}X\right)& \mathbf{P}\left({ }_{3}X{ }_{2}X\right)& \mathbf{P}\left({ }_{3}X{ }_{3}X\right)\end{array}\right|.$$

If each element of the following square matrix of order 3 given by

$$\left[\begin{array}{ccc}\mathbf{P}\left({ }_{1}X{ }_{1}X\right)& \mathbf{P}\left({ }_{1}X{ }_{2}X\right)& \mathbf{P}\left({ }_{1}X{ }_{3}X\right)\\ \mathbf{P}\left({ }_{2}X{ }_{1}X\right)& \mathbf{P}\left({ }_{2}X{ }_{2}X\right)& \mathbf{P}\left({ }_{2}X{ }_{3}X\right)\\ \mathbf{P}\left({ }_{3}X{ }_{1}X\right)& \mathbf{P}\left({ }_{3}X{ }_{2}X\right)& \mathbf{P}\left({ }_{3}X{ }_{3}X\right)\end{array}\right]$$

is coherent, then $\mathbf{P}\left(X_{12\dots 3}\right)$ is a parameter of a coherent nature. This means that $\mathbf{P}\left(X_{12\dots 3}\right)$ is a point belonging to the union of different closed convex sets. Each closed convex set is a part of a one-dimensional straight line that is bounded by two distinct endpoints. The product of a possible value for a marginal variable and a possible value for another one is a point of a one-dimensional straight line. This validation holds regardless of the particular masses that may be chosen by a given individual. The reader can verify this matter. As a consequence, the validation of the multilinear regression model treated in this paper is of a logical nature as well.

Remark 2. 

If the number of the possible values for two marginal variables is not the same, then this number can become the same. For instance, let

$$\left[\begin{array}{c}2\\ 3\\ 4\\ 5\end{array}\right],\left[\begin{array}{c}0.1\\ 0.4\\ 0.2\\ 0.3\end{array}\right]$$

and

$$\left[\begin{array}{c}1\\ 7\\ 8\end{array}\right],\left[\begin{array}{c}0.2\\ 0.6\\ 0.2\end{array}\right]$$

be two ordered pairs of vectors. The second vector of each ordered pair identifies the corresponding probabilities. If one writes

$$\left[\begin{array}{c}2\\ 3\\ 4\\ 5\end{array}\right],\left[\begin{array}{c}0.1\\ 0.4\\ 0.2\\ 0.3\end{array}\right]$$

and

$$\left[\begin{array}{c}0\\ 1\\ 7\\ 8\end{array}\right],\left[\begin{array}{c}0\\ 0.2\\ 0.6\\ 0.2\end{array}\right],$$

then the number of the possible values for the two marginal variables under consideration becomes the same. The possible values for the two marginal variables into account identify the first vector of each ordered pair.

9.2. Difference between a Multiple Random Variable and a Multivariate One

There is a fundamental logical difference between a multiple random variable and a multivariate one. For instance, if a multivariate random variable is composed of three marginal probability distributions, then the probability distribution of this variable is of a three-dimensional nature. It is then possible to consider three axes belonging to ${\mathbb{R}}^3$. They are pairwise orthogonal. Each axis of them is the support of a marginal variable. Each marginal probability distribution which is summarized gives rise to a real number. Each real number associated with a marginal variable is represented on an axis belonging to ${\mathbb{R}}^3$. Each axis of ${\mathbb{R}}^3$ is a linear subspace of it. Its dimension is equal to 1. If the probability distribution of a multivariate random variable composed of three marginal probability distributions is summarized, then an ordered triplet of real numbers belonging to ${\mathbb{R}}^3$ has to be studied. This is because the probability distribution of a multivariate random variable composed of three marginal probability distributions is of a three-dimensional nature. Whenever the probability distribution of a multivariate random variable composed of three marginal probability distributions is summarized and a real number is handled, only empirical aspects are taken into account. Nevertheless, this paper says that it is necessary to consider logical aspects as well. For this reason, the previous real number is never an aggregate measure. A multiple random variable of order 3 logically gives rise to a tensor of order 3 which is naturally an element of a linear space over $\mathbb{R}$. Thus, $3^2=9$ ordered pairs of marginal variables are dealt with. Some ordered pairs out of $3^2=9$ give rise to a bivariate variable such that the corresponding bivariate probability distribution is summarized inside a linear subspace of ${\mathbb{R}}^3$ of a two-dimensional nature. This linear subspace is generated by two linear subspaces of ${\mathbb{R}}^3$ of a one-dimensional nature. Conversely, three ordered pairs of marginal variables out of $3^2=9$ give rise to a bivariate variable such that the corresponding bivariate probability distribution is summarized inside a linear subspace of ${\mathbb{R}}^3$ of a one-dimensional nature. It is one of the three axes of ${\mathbb{R}}^3$: only the squares of the possible values for each marginal variable are taken into account, so there is no loss of information. On the other hand, the same is true if a multiple random variable is of order n, where n is an integer and one has $n=2$ or $n>3$, and the probability distribution of a multivariate random variable is of an n-dimensional nature. Within this context, the theorem mentioned at the beginning of Section 8 does not come into play.

10. Conclusions and Some Future Perspectives

Two dynamical phases depending on the state of information and knowledge associated with a given individual always identify the notion of prevision. This notion expresses the judgment of a given individual at a given time. A given individual can criticize his previous judgment. He can therefore verify whether he forgot or underestimated or overestimated some circumstances. Such circumstances could have led him to a more accurate prevision if he had exercised better thought. Nevertheless, if his previous judgment translated into quantitative terms is an element of a closed convex set, then his past prevision is never wrong. There are only natural or intrinsic errors. New pieces of information can be used to make different previsions for more or less similar future cases. It is absolutely wrong to criticize a prevision based on a specific state of information and knowledge by referring to a different state of information and knowledge. In this paper, closed convex sets play an essential role. Closed convex sets are closed line segments. Each line segment is a part of a line regression. Fair estimations are fair previsions. They are studied inside linear spaces over $\mathbb{R}$ provided with a quadratic metric. Only their dimension can be different. Since it is possible to show that an invariant or intrinsic metric can be used, only the affine properties make sense. All indices which are obtained inside linear spaces over $\mathbb{R}$ provided with a quadratic metric do not depend on the arbitrary choice of the coordinate system. They are therefore of an intrinsic nature. This paper shows that obtaining aggregate measures is meaningful. For this reason, different applications of an economic nature are mentioned. Thus, if a given individual is indifferent to the exchange of ${ }_{1}X$ for $\mathbf{P}\left({ }_{1}X\right)$ and of ${ }_{2}X$ for $\mathbf{P}\left({ }_{2}X\right)$, then the same individual has to be indifferent to the exchange of $X_{12}$ for $\mathbf{P}\left(X_{12}\right)$, where $\mathbf{P}\left(X_{12}\right)$ is an aggregate measure. $\mathbf{P}\left(X_{12}\right)$ is a real number obtained from four bivariate probability distributions. They are all summarized and decomposed. In this paper, a bivariate probability distribution is of a two-dimensional nature. A bivariate probability distribution that is summarized is not properly associated with a real number. A bivariate probability distribution that is summarized is intrinsically composed of two marginal probability distributions that are summarized. A bivariate probability distribution that is summarized can then be associated with an ordered pair of real numbers. Logical aspects of probability calculus are studied in depth. Also, they give a heuristic contribution. To determine an aggregate measure is crucial in connection with the optimization of decisions of an economic nature. Such decisions usually depend on the subjective preferences associated with a given individual, who can be interested in two or more than two components of a multiple variable. These components are studied using the techniques of multilinearity. Among other things that can represent future perspectives, it is possible to prove that closed convex sets are of an essential importance in connection with stochastic models related to fair estimations. Such models are intrinsically based on a Bayesian reinterpretation of the central limit theorem. Moreover, it is possible to make evident important aspects of time series through a reinterpretation of principal component analysis based on the techniques of multilinearity.