Non-Parametric Regression and Riesz Estimators

Round 1

Reviewer 1 Report

Dear Author

Please,  check the attached file to see the comments.

Comments for author File: Comments.pdf

Author Response

1. There is not any relevant work between 2007 and 2023.

2.  The main contribution is that through any design  matrix relying on a sample, we fit a non-parametric  Linear Regression Model. The calculation of its parameters is almost directly though observing a partition for the set of sample observations. This partition is the one defined by the subset of observations taking different positive values. The Estimators are  actually the Ordinary Least Squares' Estimators, being usual for fitting a Linear Regression model. This regression model may include data coming from both categorical variables and variables whose support is some interval of real numbers. The most relevant model relying on both categorical and non -categorical variables is the Principal Components' Anaysis. Principal Components' Analysis is not actually a Regression Model. An important contribution is that we define Goodness -of -Fitting  measures for the proposed Riesz Regression Model. These measures rely on the dimension of the sub- lattice generated by the columns of the design matrix.

3. The Linear Regression we propose is non -parametric. It is not related to specific distributions. Hence it may be used for data analysis in both actuarial and financial issues, where there is not any specific way of fitting some Linear Regression Model. This fashion of Regression is compatible to beta pricing and factor pricing, whose tretament is briefly presented in the book 'Principles of Financial Economics' by S.F. LeRoy and J. Werner. Specifically in Chapter 20 of this book, the Regression Equation proposed in Paragraph 20.7. is the one proposed by C.D. Aliprantis,  D. Harris and R. Tourky in p.434 of the paper under the title 'Riesz Estimators'.
    4. The abstract is quite different from the conclusion in the new version of the paper.
    5. Yes, we tried to be accurate.
    6. The new version of the paper contains a different abstract. We think that its form is more accurate.
   7.  Yes, the introduction is going to include all the above responses in order to make the content of the paper more clear.
   8.  Yes, we tried to be accurate.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors consider their version of a non-parametric regression model based on Reisz Estimators. The results are sound and rigorously proven. Yet the paper outline lacks clarity and the presentation is not playing justice to their accomplishment. For example, they state that "This regression model is similar to the least squares’ regression" but they do not discuss clearly why their approach might be preferable and in which situations outperforms standard estimators. They state their  "strong motivation" to include "categorical variables" but offer no indication or discussion on this topic either.

Although it is praiseworthy (and actually quite rare) that the authors explicitly state their assumptions, the reader cannot easily access the motivation and context for this work.

At another point, "CAPM Model" is briefly discussed without any reference or even to what these initials stand for. This might be evident for a specialized readership working on financial mathematics, but not for the non-specialist.

Again the numerical examples are rather rudimental, and although they are reasonable exercises they are provided devoid of context.

Finally, Section ^. Is rather laconic. Which are the new conclusions, based in what part of the paper?  For example, is the conclusion that "Riesz Estimators is (NB: replace 'is' with 'are') actually a strictly positive projection operator
is a new result or a general statement like  "categorical variables are included in this linear regression model", is this the only model that achieves that? It probably is not unique, or is it not? Some more specific references beyond the encyclopedic one [von Neumann and Morgenstern (1947)] are greatly needed to provide context and understanding of why should someone read this, by any other means OK, paper.

Author Response

    1.  The estimation of the components of the Riesz Linear Regression Model are actually the usually Least Squares' Estimators. That's what we mean.
    2.  Categorical Data are not included in usual Linear Regression Models. Moreover, Linear Regression Models do not include data coming from both categorical and non -categorical variables. The proposed model is not an abstract mathematical application, since it may be used for fitting linear regression on both categorical and categorical variables' data. Categorical values' data are studied by the multivariate logit regression, as well. The proposed model is sufficient in fitting some linear regression model in cases of samples containing both categorical variables data and data from variables whose support is some interval of the real numbers. This is a main part of motivation of the paper.
    3. The Projection Operator $\Pi$ is related to the presence of data coming from categorical variables.
    4.  Both Introduction and Conclusions Section is more clear in the new version of the paper.
    5.  Overfitting of the proposed model is indicated from the value of the adjusted $R^{2}$ measure we propose. On the other hand, the objective probability values may be estimated by the usual $\chi^{2}$ test. For this reason some previously obtained samples may be used. 
    6. The last numerical examples are sufficient for showing that objective probability vector is related to the fitting of the proposed model. Of course these numerical examples may include data coming from macroeconomic and finance -related variables. The categorical variables may be related to the classification of the sovereign bonds.
    7. CAPM as a linear regression model is related to the objective probability vector's estimation, since it is a regression model including mean values and variance values.
    8. CAPM explanation is out of the scope of the paper and we have to eliminate from the new version of the paper. Options as factors in the so -called beta pricing is related to the regression model being proposed, since option payoffs lie in the sub -lattice generated by the span of the variables included in the Riesz Estimation Regression.
    9. CAPM details are widely explained in the nice book of M. Magill and M. Quinzii ''Theory of Incomplete Markets''.  However, the book including both the details of CAPM and beta pricing is the book ''Principles of Financial Economincs'' by S. F. LeRoy and J. Werner. The last book is included in the References.
    10. The associated References about Objective Probabilities are included in the above book. The References which are beyond John von Neumann and Oskar Morgenstern establishment are going to be included in the paper. There is an essential relation between estimation of Objective Probabilities and a Bayesian Approach on them.

Author Response File: Author Response.pdf

Reviewer 3 Report

The model built and presented in "NON -PARAMETRIC REGRESSION AND RIESZ ESTIMATORS" is very interesting when seen from the point of view of its mathematical development. There is remarkable ingenuity here in the treatment of concepts and in the analytical development. And, also, some pioneering in the consideration of Riez estimators.
However, the model resulting from this construction, in my opinion, does not add anything to the many already known congeners. Namely in the estimation process, something different but similar in substance.
Thus, the content of this article is merely an analytical exercise that, although brilliant, does not significantly contribute to the advancement of knowledge. As such, I cannot recommend it for publication in Axioms.

Author Response

1.  The Projection Operator $\Pi$ defined in the paper implies the fitting of the Riesz Estimators' Linear Regression. This Operator is not directly obtained in the paper of C.D. Aliprantis et al. (2007), though authors refer to this Regression Model as a Linear Regression model in p. 439 of it. A question which arises naturally is the following one: 'Since the set of the affine Riesz Estimators lie in the sub- lattice generated by the random variables of a vector random variable, which is a unified way to specify this sub -lattice ?' The answer that we consider is implied by using the positive basis of this sub- lattice, which is not a result of us. It is a result arising from the paper of I.A. Polyrakis, 'Minimal Lattice -Subspaces' published in 1999 in Transactions of the American Mathematical Society.
2. The error term geometry is also specified by the complementary Projection Operator $I-\Pi$, where $I$ is the identity Operator and $\Pi$ is the Operator defined by the positive basis of the sub -lattice generated by the columns of the design matrix denoted by $X$ Both of these Operators imply the Goodness -of -Fit Measures of the Riesz Estimators' Regression, which are not determined previously.
3. The result that the Riesz Estimators are actually the Ordinary Least Squares' Estimators with respect to the positive basis of the sub- lattice generated by the sample indicated by the design matrix was not determined previously.
4. The cause for using finite dimensional spaces in the fitting of the Regression Model arises from the fact that any Regression Model fitting relies on some sample. In C.D. Aliprantis et al. (2007), random variables arising as elements of some infinite -dimensional $L^{2}$ space, which is the seminal approach of Linear Regression Models due to the Ordinary Least Square Estimators. On the other side, both of the algorithms RIESZVAR (i), RIESZVAR (ii) used for fitting, provide finitely -dimensional Linear Regression Equations. This is the usual form of a Linear Regression Model. Fitting way provided in the present paper indicates another way of this fitting procedure relying on positive bases, while the Regression Equation of p. 440 in the paper C.D. Aliprantis et al. (2007), linearity appears from positive parts of the associated variables. The way proposed in the present paper is independent from the size of the sample. 
5.  We do emphasize that fitting of Riesz Etimators' Regression may include only categorical variables' data. It also may include both categorical and variables' data and data coming from variables whose support is some interval of the real numbers.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Dear Authors 

Thanks for the revised copy for your paper.

The authors made all comments and revised the paper. So, I see that the paper ready for publishing.

Reviewer 3 Report

OK. As it stands, this paper, due to the additions made by the authors, has reached sufficient quality to be published in Axioms.