**1. Introduction**

In recent years, there has been a dramatic growth in the number of generalisations of well-known probability distributions. Most notable generalizations are achieved by (i) inducting power parameters in well established parent distributions, (ii) extending the classical distribution by modification in their functions, (iii) introducing special functions such as W[K(x)] as generators and (iv) by compounding of distributions. This heaped surge of generalized families is due to the flexibility in modelling phenomenons related to the changing scenarios of contemporary scientific field including demography, actuarial, survival, biological, ecological, communication theory, epidemiology and environmental sciences. However, a clear understanding of the applicability of these models in most applied areas is necessary if one is to gain insights into systems that can be modeled as random processes. The model, thus obtained, acquires improved empirical results to the real data that is collected adaptively.

Although there exist many functions which act as generators to produce flexible classes of distributions, in this project, we will emphasize generalizations in which a ratio of survival function (sf) has been used in some form, commonly known as the odd ratio. In the reference [1], a proportional odd family *viz. a viz.* the Marshall Olkin-G (MO-G) was generalized by sf *<sup>K</sup>*(*x*) = 1 − *<sup>K</sup>*(*x*), where *<sup>K</sup>*(*x*) is the distribution function (cdf) of parent distribution, with the induction of a tilt parameter. Gleaton and Lynch, in the reference [2],

**Citation:** Khan, S.; Balogun, O.S.; Tahir, M.H.; Almutiry, W.; Alahmadi, A.A. An Alternate Generalized Odd Generalized Exponential Family with Applications to Premium Data. *Symmetry* **2021**, *13*, 2064. https:// doi.org/10.3390/sym13112064

Academic Editor: Zhivorad Tomovski

Received: 19 September 2021 Accepted: 14 October 2021 Published: 1 November 2021

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used the odd function as generator when they defined a log odd family (OLL-G). In the reference [3], defined the odd Weibull family, as an asymptotically equivalent log-logistic model for larger values of *θ*, the scale parameter. The reference [4] used the Transformed -Transformer (TX) family, due to the reference [5], to define odd Weibul-G families of distribution. Since then, a myriad of distributions has been generalized using odd function. Some of the important families include [6–31], among others.

Focussing on the origins and motivations of our proposed scheme, the authors in [32] proposed the odd generalized exponential family (which we refer to OGE-G) as a better alternative to generalized exponential (GE) family using Lehmann Alternative-I (LA-I). The cdf of the two parameter OGE-G family is mentioned below:

$$F\_{\rm OGE-G}(\mathbf{x}; \mathfrak{a}, \lambda, \mathfrak{v}) = \left(1 - \mathbf{e}^{-\lambda \frac{K(\mathbf{x}; \mathfrak{v})}{\overline{K}(\mathbf{x}; \mathfrak{v})}}\right)^{\mathfrak{a}}, \quad \mathbf{x} > 0, \mathfrak{a} > 0, \lambda > 0.$$

In the reference [33], an odd family of GE was proposed so-called generalized odd generalized exponential family (which we refer to OGE1-G). The cdf of OGE1-G family is presented as:

$$F\_{\rm OGE1-G}(\mathbf{x}; \mathfrak{a}, \boldsymbol{\beta}, \boldsymbol{\psi}) = \left(1 - \mathbf{e}^{-\frac{K^{\alpha}(\mathbf{x}; \boldsymbol{\psi})}{1 - K^{\alpha}(\mathbf{x}; \boldsymbol{\psi})}}\right)^{\beta}, \quad \mathbf{x} > 0, \boldsymbol{\alpha} > 0, \beta > 0.$$

Because of its capacity to simulate variable hazard rate function (hrf) forms of all traditional types in lifetime data analysis, we believe OGE-G offers a sensible combination of simplicity and flexibility. However, the relevance of OGE1-G to lifespan modelling in domains such as reliability, actuarial sciences, informatics, telecommunications, and computational social sciences (just to highlight a few) is still debatable. According to the reference [34], the Lehmann Alternative-II (LA2) approach has received less attention. This motivated us to use LA2 approach to develop the exponentiated odd generalized exponential (OGE2-G), in the same vein as OGE-G and OGE1-G. Adhering to the framework defined in the reference [5], if *T* follows GE random variable (rv), then the cdf of OGE2-G family is mentioned below:

$$F\_{\rm OGE2-G}(\mathbf{x}; \mathbf{a}, \boldsymbol{\beta}, \boldsymbol{\psi}) = \left[1 - \mathbf{e}^{-\frac{1 - \overline{\mathbf{Z}}(\mathbf{x}; \boldsymbol{\psi})^{\mathbf{a}}}{\overline{\mathbf{Z}}(\mathbf{x}; \boldsymbol{\psi})^{\mathbf{a}}}}\right]^{\beta}, \quad \mathbf{x} > \mathbf{0}, \mathbf{a}, \beta > 0 \tag{1}$$

where *α* and *β* are shape parameter and *ψ* is the vector of baseline parameter.

Consider the following points to emphasise the model's distinctiveness; (i) In the literature, the proposed model in its current form has not been studied to the best of our knowledge, (ii) From an analytical standpoint, the OGE2-G family has a significantly better configuration and practicality than OGE-G and OGE1-G for inverted models with minimal chance to counter non-identifiability issues, (iii) The OGE2-G has several curious connections to other families. When *α* approaches 0, *<sup>F</sup>*(*x*; *α*, *β*, *ψ*) tends to GE with *λ* = 1, when *α* = 1 *<sup>F</sup>*(*x*; *α*, *β*, *ψ*) tends to OGE-G, if *α* → 0 and *β* = 1 then *<sup>F</sup>*(*x*; *α*, *β*, *ψ*) tends to odd exponential (OE) (iv) This new dimension allowed us to explore models which are naturally constituted by LA2. The generalizations, thus attained, produced skewed distributions with much heavier tails enabling its practicality in risk evaluation theory with far better results, (v) The successful application of OGE2-G family motivates future research, as it outperforms nine well-established existing models, (vi) We present a physical explanation for *X* when *α* and *β* are integers. Consider there be a parallel system consisting of *β* identically independent components. Suppose that the lifetime of a rv *Y* with a specific *<sup>K</sup>*(*x*; *ψ*) with *α* components in a series system such that the risk of failing at time *x* is

represented by the odd function as <sup>1</sup>−*<sup>K</sup>*(*<sup>x</sup>*;*ψ*)*<sup>α</sup> <sup>K</sup>*(*x*;*ψ*)*<sup>α</sup>* . Consider that the randomness of this risk is represented by the rv *X*, then we can assume the following relation holds

$$\Pr(\mathbf{Y} \le \mathbf{x}) = \Pr(X \le \frac{1 - \overline{\mathbb{K}}(\mathbf{x}; \boldsymbol{\psi})^a}{\overline{\mathbb{K}}(\mathbf{x}; \boldsymbol{\psi})^a}) = F(\mathbf{x}; \boldsymbol{\mathfrak{a}}, \boldsymbol{\beta}, \boldsymbol{\psi}) \; \mathsf{A}$$

explicitly given in Equation (1). The OGE2-G family is offered and explored in this research, emphasising its diversity and scope for application to real life phenomenons. The major features of the OGE2-G family, including the pdf, hrf, qf, and ten unique models from OGE2-G family presented in Table 1, are provided in the first half. Then, certain mathematical properties of the OGE2-G such as series expansion of the exponentiated pdf, moments, parameter estimation, order statistics, Rényi entropy, stress-strength analysis and stochastic dominance results are investigated. Furthermore, Fréchet is specified as baseline model termed as OGE2-Fréchet (denoted as OGE2Fr) and the maximum likelihood (ML) technique is then used to construct statistical applications of the special model. We choose to study OGE2Fr specifically as its nested model include inverse-Rayleigh (IR) and inverse exponential (IE), favoring its suitability over sub-models as well. It is applied to fit two sets of premium data from actuarial field. Using key performance indicators, we reveal that OGE2Fr outperforms nine competing models. A portion pertinent to specific risk measures, with an emphasis on the value at risk (VaR) and the expected shortfall (ES), is presented. Eventually, the estimation of risk measures for the examined data sets is then discussed, with the proposed methodology yielding a rather satisfying result. Equation (1) can be useful in modelling real life survival data with different shapes of hrf. Table 1 lists <sup>1</sup>−*<sup>K</sup>*(*<sup>x</sup>*;*ψ*)*<sup>α</sup> <sup>K</sup>*(*x*;*ψ*)*<sup>α</sup>* and the corresponding parameters for some special distributions which are considered to be the potential sub-models of OGE2-G family.


**Table 1.** Distributions and corresponding <sup>1</sup>−*<sup>K</sup>*(*<sup>x</sup>*;*ψ*)*<sup>α</sup> <sup>K</sup>*(*x*;*ψ*)*<sup>α</sup>* functions.

The following is a breakdown of how the paper is constructed. In Section 2, we acquaint the readers to the new family with basic properties and ten potential baseline models which can become members of OGE2-G family. Section 3 is comprised of the mathematical properties of the OGE2-G family. Section 4 progresses by taking Fréchet (Fr) as sub-model to propose OGE2Fr and related statistical and inferential properties. Section 5 specifies two applications of actuarial data sets with emphasis on risk evaluation (premium returns) and the proposed model's veracity is established. Furthermore, the

model is applied to compute some actuarial measures. Section 6 is the final section, with some annotations and useful insights.

### **2. The OGE2G Family**

In this segment, basic statistical properties of the newly proposed family characterized by the cdf, in Equation (1) are presented. Functional forms of ten sub models are also defined.

### *2.1. Definition of pdf and hrf*

The pdf in agreemen<sup>t</sup> with Equation (1) is given as (2).

$$\left(f\_{\text{OGE2}-G}(\mathbf{x}; \mathfrak{a}, \mathfrak{k}, \mathfrak{\boldsymbol{\psi}})\right)^{-} = \left. a \, \mathfrak{k} \, k(\mathbf{x}; \mathfrak{\boldsymbol{\psi}}) \, \overline{\mathbf{K}}(\mathbf{x}; \mathfrak{\boldsymbol{\psi}})^{-a-1} \, \mathbf{e}^{-\left\{\frac{1-\overline{\mathbf{K}}(\mathbf{x}; \mathfrak{\boldsymbol{\psi}})^{a}}{\overline{\mathbf{K}}(\mathbf{x}; \mathfrak{\boldsymbol{\psi}})^{a}}\right\} \left[1-\mathbf{e}^{-\left\{\frac{1-\overline{\mathbf{K}}(\mathbf{x}; \mathfrak{\boldsymbol{\psi}})^{a}}{\overline{\mathbf{K}}(\mathbf{x}; \mathfrak{\boldsymbol{\psi}})^{a}}\right\} \right]^{\beta-1} \tag{2}$$

Using the results defined in Equations (1) and (2), the hrf is defined as

$$\begin{split} \tau(\mathbf{x}; \boldsymbol{a}, \boldsymbol{\beta}, \boldsymbol{\psi}) &= \quad \frac{f\_{\text{OGE2}-G}(\mathbf{x}; \boldsymbol{a}, \boldsymbol{\beta}, \boldsymbol{\psi})}{1 - F\_{\text{OGE2}-G}(\mathbf{x}; \boldsymbol{a}, \boldsymbol{\beta}, \boldsymbol{\psi})} \\ &= \quad \mathbf{a} \boldsymbol{\beta} \, k(\mathbf{x}; \boldsymbol{\psi}) \overline{\mathcal{K}}(\mathbf{x}; \boldsymbol{\psi})^{-\mathbf{a}-1} \mathbf{e}^{-\left\{\frac{1-\overline{\mathcal{K}}(\boldsymbol{x}; \boldsymbol{\psi})^{\mathbf{a}}}{\overline{\mathcal{K}}(\boldsymbol{x}; \boldsymbol{\psi})^{\mathbf{a}}}\right\}} \left[1 - \mathbf{e}^{-\left\{\frac{1-\overline{\mathcal{K}}(\boldsymbol{x}; \boldsymbol{\psi})^{\mathbf{a}}}{\overline{\mathcal{K}}(\boldsymbol{x}; \boldsymbol{\psi})^{\mathbf{a}}}\right\}}\right]^{\beta - 1} \\ & \times \left[1 - \left(1 - \mathbf{e}^{-\left\{\frac{1-\overline{\mathcal{K}}(\boldsymbol{x}; \boldsymbol{\psi})^{\mathbf{a}}}{\overline{\mathcal{K}}(\boldsymbol{x}; \boldsymbol{\psi})^{\mathbf{a}}}\right\}\right)^{\beta - 1}\right] \end{split} \tag{3}$$

The hazard rate is just a calculation of the change in survivor rate per unit of time. Hence, its importance in reliability and survival analysis is crucial. The hrf has some characteristic shapes which include monotonic (increasing, decreasing), non-monotonic (bathtub or upside down bathtub) or constant. Standard statistical distribution yield maximum three shapes, but OGE2-G family can yield a diverse range of shapes (including increasing-decreasing-increasing) depending upon the choice of special model. For further details on hrf, see [35].

### *2.2. Quantile Function and Potential Sub-Models*

The OGE2-G family may be readily approximated by reversing Equation (1) as shown below: If indeed the distribution of *u* is uniform *u*(0, <sup>1</sup>), therefore

$$\mathbf{x} = Q\_K \left[ 1 - \left\{ 1 - \log \left( 1 - u^{1/\beta} \right) \right\}^{-1/a} \right]. \tag{4}$$

Equation (4) can be useful to define statistical measures such as median, skewness, and kurtosis based on quartiles, deciles, or percentiles. These measures facilitates to concisely define the skewness and kurtosis measures which are significant tool to comprehend the shape(s) of the distribution.

Theorem 1 shows how the OGE2 family is related to other distributions.

\*\*Theorem 1.\*\*  $Let \ X \sim \text{OGE2-G}(\mathfrak{a}, \beta; \mathfrak{\psi})$ , then

 $(a) \text{ If } Y = 1 - \overline{\mathcal{K}}(\mathfrak{x}; \mathfrak{\psi})^{\mathfrak{a}}, \text{ then } F\_Y(y) = \left(1 - \mathbf{e}^{-\frac{\underline{\chi}}{1 - \underline{\chi}}}\right)^{\mathfrak{f}}, \quad 0 < y < 1, \text{ and}$ 
 $(b) \text{ If } Y = \frac{1 - \overline{\mathcal{K}}(\mathfrak{x}; \mathfrak{\psi})^{\mathfrak{a}}}{\overline{\mathcal{K}}(\mathfrak{x}; \mathfrak{\psi})^{\mathfrak{a}}}, \text{ then } Y \sim \text{GE}(1, \beta).$ 

### **3. Mathematical Properties of OGE2-G Family**

To capture the family's modelling capacity, numerous mathematical features of the OGE2-G are examined in this section. Some of the key results established in this section are then applied in Section 5.

### *3.1. Linear Expansion of cdf*

We provide a useful expansion for (1) in terms of linear combinations of exp-G density functions using the following series expansion as

$$(1-z)^{\eta-1} = \sum\_{i=0}^{\infty} \frac{(-1)^i \Gamma(\eta)}{i! \Gamma(\eta-i)} z^i,$$

whereas the expansion holds for all |*z*| < 1 and *η* > 0 a non-integer value. Then, the cdf of OGE2-G class in (1) can indeed be phrased with

$$F(\mathbf{x}) = \sum\_{i=0}^{\infty} \frac{(-1)^i \Gamma(\beta + 1)}{i! \, \Gamma(\beta + 1 - i)} \mathbf{e}^{-i} \left(\frac{\frac{1 - \mathbb{E}^{\mathbf{d}}}{\mathbb{E}^{\mathbf{d}}}}{\mathbb{E}^{\mathbf{d}}}\right) . \tag{5}$$

Using series expansion and power series expansion in Equation (5), will yield the following cdf

$$F(\mathbf{x}) = F(\mathbf{x} : \mathfrak{a}, \boldsymbol{\beta}, \boldsymbol{\psi}) = \sum\_{\ell=0}^{\infty} \mathfrak{J}\_{\ell} H\_{\ell}(\mathbf{x}) \, , \tag{6}$$

where *<sup>H</sup>*(*x*) = *<sup>K</sup>*(*x*; *ψ*) (for ≥ 1) denotes the cdf of exp-G distribution with power parameter and

$$\xi\_{\ell} = (-1)^{\ell} \sum\_{i,j=1}^{\infty} \sum\_{k=0}^{j} (-1)^{i+j+k} \frac{\Gamma(\beta+1)}{i! j! \Gamma(\beta+1-i)} \binom{j}{k} \binom{\alpha(k-i)}{\ell} \dots$$

Through differentiating Equation (6) the OGE2-G family density, we may express it as a combination of exp-G densities.

$$f(\mathbf{x}:\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\psi}) = \sum\_{i,j=1}^{\infty} \xi\_{\ell}^{\boldsymbol{\alpha}} h\_{\ell}(\mathbf{x})\,. \tag{7}$$

where *h*(*x*) = *<sup>K</sup>*−<sup>1</sup>(*x*; *ψ*) *k*(*x*; *ψ*) is the exp-G pdf with power parameter . As a result, numerous features of the proposed model may be deduced from the exp-G distribution's attributes. Most modern computation frameworks, such as MathCad, Maple, Mathematica, and Matlab, can efficiently handle the formulas derived throughout the article, which can currently operate using the use of analytic formulations of enormous size and complexity.

### *3.2. Numerous Types of Moments*

The fundamental formula for the *p*th moment of *X* is supplied by (7) as

$$
\mu\_p' = \sum\_{i,j=0}^{\infty} \xi\_\ell \, E(X\_\ell^p). \tag{8}
$$

where <sup>E</sup>(*Xp* ) = ∞0 *<sup>x</sup>ph*(*x*)*dx*. Setting *p* = 1 in (8) can provide explicit expression for the mean of several parent distributions.

A another expression for *μp* is taken from (8) as far as the baseline qf is concerned

$$
\mu\_p' = \sum\_{i,j=0}^{\infty} \xi\_\ell \,\,\, \pi(p,\ell-1). \tag{9}
$$

where *<sup>τ</sup>*(*p*, − 1) = 10 *QG*(*u*)*<sup>p</sup> <sup>u</sup>du*.

The central moments (*μp*) and cumulants (*<sup>κ</sup>p*) of *X* can follow from Equation (8) as *μp* = <sup>∑</sup>*pk*=<sup>0</sup> (*pk*)(−<sup>1</sup>)*<sup>k</sup> μp*1 *<sup>μ</sup>p*−*<sup>k</sup>* and *κs* = *μs* − ∑*<sup>s</sup>*−<sup>1</sup> *k*=1 (*<sup>s</sup>*−<sup>1</sup> *<sup>k</sup>*−<sup>1</sup>) *κk <sup>μ</sup>s*−*k*, respectively, where *κ*1 = *<sup>μ</sup>*1.

The *r*th lower incomplete moment of *X* can be determined from Equation (7) as

$$m\_r(y) = \sum\_{i,j=0}^{\infty} \xi\_\ell^x \int\_0^{G(y)} Q\_G(u)^r \, u^{l-1} d\_u. \tag{10}$$

For most *G* distributions, the final integral may be calculated.

### *3.3. Inference Related to OGE2 Family*

The strategy of maximum likelihood (MLL) approach is used to estimate the unknown parameters of the new class. Let *x*1, ... , *xn* be *n* observations from the OGE2-G density class (2) with parameter vector Θ = (*<sup>α</sup>*, *β*, *ψ*). Then the likelihood function L(*<sup>α</sup>*, *β*, *δ*) on the domain Θ is defined as

$$\begin{aligned} \mathcal{L} &= \left[ n \log(a) + n \log(\beta) + \sum\_{i=1}^{n} \log k(\mathbf{x}\_i; \boldsymbol{\psi}) - (n+1) \log \overline{\mathcal{K}}(\mathbf{x}\_i; \boldsymbol{\psi}) \right] \\ &- V(\mathbf{x}\_i; \boldsymbol{a}, \boldsymbol{\psi}) + (\beta - 1) \sum\_{i=1}^{n} \log \left[ 1 - \mathbf{e}^{-V(\mathbf{x}; \boldsymbol{a}, \boldsymbol{\psi})} \right], \end{aligned} \tag{11}$$

where *<sup>V</sup>*(*xi*; *α*, *ψ*) = <sup>1</sup>−*<sup>K</sup>*(*xi*;*ψ*)*<sup>α</sup> <sup>K</sup>*(*xi*;*ψ*)*<sup>α</sup>*

> The elements of the score vector *U*(Θ) are as described in the following:

.

*Uα* = *n α* − *n* ∑ *i*=1 log *<sup>K</sup>*(*xi*; *ψ*) − *<sup>V</sup>*(*α*)(*xi*; *α*, *ψ*) + *n* ∑ *i*=1 *β* − <sup>1</sup>e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)*V*(*α*)(*xi*; *α*, *ψ*) 1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*) , *Uβ* = *nβ* + *n*∑*i*=1 log'1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)(, *<sup>U</sup>ψ* = *n* ∑ *i*=1% *k* (*xi*; *ψ*) *k*(*xi*; *ψ*) & + (*α* + 1) *n*∑*i*=1% *k*(*xi*; *ψ*)*k* (*xi*; *ψ*) *<sup>K</sup>*(*xi*; *ψ*) & − *V*(*ξk* )(*xi*; *α*, *ψ*) +(*β* − 1) *n* ∑ *i*=1 /e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)*V*(*ξ<sup>k</sup>* )(*xi*; *α*, *ψ*) 1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*) 0 ,

where *<sup>V</sup>*(*α*)(.) and *V*(*ψk* )(.) means the derivative of the function *V* with respect to *α* and *ψ*, respectively.

The next elements are produced by the components of the score vector *J*(Θ).

*Jαα* = − *n α*<sup>2</sup> + *n* ∑ *i*=1 *k*(*xi*; *α*, *β*, *ψk*)*V*(*ψ<sup>k</sup>* ) *K* ¯(*xi*; *α*, *β*, *ψk*) − *V*(*<sup>α</sup> α*) + (*β* − 1) × *n* ∑ *i*=1 e <sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)*V*(*xi*; *α*, *α*) + e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*)(*V*(*xi*; *α*))<sup>2</sup> (1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*))<sup>2</sup> , *Jαβ* = *n* ∑ *i*=1 e <sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*) *V <sup>α</sup>*(*xi*; *α*, *ψ*) [1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*)]<sup>2</sup> , *Jαψ* = *n* ∑ *i*=1 *k*(*xi*; *α*, *β*, *ψ*)*k*(*xi*; *α*, *β*, *ψ*) *K* ¯(*xi*; *α*, *β*, *ψ*) − *V*(*<sup>α</sup> <sup>ψ</sup>*)(*xi*; *α*, *ψ*)+(*β* − 1) × *n* ∑ *i*=1 1e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)\*<sup>2</sup>*V*(*α*)(*xi*; *α*, *ψ*)*V*(*ψ*)(*xi*; *α*, *ψ*) − e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)*V*( *αψ*)(*xi*; *α*, *ψ*) (1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*))<sup>2</sup> −(*β* − 1) *n* ∑ *i*=1 e <sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*β*,*ψ*)*V*(*α*)(*xi*; *α*, *ψ*)*V*(*ψ*)(*xi*; *α*, *ψ*) (1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*))<sup>2</sup> +(*β* − 1) *n* ∑ *i*=1 1e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)\*<sup>2</sup>*V*(*α*)(*xi*; *α*, *ψ*)*V*(*<sup>ψ</sup>*)(*xi*; *α*, *ψ*) (1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*))<sup>2</sup> , *Jββ* = − *nβ* 2 , *Jβψ* = *n* ∑ e <sup>−</sup>*<sup>V</sup>*(*xi*;*α*, *<sup>ψ</sup>*)*V*(*ψ*)(*xi*; *α*, *ψ*) 1<sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*, *ψ*) ,

*Jψψ* = *n* ∑ *i*=1 *<sup>k</sup>*(*xi*;*<sup>ψ</sup>*)*k*(*xi*;*<sup>ψ</sup>*) − 1*k*(*xi*;*<sup>ψ</sup>*)\*<sup>2</sup> 1*k*(*xi*;*<sup>ψ</sup>*)\*<sup>2</sup> + (*α* + 1) *n* ∑ *i*=1 % *<sup>k</sup>*(*xi*;*<sup>ψ</sup>*)*k*(*xi*;*<sup>ψ</sup>*){−*k*(*xi*;*<sup>ψ</sup>*)}<sup>2</sup> *K*¯ <sup>2</sup>((*xi*;*<sup>ψ</sup>*)) & −*V*(*<sup>ψ</sup> ψ*)(*xi*;*<sup>ψ</sup>*) − (*β* − 1) *n* ∑ *i*=1 1e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)*V*(*ψ*)(*xi*; *α*, *ψ*)\*<sup>2</sup> (1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*))<sup>2</sup> −(*β* − 1) *n* ∑ *i*=1 e <sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)*V*(*ψ <sup>ψ</sup>*)(*xi*; *α*, *ψ*) + e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*)[*V*(*ψ*)(*xi*; *α*, *ψ*)]<sup>2</sup> (1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*))<sup>2</sup> +(*β* − 1) *n* ∑ *i*=1 'e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)(<sup>2</sup>*V*(*<sup>ψ</sup> <sup>ψ</sup>*)(*xi*; *α*, *ψ*) + 1e<sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*ψ*)*V*(*ψ*)(*xi*; *α*, *ψ*)\*<sup>2</sup> (1 − <sup>e</sup><sup>−</sup>*<sup>V</sup>*(*xi*;*α*,*<sup>ψ</sup>*))<sup>2</sup> ,

where *V*(*α <sup>α</sup>*)(.) is the derivative of *<sup>V</sup>*(*α*)(.) with respect to *α*, *V*(*α <sup>ψ</sup>*)(.) is the derivative of *V <sup>α</sup>*(.) with respect to *ψk* and *V ψ <sup>ψ</sup>*(.) is the derivative of *<sup>V</sup>*(*ψ*)(.) with respect to *ψ*.
