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 identical units undergoing a lifetime test. Let be the results of lifetimes from distributions with CDF and PDF . Let be an integer an integer and suppose are time points. We have three cases as follows:
If the failure occurs before the time point , the experiment will be terminated at this time.
If the failure occurs between the time points and , then the experiment will be terminated at the time of the failure, .
If the failure occurs after the time point , the experiment will be terminated at time point .
This type of censoring, while aiming for a minimum number of failures,
a, guarantees that the experiment will be completed by time
. Therefore,
, 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 , 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:
where ,
- Case II:
where ,
- Case III:
where .
Note that and are the number of observed failures up to time points and , respectively. Then, for the G-Type-II HCS, the likelihood functions for the three different cases described above are as follows:
Case III
where
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)
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:
and the probability density function (PDF)
can be given by:
where
is a location parameter,
is a positive scale parameter,
is the shape parameter, and the values of
x are defined by:
The distribution in Equation (
1) is known as a generalized extreme value (GEV) distribution under linear normalization. We denote it by
The Gumbel probability distribution in Equations (1) and (2) as
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
.
Provost et al. [
8] proposed the q-GEVL distribution and q-Gumbel distribution (obtained by letting
in the q-GEVL model), and the corresponding distributions are provided by:
and
where the values of
x can be determined by:
Figure 2 refers to the cumulative distribution and density function of q-GEVL distribution for
.
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)
. It is written as:
The expectancy of a random variable is a statistic that has recently gained the interest of investigators.
The Shannon entropy of GEVL family is well known as:
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
family as follows:
where
is the Euler–Macheronic constant.
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
and
. These matrices are denoted by
and
, respectively, where
and
. Then, the
total observed information matrix associated with the GEVL distribution is given by
, whereas their parameters are replaced by their MLEs where
with
and so on.
The
total observed information matrix associated with the q-distribution
is given by
where in the parameters are replaced by their MLEs where
with
where
and so on.
Under standard regularity conditions, asymptotically follows the multivariate normal distribution , and the asymptotic distribution of is . 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
, that is
by
, one would have the following approximate
confidence intervals for the parameters of the q-GEVP distributions:
where
denotes the
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:
The basic statistics for the dataset are illustrated in
Table 3.
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 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
, and
a in each case as shown in
Table 5:
Also, the confidence intervals for the parameters are determined for the GEVL distribution in
Table 6.