Applying Generalized Type-II Hybrid Censored Samples on Generalized and q-Generalized Extreme Value Distributions under Linear Normalization

Abstract

The generalized extreme value (GEV) distributions have wide applications for describing a variety of random events, such as those that occur during specific survival, financial, or reliability investigations. Also, the q-analogues of GEV distributions, called (q-GEVs), are characterized by their ability to provide more flexibility for modeling, which is due to the influence of the q parameter. In this study, we estimated the parameters of generalized and q-generalized extreme value distributions under linear normalization, called GEVL and q-GEVL, respectively. These parameters were estimated using the maximum likelihood estimator method and are based on the generalized type-II hybrid censored sample (G-Type-II HCS). The confidence intervals for these parameters were evaluated. Also, Shannon entropy was estimated for GEVL and q-GEVL distributions. The accuracy of these parameters and the performance of estimators were demonstrated through a real-life example and a simulation study.

1. Introduction

Asymmetrical models, such as Gumbel, logistic, Weibull, and generalized extreme value (GEV) distributions, have been widely used to describe a variety of random events, such as those that may arise during specific survival, financial, or reliability investigations. The Gumbel probability distribution is used to analyze and model the behavior of random phenomena in many fields, such as engineering, business, biology, management, sports, and economics [1]. We can find many examples of the Gumbel probability distribution, also known as the double exponential probability distribution, in [2,3,4,5,6,7].

In order to boost the flexibility of modeling, Provost et al. [8] created q-analogues of the generalized extreme value (q-GEVs) and Gumbel distributions.

A hybrid censoring technique, which combines Type I and Type II censoring schemes, has been proposed for adjustable efficiency levels or termination times [9]. In order to clarify the concept of the censored sample (G-Type-II HCS), we propose the following experiment.

Consider a life-testing experiment that begins with $\nu$ identical units undergoing a lifetime test. Let $x_1,x_2,\dots ,x_{\nu }$ be the results of lifetimes from distributions with CDF $\left(F\right(x\left)\right)$ and PDF $\left(f\right(x\left)\right)$. Let $a\in 1,2,\dots ,\nu$ be an integer an integer and suppose $T_1<T_2\in (0,\infty )$ are time points. We have three cases as follows:

• If the $a^{th}$ failure occurs before the time point $T_1$, the experiment will be terminated at this time.
• If the $a^{th}$ failure occurs between the time points $T_1$ and $T_2$, then the experiment will be terminated at the time of the failure, $x_a$.
• If the $a^{th}$ failure occurs after the time point $T_2$, the experiment will be terminated at time point $T_2$.

This type of censoring, while aiming for a minimum number of failures, a, guarantees that the experiment will be completed by time $T_2$. Therefore, $T_2$, known as the absolute maximum time of the experiment, is not exceeded [10].

The maximum time for the experiment is fixed using the G-Type-II HCS is $T_2$, and this is an advantage from an experimental point of view. One of the following cases are observed using G-Type-II hybrid censoring sample:

Case I:

$\{x_{1:\nu }<x_{2:\nu }<\dots <x_{a:\nu }<\dots <x_{{\nu }_1}\le T_1\},$ where $x_{a:\nu }<T_1$,

Case II:

$\{x_{1:\nu }<x_{2:\nu }<\dots <T_1<\dots <x_{a:\nu }\le T_2\},$ where $T_1<x_{a:\nu }<T_2$,

Case III:

$\{x_{1;\nu }<x_{2:\nu }<\dots <T_{1:\nu }<\dots <x_{{\nu }_2}\le T_2\},$ where $x_{a:\nu }>T_2$.

Note that ${\nu }_1$ and ${\nu }_2$ are the number of observed failures up to time points $T_1$ and $T_2$, respectively. Then, for the G-Type-II HCS, the likelihood functions for the three different cases described above are as follows:

Case I

$$\frac{\nu !}{(\nu -{\nu }_1)!}\prod _{i=1}^{{\nu }_1}f\left(x_{i:\nu }\right){[S\left(T_1\right)]}^{\nu -{\nu }_1};{\nu }_1=a,a+1,\dots ,or\nu ,$$

Case II

$$\frac{\nu !}{(\nu -a)!}\prod _{i=1}^{a}f\left(x_{i:r}\right){[S\left(x_a\right)]}^{\nu -a},$$

Case III

$$\frac{\nu !}{(\nu -{\nu }_2)!}\prod _{i=1}^{n_2}f\left(x_{i:\nu }\right){[S\left(T_2\right)]}^{\nu -{\nu }_2};{\nu }_2=0,1,2,\dots ,or(a-1),$$

where $S\left(x\right)$ is the survival function.

Entropy was initially developed by Clausius et al. [11] in the context of information theory. He created a new route for the advancement of thermodynamics by using the idea of entropy to represent the second rule of thermodynamics quantitatively. This notion was continued by Shannon [12], and ever since then it has been used in a variety of domains, including economics and image and signal processing. On entropy estimation for various distributions, several papers have been provided. The entropy of the Weibull distribution with progressive censoring was studied by Naif and Malyk [13]. The entropy of the Rayleigh distribution based on the doubly generalized G-Type-II HCS was evaluated by Cho et al. [14]. Cho et al. [15] estimated the entropy of Weibull distribution using a generalized progressively censored sample. Ahmad [16] constructed estimators for entropy function of the Fréchet distribution based on the extended type I hybrid censored samples. The estimators for entropy function of the Lomax distribution with extended type I hybrid censored samples were developed by Mahmoud et al. [17].

In this study, we constructed maximum likelihood estimation to evaluate the parameters of the family of GEVL and q-GEVL distributions using the G-Type-II HCS scheme, to ensure applicability to Shannon entropy. Also, the confidence intervals for the parameters of GEVL and q-GEVL distributions were determined. Section 2 presents the GEVL and q-GEVL distributions and their respective entropy functions. The purpose of this section is to identify these distributions and to provide a detailed description of their entropy functions. In Section 3, we obtain the maximum likelihood estimation for the parameters of GEVL based on the G-Type-II HCS scheme. Also, the simulation of this procedure and calculation of the Shannon entropy are described. In Section 4, we evaluate the maximum likelihood estimation for the parameters of q-GEVL based on the G-Type-II HCS scheme. Also, the simulation of this procedure and calculation of the Shannon entropy are described. In Section 5, the confidence intervals for the parameters of GEVL and q-GEVL are determined. After that, the Conclusion Section (Section 6) is presented.

2. The Family of GEVL and q-GEVL Distributions

The limit of the cumulative density function (CDF) $L_{\varphi }$ is described by the extremal types theorem as having one type of three types [18]. The three types, which are together grouped in the family below, are frequently referred to as the Gumbel, Fréchet, and Weibull types:

$$L_{\varphi }(x;\alpha ,\beta ,\varphi )=\left\{\begin{array}{l}exp\{-{(1+\varphi (\frac{x-\alpha }{\beta }))}^{\frac{-1}{\varphi }}\},\varphi \ne 0,\\ exp\{-exp(-\frac{x-\alpha }{\beta })\},\varphi \to 0,\end{array}\right.$$

(1)

and the probability density function (PDF) $l_{\varphi }$ can be given by:

$$l_{\varphi }(x;\alpha ,\beta ,\varphi )=\left\{\begin{array}{l}\frac{1}{\beta }exp\{-{(1+\varphi (\frac{x-\alpha }{\beta }))}^{\frac{-1}{\varphi }}\}\times {(1+\varphi (\frac{x-\alpha }{\beta }))}^{\frac{-1}{\varphi }-1},\varphi \ne 0,\\ \frac{1}{\beta }exp\{-exp(-(\frac{x-\alpha }{\beta }))\}\times exp(-(\frac{x-\alpha }{\beta })),\varphi \to 0,\end{array}\right.$$

(2)

where $\alpha$ is a location parameter, $\beta$ is a positive scale parameter, $\varphi$ is the shape parameter, and the values of x are defined by:

$$x\in \left\{\begin{array}{l}(\alpha -\frac{\beta }{\varphi },\infty ),\varphi >0,\\ (-\infty ,\infty ),\varphi \to 0,\\ (-\infty ,\alpha -\frac{\beta }{\varphi }),\varphi <0.\end{array}\right.$$

The distribution in Equation (1) is known as a generalized extreme value (GEV) distribution under linear normalization. We denote it by $GEVL(x;\alpha ,\beta ,\varphi ).$ The Gumbel probability distribution in Equations (1) and (2) as $\varphi \to 0$ is used to analyze and model the behavior of random phenomena in many fields. Bashir et al. [19] examined and contrasted three estimation methods used to approximate the parameter values for simulated observations taken from the GEVL distribution. Figure 1 refers to the cumulative distribution and density function of GEVL distribution for $\varphi \to 0$.

Provost et al. [8] proposed the q-GEVL distribution and q-Gumbel distribution (obtained by letting $\varphi \to 0$ in the q-GEVL model), and the corresponding distributions are provided by:

$$Ł(x;\alpha ,\beta ,\varphi ,q)=\left\{\begin{array}{c}{[1+q{(\varphi (\frac{x-\alpha }{\beta })+1)}^{-\frac{1}{\varphi }}]}^{-\frac{1}{q}},\varphi \ne 0,q\ne 0\hfill \\ {(1+q{e}^{-(\frac{x-\alpha }{\beta })})}^{-\frac{1}{q}},\varphi \to 0,q\ne 0\hfill \end{array}\right.$$

(3)

and

$$ł(x;\alpha ,\beta ,\varphi ,q)=\left\{\begin{array}{c}\frac{1}{\beta }{(1+\varphi (\frac{x-\alpha }{\beta }))}^{\frac{-1}{\varphi }-1}\times {[1+q{(\varphi (\frac{x-\alpha }{\beta })+1)}^{-\frac{1}{\varphi }}]}^{-\frac{1}{q}-1},\varphi \ne 0,q\ne 0\hfill \\ {(1+q{e}^{-(\frac{x-\alpha }{\beta })})}^{-\frac{1}{q}-1}\frac{1}{\beta }{e}^{-(\frac{x-\alpha }{\beta })},\varphi \to 0,q\ne 0,\hfill \end{array}\right.$$

(4)

where the values of x can be determined by:

$$x\in \left\{\begin{array}{ll}(-\infty ,\infty )& \varphi \to 0,q>0,\\ (\frac{\frac{\alpha }{\beta }+ln(-q)}{\beta },\infty )& \varphi \to 0,q<0.\end{array}\right.$$

Figure 2 refers to the cumulative distribution and density function of q-GEVL distribution for $\varphi \to 0$.

The differential entropy is a measure of uncertainty and is defined as follows:

Let X be an absolutely continuous random variable with probability density function (PDF) $f\left(x\right)$. It is written as:

$$H\left(X\right)=-{\int }_{S}f\left(x\right)logf\left(x\right)dx$$

(5)

The expectancy of a random variable $(-logf\left(X\right))$ is a statistic that has recently gained the interest of investigators.

The Shannon entropy of GEVL family is well known as:

$$H\left(X\right)=log\beta +(\varphi +1)\gamma +1$$

(6)

The Shannon entropy of each type in Equation (6) is evaluated by Ravi and Saeb [20].

On the other hand, Eliwa, et al. [21], evaluated the Shannon entropy of $q-GEVL$ family as follows:

$$H\left(X\right)=log\beta +(\varphi +1)\gamma +(1+q)[1-\sum _{n=2}^{\infty }{(-1)}^{n+1}{q}^{n-1}\Gamma (n-1)],$$

(7)

where $\gamma$ is the Euler–Macheronic constant.

3. Maximum Likelihood Estimation for the Family of GEVL Based on G-Type-II HCS

3.1. Estimation of GEVL Parameters through G-Type-II HCS

In order to estimate the parameters of the family of generalized extreme distribution under linear normalization, whose cumulative function and density function as in Equations (1) and (2) is based on the G-Type-II HCS, we suppose that ${\nu }_1$ and ${\nu }_2$ denote the number of failures that occur by time points $T_1$ and $T_2$, respectively. Then, based on the three forms of the G-Type-II HCS, the likelihood function will take one of the following forms:

Case I

$$L_I(\alpha ,\beta ,\varphi )=\frac{\nu !}{(\nu -{\nu }_1)!}\prod _{i=1}^{{\nu }_1}\frac{1}{\beta }{e}^{-{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }-1}{[1-e^{-{[1+\varphi \frac{T_1-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}]}^{\nu -{\nu }_1},$$

Case II

$$L_{II}(\alpha ,\beta ,\varphi )=\frac{\nu !}{(\nu -a)!}\prod _{i=1}^a\frac{1}{\beta }{e}^{-{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }-1}{[1-e^{-{[1+\varphi \frac{x_r-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}]}^{\nu -a},$$

Case III

$$L_{III}(\alpha ,\beta ,\varphi )=\frac{\nu !}{(\nu -{\nu }_2)!}\prod _{i=1}^{{\nu }_2}\frac{1}{\beta }{e}^{-{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }-1}{[1-e^{-{[1+\varphi \frac{T_2-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}]}^{\nu -{\nu }_2}.$$

The log likelihood functions are:

Case I

$$\begin{gathered} {\ell }_I(\alpha ,\beta ,\varphi )=E_1-{\nu }_{1}log\beta -\sum _{i=1}^{{\nu }_1}{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }}-(\frac{1}{\varphi }+1)\sum _{i=1}^{{\nu }_1}log[1+\varphi \frac{x_i-\alpha }{\beta }] \\ +(\nu -{\nu }_1)log[1-e^{-{[1+\varphi \frac{T_1-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}]], \end{gathered}$$

Case II

$$\begin{gathered} {\ell }_{II}(\alpha ,\beta ,\varphi )=E_2-alog\beta -\sum _{i=1}^a{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }}-(\frac{1}{\varphi }+1)\sum _{i=1}^{r}log[1+\varphi \frac{x_i-\alpha }{\beta }] \\ +(\nu -a)log[1-e^{-{[1+\varphi \frac{x_a-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}]], \end{gathered}$$

Case III

$$\begin{gathered} {\ell }_{III}(\alpha ,\beta ,\varphi )=E_3-{\nu }_{2}log\beta -\sum _{i=1}^{{\nu }_2}{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }}-(\frac{1}{\varphi }+1)\sum _{i=1}^{{\nu }_2}log[1+\varphi \frac{x_i-\alpha }{\beta }] \\ +(\nu -{\nu }_2)log[1-e^{-{[1+\varphi \frac{T_2-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}]], \end{gathered}$$

where $E_1$, $E_2$, and $E_3$ are normalizing constants that do not depend on the parameters. We can rewrite cases I, II, and III as a single formula as follows:

$$\begin{gathered} \ell (\alpha ,\beta ,\varphi )=E-dlog\beta -\sum _{i=1}^d{[1+\varphi \frac{x_i-\alpha }{\beta }]}^{-\frac{1}{\varphi }}-(\frac{1}{\varphi }+1)\sum _{i=1}^{d}log[1+\varphi \frac{x_i-\alpha }{\beta }] \\ -(\nu -d)log[1-e^{-{[1+\varphi \frac{s-\alpha }{\beta }]}^{-\frac{1}{\varphi }}}]], \end{gathered}$$

where $E=E_1,d={\nu }_{1}ands=T_1$ for case I; $E=E_2,d=aands=x_a$ for case II; and $E=E_3,d={\nu }_{2}ands=T_2$ for case III.

The corresponding log likelihood equations are:

$$\frac{\partial \ell }{\partial \alpha }=-\frac{1}{\beta }\sum _{i=1}^d{A}^{-(\frac{1}{\varphi }+1)}+\sum _{i=1}^d\frac{\varphi +1}{\beta }A{1}^{-1}+\frac{(n-d)A{1}^{-(\frac{1}{\varphi }+1)}{e}^{A{1}^{-\frac{1}{\varphi }}}}{\beta [1-e^{A{1}^{-\frac{1}{\varphi }}}]}=0$$

$$\begin{gathered} \frac{\partial \ell }{\partial \beta }=\frac{-d}{\beta }+\frac{\varphi +1}{{\beta }^2}\sum _{i=1}^d\frac{(x_i-\alpha )}{A}-\frac{1}{{\beta }^2}\sum _{i=1}^d(x_i-\alpha )Y^{-(\frac{1}{\varphi }+1)}\varphi \\ -\frac{(n-d)(s-\alpha )A{1}^{-(\frac{1}{\varphi }+1)}{e}^{-A{1}^{-\frac{1}{\varphi }}}}{{\beta }^2(1-e^{A{1}^{-\frac{1}{\varphi }}})}=0 \end{gathered}$$

$$\begin{gathered} \frac{\partial \ell }{\partial \varphi }=\sum _{i=1}^d\frac{A^{-\frac{1}{\varphi }}log\left(A\right)}{{\varphi }^2}+\frac{1}{\beta \varphi }\sum _{i=1}^d\frac{(x_i-\alpha )}{A^{-(\frac{1}{\varphi })}}+\frac{1}{{\varphi }^2}\sum _{i=1}^{d}logA-\frac{1}{\beta }\sum _{i=1}^d\frac{(\frac{1}{\varphi }+1)(x_i-\alpha )}{A} \\ -\frac{(\nu -d)A{1}^{-\frac{1}{\varphi }}{e}^{-A{1}^{-\frac{1}{\varphi }}}log\left(A1\right)}{{\varphi }^2(1-A{1}^{-\frac{1}{\varphi }})}+\frac{(s-\alpha )A{1}^{-(\frac{1}{\varphi }+1)}}{\beta \varphi }=0, \end{gathered}$$

(8)

where

$$A=[1+\varphi \frac{x_i-\alpha }{\beta }],$$

$$A1=[1+\frac{\varphi (s-\alpha )}{\beta }].$$

The systems specified by Equation (8) yields the maximum likelihood estimates for the parameters of the family of GEVL which follow the G-Type-II HCS. Since this equation cannot be solved analytically, the Newton–Raphson technique will be applied.

3.2. Simulation Study

A simulation study was used to demonstrate the performance of the estimators produced in the preceding section. We used the family of GEVL based on the G-Type-II HCS with $\alpha =6$, $\beta =1$ and $\varphi =0.5$ to simulate a small random sample of size $\nu =20$:

$$4,5.1555,5.3180,5.4521,5.5765,5.6986,5.8227,5.9520,6.0894,6.2382,6.4022,$$

$$6.5867,6.7983,7.0472,7.3488,7.7288,8.2339,8.9611,10.1616,12.8308$$

We applied these data to the G-Type-II HCS by solving the nonlinear systems that are specified in Equation (8) and using the Newton–Raphson technique, and MATLAB (Version 2021) was used for estimation. Then, we used Equation (6) to calculate entropy. The maximum likelihood estimations (MLEs) of the parameters and entropy of the GEVL with the G-Type-II HCS were yielded by the proposed values of $T_1,T_2$, and a in each case as shown in

Table 1. MLEs for $\alpha ,\beta ,\varphi$ and estimated entropy of GEVL with the G-Type-II HCS.

	${\mathit{T}}_1$	${\mathit{T}}_2$	a	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	$\widehat{\mathit{\varphi }}$	$\widehat{\mathit{H}}$
Case I	6	7	3	6.1	1.1699	−0.442	1.479
Case II	6	8	9	5.946	0.694	−0.8091	0.7449
Case III	7	9	18	5.2156	0.9455	0.3014	1.6951

.

4. Maximum Likelihood Estimation for the Family of q-GEVL Based on G-Type-II HCS

4.1. Estimation of q-GEVL Parameters through G-Type-II HCS

In order to estimate the parameters of the family of q-GEVL, whose cumulative function and density function as in Equations (3) and (4) is based on the G-Type-II HCS, we used ${\nu }_1$ and ${\nu }_2$ denoted the number of failures that occurred by time points $T_1$ and $T_2$, respectively. Then, based on the three forms of the G-Type-II HCS, the likelihood function will take one of the following forms:

Case I

$$L_I(\alpha ,\beta ,\varphi ,q)=\frac{\nu !}{(\nu -{\nu }_1)!}\prod _{i=1}^{{\nu }_1}\frac{1}{\beta }{(1+\varphi (\frac{x_i-\alpha }{\beta }))}^{\frac{-1}{\varphi }-1}\times {[1+q{(\varphi (\frac{x_i-\alpha }{\beta })+1)}^{-\frac{1}{\varphi }}]}^{-\frac{1}{q}-1}$$

$${[1-{\{1+q{(\varphi (\frac{T_1-\alpha }{\beta })+1)}^{-\frac{1}{\varphi }}\}}^{-\frac{1}{q}}]}^{\nu -{\nu }_1},$$

Case II

$$L_{II}(\alpha ,\beta ,\varphi ,q)=\frac{\nu !}{(\nu -a)!}\prod _{i=1}^a\frac{1}{\beta }{(1+\varphi (\frac{x_i-\alpha }{\beta }))}^{\frac{-1}{\varphi }-1}\times {[1+q{(\varphi (\frac{x_i-\alpha }{\beta })+1)}^{-\frac{1}{\varphi }}]}^{-\frac{1}{q}-1}$$

$${[1-{\{1+q{(\varphi (\frac{x_a-\alpha }{\beta })+1)}^{-\frac{1}{\varphi }}\}}^{-\frac{1}{q}}]}^{\nu -a},$$

Case III

$$L_{III}(\alpha ,\beta ,\varphi ,q)=\frac{\nu !}{(\nu -{\nu }_2)!}\prod _{i=1}^{{\nu }_2}\frac{1}{\beta }{(1+\varphi (\frac{x_i-\alpha }{\beta }))}^{\frac{-1}{\varphi }-1}\times {[1+q{(\varphi (\frac{x_i-\alpha }{\beta })+1)}^{-\frac{1}{\varphi }}]}^{-\frac{1}{q}-1}$$

$${[1-{\{1+q{(\varphi (\frac{T_2-\alpha }{\beta })+1)}^{-\frac{1}{\varphi }}\}}^{-\frac{1}{q}}]}^{\nu -{\nu }_2}.$$

The log likelihood functions are:

Case I

$${\ell }_I(\alpha ,\beta ,\varphi ,q)=E_1-{\nu }_{1}log\beta -(1+\frac{1}{\varphi })\sum _{i=1}^{{\nu }_1}log(1+\varphi (\frac{x_i-\alpha }{\beta })$$

$$-(\frac{1}{q}+1)\sum _{i=1}^{{\nu }_1}log[1+q(1+\varphi \frac{x_i-\alpha }{\beta })-\frac{1}{\varphi }]$$

$$+(\nu -{\nu }_1)log[1-[1+q(1+\varphi \frac{T_1-\alpha }{\beta })-\frac{1}{\varphi }]-\frac{1}{q}],$$

Case II

$${\ell }_{II}(\alpha ,\beta ,\varphi ,q)=E_2-alog\beta -(1+\frac{1}{\varphi })\sum _{i=1}^{a}log(1+\varphi (\frac{x_i-\alpha }{\beta })$$

$$-(\frac{1}{q}+1)\sum _{i=1}^{a}log[1+q(1+\varphi \frac{x_i-\alpha }{\beta })-\frac{1}{\varphi }]$$

$$+(\nu -a)log[1-[1+q(1+\varphi \frac{x_a-\alpha }{\beta })-\frac{1}{\varphi }]-\frac{1}{q}],$$

Case III

$${\ell }_{III}(\alpha ,\beta ,\varphi ,q)=E_3-{\nu }_{2}log\beta -(1+\frac{1}{\varphi })\sum _{i=1}^{{\nu }_2}log(1+\varphi (\frac{x_i-\alpha }{\beta })$$

$$-(\frac{1}{q}+1)\sum _{i=1}^{{\nu }_2}log[1+q(1+\varphi \frac{x_i-\alpha }{\beta })-\frac{1}{\varphi }]$$

$$+(\nu -{\nu }_2)log[1-[1+q(1+\varphi \frac{T_2-\alpha }{\beta })-\frac{1}{\varphi }]-\frac{1}{q}],$$

where $E_1$, $E_2$, and $E_3$ are normalizing constants that do not depend on the parameters. We can rewrite the cases I, II, and III as a single formula as follows:

$$\ell (\alpha ,\beta ,\varphi ,q)=E-dlog\beta -(1+\frac{1}{\varphi })\sum _{i=1}^{d}log(1+\varphi (\frac{x_i-\alpha }{\beta })$$

$$-(\frac{1}{q}+1)\sum _{i=1}^{d}log[1+q(1+\varphi \frac{x_i-\alpha }{\beta })-\frac{1}{\varphi }]$$

$$+(\nu -d)log[1-[1+q(1+\varphi \frac{s-\alpha }{\beta })-\frac{1}{\varphi }]-\frac{1}{q}],$$

where $E=E_1,d={\nu }_{1}ands=T_1$ for case I; $E=E_2,d=aands=x_a$ for case II; and $E=E_3,d={\nu }_{2}ands=T_2$ for case III.

The corresponding log likelihood equations are:

$$\frac{\partial \ell }{\partial \alpha }=\sum _{i=1}^d\frac{\varphi +1}{\beta A}-\frac{1+q}{\beta }\sum _{i=1}^d\frac{A^{-(\frac{1}{\varphi }+1)}}{1+q{A}^{-\frac{1}{\varphi }}}+\frac{(\nu -d)A{1}^{-(\frac{1}{\varphi }+1)}{(1+qA{1}^{-\frac{1}{\varphi }})}^{-(\frac{1}{q}+1)}}{\beta [1-{(1+qA{1}^{-\frac{1}{\varphi }})}^{-\frac{1}{q}}]}=0$$

$$\frac{\partial \ell }{\partial \beta }=\frac{-d}{\beta }+\frac{\varphi +1}{{\beta }^2}\sum _{i=1}^d\frac{(x_i-\alpha )}{A}-\frac{1+q}{{\beta }^2}\sum _{i=1}^d\frac{(x_i-\alpha )A^{-(\frac{1}{\varphi }+1)}}{1+q{A}^{-\frac{1}{\varphi }}}$$

$$-\frac{(\nu -d)(s-\alpha )A{1}^{-(\frac{1}{\varphi }+1)}{[1+qA{1}^{-\frac{1}{\varphi }}]}^{-(\frac{1}{q}+1)}}{{\beta }^2{(1-{(1+qA1)}^{-\frac{1}{\varphi }})}^{-\frac{1}{q}})}=0$$

$$\frac{\partial \ell }{\partial \varphi }=\sum _{i=1}^d\frac{log\left(A\right)}{{\varphi }^2}-\sum _{i=1}^d\frac{(\frac{1}{\varphi })(x_i-\alpha )}{\beta A}$$

$$-\sum _0^d\frac{(1+q)}{(1+q{A}^{-\frac{1}{\varphi }})}[\frac{A^{-\frac{1}{\varphi }}logA}{{\varphi }^2}-\frac{(x_i-\alpha )A^{-(\frac{1}{\varphi }+1)}}{\varphi \beta }]$$

$$+\frac{(\nu -d){[1+qA{1}^{-\frac{1}{\varphi }}]}^{-(\frac{1}{q}+1)}}{{[1-{[1+qA1]}^{-\frac{1}{\varphi }}]}^{-\frac{1}{q}}]}[\frac{A{1}^{-\frac{1}{\varphi }}log\left(A1\right)}{{\varphi }^2}-\frac{(s-\alpha )A{1}^{-(\frac{1}{\varphi }+1)}}{\varphi \beta }]=0$$

$$\frac{\partial \ell }{\partial q}=\sum _{i=1}^d\frac{log\left(A\right)}{{\varphi }^2}-\sum _{i=1}^d\frac{(\frac{1}{\varphi }+1)(x_i-\alpha )}{\beta A}-\sum _{i=1}^d\frac{(q+1)}{[1+q{A}^{-\frac{1}{\varphi }}]}[\frac{A^{-\frac{1}{\varphi }}log\left(A\right)}{{\varphi }^2}-\frac{(x_i-\alpha )A^{-(\frac{1}{\varphi }+1)}}{\varphi \beta }]$$

$$+\frac{(\nu -d){(1+qA{1}^{-\frac{1}{\varphi }})}^{-(\frac{1}{q}+1)}}{(1-{(1+qA{1}^{-\frac{1}{\varphi }})}^{-\frac{1}{q}})}[\frac{A{1}^{-\frac{1}{\varphi }}log\left(A1\right)}{{\varphi }^2}-\frac{(s-\alpha )A{1}^{-(\frac{1}{\varphi }+1)}}{\varphi \beta }],$$

(9)

where

$$A=[1+\varphi \frac{x_i-\alpha }{\beta }],$$

$$A1=[1+\frac{\varphi (s-\alpha )}{\beta }].$$

Systems that are specified by Equation (9) yield the maximum likelihood estimations for the parameters of the family of q-generalized extreme value distribution under linear normalization based on the G-Type-II HCS. Since this equation cannot be solved analytically, the Newton–Raphson technique will be applied.

4.2. Simulation Study

A simulation study was used to demonstrate the performance of the estimators produced in the preceding section. We used the family of q-GEVL distribution based on the G-Type-II HCS with $\alpha =5$, $\beta =1$, $\varphi =0.5$ and $q=0.5$ to simulate a small random sample of size $\nu =20$ based on the G-Type-II HCS:

$$0.5,3,3.7590,3.9617,4.1244,4.2720,4.4142,4.5563,4.7021,4.8551,5.0188,5.1974,$$

$$5.3959,5.6216,5.8847,6.2007,6.5956,7.1163,7.8607,9.0806,11.7742$$

We applied these data to the G-Type-II HCS by solving the nonlinear systems that are specified in Equation (9) and using the Newton–Raphson technique, and MATLAB (Version 2021) was used for estimation. Then, we used Equation (6) to calculate entropy. The maximum likelihood estimates (MLEs) of the parameters and entropy of the q-GEVL are yielded by proposed values of $T_1,T_2$ and a in each case as shown in

Table 2. MLEs for $\alpha ,\beta ,\varphi ,q$ and estimated entropy of the q-GEVL with the G-Type-II HCS.

	${\mathit{T}}_1$	${\mathit{T}}_2$	a	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	$\widehat{\mathit{\varphi }}$	$\widehat{\mathit{q}}$	$\widehat{\mathit{H}}$
Case I	6	8	4	5.169	0.858	−0.0415	1.048	2.269
Case II	5	7	13	5.106	0.648	−0.1299	1.1601	1.95
Case III	5	8	20	5.1686	1.0794	0.0282	0.9075	2.494

.

5. Confidence Intervals for the Parameters of the Proposed Procedure

To estimate the approximation confidence intervals for the parameters of the GEVL and q-GEVL distributions based on the G-Type-II HCS, we need the observed information matrices of degrees $3\times 3$ and $4\times 4$. These matrices are denoted by $I\left({\mathrm{\Theta }}_1\right)$ and $I\left({\mathrm{\Theta }}_2\right)$, respectively, where ${\mathrm{\Theta }}_1=(\alpha ,\beta ,\varphi )$ and ${\mathrm{\Theta }}_2=(\alpha ,\beta ,\varphi ,q)$. Then, the $3\times 3$ total observed information matrix associated with the GEVL distribution is given by $I\left({\mathrm{\Theta }}_1\right)$, whereas their parameters are replaced by their MLEs where

$$I\left({\mathrm{\Theta }}_1\right)=\left(\begin{array}{ccc}{I}_{\alpha \alpha }& I_{\alpha \beta }& I_{\alpha \varphi }\\ I_{\beta \alpha }& I_{\beta \beta }& I_{\beta \varphi }\\ I_{\varphi \alpha }& I_{\varphi \beta }& I_{\varphi \varphi }\end{array}\right)$$

with

$$I_{\alpha \alpha }=\frac{{\partial }^2\ell }{\partial {\alpha }^2}=-\frac{3(\varphi +1)}{{\beta }^2}\sum _{i=1}^d{A}^{\frac{-1}{\varphi }}+\sum _{i=1}^d\frac{2\varphi (\varphi +1)}{A^2{\beta }^2}-\frac{(n-d)}{{\beta }^2(1-e^{-A^{frac-1\varphi }})}[\frac{e^{-2A{1}^{\frac{-1}{\varphi }}}A{1}^{-2(1+\frac{1}{\varphi })}}{(1-e^{-A{1}^{\frac{-1}{\varphi }}})}$$

$$-A{1}^{-2(1+\frac{1}{\varphi })}-(\varphi +1)A{1}^{-(2+\frac{1}{\varphi })}]$$

$$I_{\beta \beta }=\frac{{\partial }^2\ell }{\partial {\beta }^2}=\frac{4d}{{\beta }^2}-\sum _{i=1}^d\frac{(x_i-\alpha )}{{\beta }^3}[6A^{-(1+\frac{1}{\varphi })}+\frac{2(\varphi +1)(x_i-\alpha )}{A^2\beta }-\frac{A^{-(1+\frac{1}{\varphi })}(\varphi +1)(x_i-\alpha )}{\beta }$$

$$-\frac{4(\varphi +1)}{A}]-\frac{(\nu -d)(s-\alpha )e^{-A{1}^{-\frac{1}{\varphi }}}A{1}^{-(1+\frac{1}{\varphi })}}{{\beta }^3(1-e^{-A{1}^{-\frac{1}{\varphi }}})}[2+\frac{(s-\alpha )A{1}^{-(1+\frac{1}{\varphi })}}{\beta }+\frac{(s-\alpha )e^{-A{1}^{-\frac{1}{\varphi }}}A{1}^{-(1+\frac{1}{\varphi })}}{\beta (1-e^{-A{1}^{-\frac{1}{\varphi }}})}$$

$$-\frac{(\varphi +1)(s-\alpha )}{\beta }]$$

$$I_{\varphi \varphi }=\frac{{\partial }^2\ell }{\partial {\varphi }^2}=\sum _{i=1}^d[\frac{6A^{\frac{-1}{\varphi }}}{{\varphi }^3}-\frac{4log\left(A\right)}{{\varphi }^3}-\frac{6(x_i-\alpha )A^{-(1+\frac{1}{\varphi })}}{\beta {\varphi }^2}+\frac{4(x_i-\alpha )}{A\beta {\varphi }^2}+\frac{2{(x_i-\alpha )}^2(\varphi +1)}{\beta \varphi A^2}$$

$$+\frac{3(x_i-\alpha )A^{-(2+\frac{1}{\varphi })}(\varphi \alpha -\varphi x_i+{\varphi }^2\alpha +x_i{\varphi }^2+A\beta log\left(A\right))}{{\beta }^2{\varphi }^3}$$

$$-\frac{3log\left(A\right)A^{-(1+\frac{1}{\varphi })}(\varphi \alpha -\varphi x_i+A\beta log\left(A\right))}{\beta {\varphi }^4}]$$

$$-\frac{(\nu -d)e^{-A{1}^{-\frac{1}{\varphi }}}}{{\varphi }^4{\beta }^2(1-e^{-A{1}^{-\frac{1}{\varphi }}})}[A^{-2(1+\frac{1}{\varphi })}(\varphi \alpha -\varphi s+A1\beta {log}^{2}A1)(1+\frac{e^{-\frac{1}{\varphi }}}{(1-e^{-A{1}^{-\frac{1}{\varphi }}})})$$

$$-A^{-(2+\frac{1}{\varphi })}({\varphi }^2({\alpha }^2+\varphi {\alpha }^2{s}^2+\varphi s^2-2\varphi \alpha s-2\alpha \varphi s)+A1\beta (A1\beta {log}^2\left(A1\right)-2\alpha {\varphi }^2+2s{\varphi }^2$$

$$-2A1{\beta }^{2}log\left(A1\right)-2\varphi \alpha log\left(A1\right)-2s\beta log\left(A1\right)\left)\right)]$$

and so on.

The $4\times 4$ total observed information matrix associated with the q-distribution $(\varphi \rightarrowtail 0)$ is given by $I\left({\mathrm{\Theta }}_2\right),$ where in the parameters are replaced by their MLEs where

$$I\left({\mathrm{\Theta }}_2\right)=\left(\begin{array}{cccc}{I}_{\alpha \alpha }& I_{\alpha \beta }& I_{\alpha \varphi }& I_{\alpha q}\\ I_{\beta \alpha }& I_{\beta \beta }& I_{\beta \varphi }& I_{\beta q}\\ I_{\varphi \alpha }& I_{\varphi \beta }& I_{\varphi \varphi }& I_{\varphi q}\\ I_{q\alpha }& I_{q\beta }& I_{q\varphi }& I_{qq}\end{array}\right)$$

with

$$I_{\alpha \alpha }=\frac{{\partial }^2{\ell }^{\ast }}{\partial {\alpha }^2}=\sum _{i=1}^d\frac{3\varphi (\varphi +1)}{A^2{\beta }^2}+\sum _{i=1}^d\frac{2(1+q)A^{-(\frac{1}{\varphi })}}{{\beta }^2(1+q{A}^{-\frac{1}{\varphi }})}[\frac{q{A}^{-(\frac{1}{\varphi })}}{(1+q{A}^{-\frac{1}{\varphi }})}-(1+\varphi )A^{-1}]$$

$$-\frac{(n\nu -d)A{1}^{-(\frac{1}{\varphi })}{(1+q{A}^{-\frac{1}{\varphi }})}^{-(\frac{1}{q})}}{{\beta }^2(1-{(1+q{A}^{-\frac{1}{\varphi }})}^{-\frac{1}{q}})}[\frac{A{1}^{-(\frac{1}{\varphi })}{(1+q{A}^{-\frac{1}{\varphi }})}^{-(\frac{1}{q})}}{(1-{(1+q{A}^{-\frac{1}{\varphi }})}^{-\frac{1}{q}})}$$

$$-(\varphi +1)A{1}^{-1}-(q+1)A{1}^{-(\frac{1}{\varphi }+1)}{(1+qA{1}^{-\frac{1}{\varphi }})}^{-1}]$$

$$I_{\beta \beta }=\frac{{\partial }^2{\ell }^{\ast }}{\partial {\beta }^2}=\frac{4d}{{\beta }^2}+\sum _{i=1}^d\frac{(x_i-\alpha )}{{\beta }^3}[\frac{3\varphi (\varphi +1)(x_i-\alpha )}{A^2\beta }-\frac{6(\varphi +1)}{A{\beta }^3}+\frac{2q{A}^{-2(\frac{1}{\varphi })}(1+q)(x_i-\alpha )}{\beta {(1+q{A}^{-\frac{1}{\varphi }})}^2}$$

$$+\frac{4A^{-(\frac{1}{\varphi }+2)}(1+q)(x_i-\alpha )}{\beta (1+q{A}^{-\frac{1}{\varphi }})}]-\frac{(n-d)(s-\alpha )A{1}^{-(\frac{1}{\varphi }+1)}{(1+q{A}^{-\frac{1}{\varphi }})}^{-(\frac{1}{q}+1)}}{{\beta }^3(1-{(1+qA{1}^{-\frac{1}{\varphi }})}^{-\frac{1}{q}})}$$

$$[2-\frac{(s-\alpha ){(1+q{A}^{-\frac{1}{\varphi }})}^{-(\frac{1}{q}+1)}A{1}^{-(\frac{1}{\varphi }+1)}}{\beta (1-{(1+qA{1}^{-\frac{1}{\varphi }})}^{-\frac{1}{q}})}-\frac{(s-\alpha )(\varphi +1)A{1}^{-1}}{\beta }$$

$$-\frac{(q+1)(s-\alpha )A{1}^{-(\frac{1}{\varphi }+1)}{(1+q{A}^{-\frac{1}{\varphi }})}^{-1}}{\beta }]$$

$$I_{\varphi \varphi }=\frac{{\partial }^2{\ell }^{\ast }}{\partial {\varphi }^2}=2(q+1)\sum _{i=1}^d[\frac{F_{x}1}{A^2{\beta }^2{\varphi }^4(q+A^{\frac{1}{\varphi }})}+\frac{q{(\varphi \alpha -\varphi x_i+A\beta log\left(A\right))}^2}{A^2{\beta }^2{\varphi }^4(q+A^{(}\frac{1}{\varphi }){)}^2}]$$

$$+\frac{(\nu -d)A^{-(2+\frac{1}{\varphi })}}{{\beta }^2{\varphi }^4(1-{(1+qA{1}^{-\frac{1}{\varphi }})}^{-\frac{1}{q}}){(1+qA{1}^{-\frac{1}{\varphi }})}^{-(1+\frac{1}{q})}}$$

$$[F_{s}1-\frac{{(\varphi \alpha -\varphi s+A1\beta log\left(A1\right))}^2}{(1-{(1+qA{1}^{-\frac{1}{\varphi }})}^{-\frac{1}{q}}){(1+qA{1}^{-\frac{1}{\varphi }})}^{-(1+\frac{1}{q})}}-\frac{(q+1)A{1}^{-\frac{1}{\varphi }{(\varphi \alpha -\varphi s+A1\beta log\left(A1\right))}^2}}{{(1+qA{1}^{-\frac{1}{\varphi }})}^{-1}}]$$

where

$$F_{x}1={\varphi }^2{\alpha }^2+{\varphi }^3{\alpha }^2+{\varphi }^2{x}_i^2+{\varphi }^3{x}_i^2-2{\varphi }^2\alpha x_i-2{\varphi }^3\alpha x_i$$

$$+A^2{\beta }^{2}log{\left(A\right)}^2-A\beta {\varphi }^2\alpha +2A\beta {\varphi }^2{x}_i-2A^2{\beta }^2\varphi log\left(A\right)+2\beta \varphi \alpha log\left(A\right)-2A\beta \varphi x_{i}log\left(A\right),$$

$$F_{s}1={\varphi }^2{\alpha }^2+{\varphi }^3{\alpha }^2+{\varphi }^2{s}^2+{\varphi }^3{s}^2-2{\varphi }^2\alpha s{\varphi }^3\alpha x_i$$

$$+A{1}^2{\beta }^{2}log{\left(A1\right)}^2-A1\beta {\varphi }^2\alpha +2A1\beta {\varphi }^{2}s$$

$$-2A{1}^2{\beta }^2\varphi log\left(A1\right)+2\beta \varphi \alpha log\left(A1\right)-2A\beta \varphi slog\left(A\right),$$

and so on.

Under standard regularity conditions, $({\mathrm{\Theta }}_1-\widehat{{\mathrm{\Theta }}_1})$ asymptotically follows the multivariate normal distribution $N_3(o,-I{(\widehat{{\mathrm{\Theta }}_1})}^{-1})$, and the asymptotic distribution of $({\mathrm{\Theta }}_2-\widehat{{\mathrm{\Theta }}_2})$ is $N_4(o,-I{(\widehat{{\mathrm{\Theta }}_2})}^{-1})$. These distributions can be utilized to construct the approximation confidence intervals for the model parameters.

Thus, denoting for example the total observed information matrix evaluated at $\widehat{{\mathrm{\Theta }}_i}$, that is $-I(\widehat{{\mathrm{\Theta }}_i}),i=1,2$ by $-\widehat{I}$, one would have the following approximate $100(1-\alpha )\%$ confidence intervals for the parameters of the q-GEVP distributions:

$$\widehat{\alpha }\pm z_{\frac{\alpha }{2}}\sqrt{{(-{\widehat{I}}^{-1})}_{\alpha \alpha }},\widehat{\beta }\pm z_{\frac{\alpha }{2}}\sqrt{{(-{\widehat{I}}^{-1})}_{\beta \beta }},$$

$$\widehat{\varphi }\pm z_{\frac{\alpha }{2}}\sqrt{{(-{\widehat{I}}^{-1})}_{\varphi \varphi }},\widehat{q}\pm z_{\frac{\alpha }{2}}\sqrt{{(-{\widehat{I}}^{-1})}_{qq}},$$

where $z_{\frac{\alpha }{2}}$ denotes the $100{(1-\frac{\alpha }{2})}^{th}$ percentile of the standard normal distribution.

Real-Life Example

The following genuine dataset, which was provided by Cooray and Ananda [22], shows the stress–rupture life of Kevlar 49/epoxy strands when they are continuously compressed at a 90 percent stress level until they all rupture:

$$0.01,0.08,0.09,0.09,0.10,0.02,0.02,0.03,0.03,0.04,0.05,0.43,0.52,0.54,0.56,0.60,0.60,1.00,0.06,$$

$$1.34,0.10,1.45,1.50,1.51,0.63,0.72,0.99,1.52,1.53,1.54,1.54,1.55,1.58,4.20,4.69,7.89,0.07,0.07,$$

$$0.36,0.38,0.40,0.65,0.67,0.68,0.79,0.80,0.80,0.83,0.72,0.42,0.12,0.13,0.18,0.19,0.20,0.23,0.24,$$

$$1.01,1.02,1.03,0.72,0.73,0.79,0.85,0.90,0.92,0.95,1.05,0.11,0.24,0.29,0.34,0.35,1.10,1.10,1.11,$$

$$1.15,1.18,1.20,1.29,1.31,0.11,0.01,0.02,1.40,1.43,1.33.$$

The basic statistics for the dataset are illustrated in

Table 3. Basic statistics.

Mean	Median	Variance	Standard Deviation	Minimum	Maximum	Range	Quantiles
0.613753	0.68	1.16632	1.079963	0.01	7.89	7.88	(0.155, 0.68, 1.105)

.

Using the K-S, Akaike information criterion (AIC), corrected AIC (AICC) and Bayesian information criterion (BIC) methods for testing the goodness of fit of the data quality (for more information, see [23] and [24]), we note from

Table 4. The goodness-of-fit methods and MEL estimators of the given data.

	Goodness of Fit				Parameters
Distributions	K-S *	AIC	AICC	BIC	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	$\widehat{\mathit{\varphi }}$	$\widehat{\mathit{q}}$
GEVL	0.1162	175.097	175.585	184.961	0.5194	0.4648	0.27648	-
q-GEVL	0.11202	533.345	534.433	543.808	0.5221	0.4650	0.2516	3.1 × 10${}^{-6}$

* Critical value at 0.01 = 0.1894

that the presence of the new parameters (q) has created an inconvenience during the application.

Table 4 refers to the result of these methods (the goodness-of-fit tests) and the MLEs for the given data.

We applied these data to the G-Type-II HCS by solving the nonlinear systems that are specified in Equation (9) and using the Newton–Raphson technique, and MATLAB (Version 2021) was used for estimation. Then, we used Equation (6) to evaluate the entropy. The maximum likelihood estimations (MLEs) of the parameters of GEVL and q-GEVL are yielded by proposed values of $T_1,T_2$, and a in each case as shown in

Table 5. MLEs for $\alpha ,\beta ,\varphi$, and q of the GEVL and q-GEVL with the G-Type-II HCS.

				GEVL			q-GEVL
	${\mathit{T}}_{\mathbf{1}}$	${\mathit{T}}_{\mathbf{2}}$	$\mathit{a}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	$\widehat{\mathit{\varphi }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	$\widehat{\mathit{\varphi }}$	$\widehat{\mathit{q}}$
Case I	0.14	0.2	21	0.0416	0.0108	−0.0516	0.1116	0.0789	−0.157	−0.3034
Case II	0.19	0.7	38	0.5044	0.0101	0.0096	0.2756	0.2868	−0.066	−0.2474
Case III	0.8	3	85	1.1063	0.0101	0.0009	0.8644	0.0424	0.0404	0.0389

:

Also, the confidence intervals for the parameters are determined for the GEVL distribution in

Table 6. The confidence intervals (CIs) for the parameters for the GEVL distribution.

	${\mathit{T}}_1$	${\mathit{T}}_2$	a	$\mathbf{CI}(\widehat{\mathit{\alpha }})$	$\mathbf{CI}(\widehat{\mathit{\beta }})$	$\mathbf{CI}(\widehat{\mathit{\varphi }})$
Case I	0.14	0.2	21	[−0.2147, 0.2979]	[−0.07, 0.0915]	[−1.3825, 1.2793]
Case II	0.19	0.7	38	[−0.1838, 1.1926]	[−0.0095, 0.0297]	[−0.009, 0.001]
Case III	0.8	3	85	[−0.2615, 2.4741]	[−0.0095, 0.0297]	[−0.0001, 0.001]

.

6. Conclusions

In this study, we estimated the parameters of GEVL and q-GEVL distributions based on the generalized type-II hybrid censored sample (G-Type-II HCS). We estimated these parameters using the maximum likelihood method. The obtained results have been used for estimation Shannon entropy for these distributions. Also, the confidence intervals for these parameters were computed. The simulation system served as an illustration for the investigation. Additionally, it was used to model a real-life example, and, after verifying that the data fit with the suggested distributions, an estimation of the parameters was carried out. For an example of GEVL distribution, confidence interval computation was performed.