1. Introduction
The extreme value theory (EVT) is fundamental to various sectors, including environmental sciences, engineering, finance, insurance, and others. In finance, EVT is especially important for studying the changes in price for many areas, like financial assets or market indices. Forecasting for a better understanding of these extremes is important for effective risk management and strategy formulation. The importance of EVT lies in modeling the distribution tail, which is important in evaluating the risk of rare events. This is especially important in many aspects; it is crucial in market crashes or unexpected increases, where typical financial models may not appropriately capture the risk or possible effects. Ref. [
1] is an excellent resource that provides in-depth insights into EVT and its applications to financial data, providing a comprehensive understanding of its methodologies and implications in the financial sector. EVT’s primary goal is to provide a probabilistic description of extreme occurrences within a series of random events. The core concepts of EVT, as originally outlined by [
2], form the cornerstone of the conventional EVT framework.
EVT identifies three forms of extreme value distributions: Frechet, Gumbel, and inverse Weibull. The Frechet distribution has an unbounded, hefty upper tail, indicating that it can handle very big numbers. Moreover, it is especially beneficial for modeling data with extreme maxima that are much higher than the average. Meanwhile, the Gumbel distribution is famous for its lighter upper tail, particularly when compared to the Frechet distribution. The Gumbel distribution is frequently used in situations when the distribution of extremes lacks a heavy tail. Due to the inverse Weibull distribution being known to have a finite upper tail, this feature makes it appropriate for simulating situations for values to have a clear upper bound. Generalized extreme value liner distributions (GEVL) is a unified framework that combines these three types of distributions, offering a comprehensive approach for modeling extremes. This versatility makes EVT a powerful tool in statistical analysis, especially in fields where understanding and predicting extreme events is crucial. Several authors have delved into studies concerning GEVL distributions, including [
3,
4,
5,
6,
7,
8] among others.The GEVL probability density function (PDF) and cumulative distribution function (CDF) are given, respectively, as,
and the support of the distribution is
Hence, indicates several parameter sets based on the value. when , suggesting a three-parameter model. However, when approaches zero, , indicating a two-parameter model. is a location parameter, is a scale parameter, and is the shape parameter that affects the distribution’s tail behavior. The values of , whether they be approaching 0, are positive, or are negative, define sub-models representing the Gumbel, Frechét, and Weibull distributions mentioned earlier.
Marshall and Olkin (1997) proposed a technique for modifying any distribution by adding a new parameter, and it is considered a very helpful tool for academics, providing a methodical approach for refining and adapting current distribution models to better suit a wide range of practical applications. Their technology has been widely applied in a variety of industries where an accurate model of data distribution is required. Ref. [
9] emphasizes their substantial contribution to statistical theory, especially in extending families of distributions. According to [
10], the Marshall–Olkin family of extended distributions is widely applicable and valuable in statistical analysis. They initiated their method by taking a base survival function (SF) and PDF, denoted as
and
, respectively, to construct a new survival function and PDF as
respectively, where,
and
, which is known as Marshall and Olkin extension. Many authors utilized the Marshall–Olkin method as an extension to the parent distribution, such as [
11,
12,
13,
14].
Because of their robust asymptotic features, maximum likelihood estimation (MLE) and Bayesian approaches have become widely known and used for parameter estimation. MLE treats parameters as fixed but unknown values, which is often useful in estimating measures like means or variances. On the other hand, Bayesian approaches estimate parameters using known prior distributions. This fundamental difference allows Bayesian analysis to incorporate prior knowledge or beliefs about the parameters, making it an exceedingly flexible and powerful tool in statistical inference. For those interested in a deeper understanding of these methodologies and their applications, the work by [
15] is an excellent resource.
This paper uses the artificial intelligence algorithm in the estimation process due to the limitation of the traditional numerical method in dealing with complex systems of equations. The genetic algorithms (GAs) fall under the broader category of evolutionary computation. GAs employ mechanisms akin to natural selection. A genetic algorithm (GA) encodes potential solutions to a problem as ’chromosomes,’ which undergo iterative processes of selection, crossing, and mutation. Through these iterations, the population evolves, ideally leading to increasingly optimal solutions. The algorithm’s strength lies in its ability to search through vast and complex solution spaces, making it applicable to a wide variety of problems in optimization and machine learning.
Refs. [
16,
17] offer deep insights into GAs. Ref. [
16] provides a comprehensive overview of GAs, delving into their mechanisms, applications, and theoretical underpinnings. Ref. [
17] focuses on the practical applications of GAs, offering insights into how these algorithms can be implemented and optimized for various real data problems.
In many experiments, observing the failure of all units under study can be a challenge due to many constraints, such as time limitations, budgetary restrictions, or other practical limitations. To address this problem, censoring schemes are commonly utilized. Censoring in statistical analysis is a technique primarily utilized in the fields of reliability testing and survival analysis.Censoring allows researchers to make inferences about the entire data set, even for units with unknown failure times. It is particularly useful in many fields, such as reliability engineering, where testing until the failure of all components may be impractical or impossible, and in medical research, where patients may leave a study before its conclusion or the event of interest (like recovery or relapse) has not occurred before the study ends.
The type II progressive censored scheme is a popular censoring strategy for parameter estimation research. Type II progressive censored data is described as follows: Consider an experiment with
N items. Only a preset number, designated as
n (where
), are observed until they fail. When the first failure donated by
occurs, a certain number of items
are randomly eliminated from the items
. Subsequently, when the second failure
occurs, another set of items
is randomly eliminated from the remaining
items, and this process continues. This process repeats until the
nth failure
occurs at this point, and the experiment is ended. The type-II progressive censored technique enables researchers to collect and evaluate data without censoring failure times across all objects in the experiment. For more comprehensive information on this scheme, refer to the work by [
18]. Also, many others have used type-II progressive censored scheme in their papers, such as [
19,
20,
21,
22,
23].
This paper’s structure is methodically arranged into sections, beginning with a discussion on the Marshall–Olkin extended generalized extreme value under linear normalization distribution, addressing PDF and CDF along with type-II progressive censoring schemes, in
Section 2. It progresses to parameter estimation using type-II progressively censored samples; exploring both point estimates and confidence intervals is discussed in
Section 3. In
Section 4, Bayesian estimation techniques are detailed, followed by
Section 5, which gives the simulation analysis demonstrating theoretical applications. Then, in
Section 6, practical applications are exemplified through real data analysis. Finally,
Section 7 presents the paper’s conclusions.
4. Bayesian Estimation
We discuss the Bayesian estimation for the MO-GEVL distribution parameters under the three cases proposed in this section. We combine both the observed sample data and the prior knowledge about the sample distribution. Combining the prior probabilities and available information on the population distribution provides a more subjective and informed perspective on unknown parameters. We next explore parameter estimation using Bayesian methods, focusing on two loss functions: square loss and LINEX loss. Our analysis covers two different scenarios: one with informative priors, which rely on specific prior knowledge, and another with non-informative priors, which assume minimal prior information. By integrating these loss functions and types of prior information, we thoroughly examine parameter estimation.
Informative priors. Assume that the unknown parameters in every scenario investigated are independent of one another. The approach of selecting parameter priors for informative cases depends on the parameter validation region, as introduced by [
28,
29]. So, we considered the priors PDF of the parameters
follow an exponential distribution with the hyper-parameters
, while the random variable
P follows a beta distribution with parameters
. Then, the prior PDF of the parameters are
and
where
is the beta function. The hyperparameters (
,
,
if
) are estimated by the same method given in [
30]. Then, the joint PDF of hyperparameters for MO-GEVL could be written as
where
if
and
if
. It is easy to obtain the joint PDF of hyperparameters for MO-Gambul by putting
into Equation (
50) equal to 0.
Non-informative prior. For this scenario all parameters prior PDFs are assumed to be equal to 1.
According to the informative prior, the posterior PDFs
of MO-GEVL for
and
, respectively, are given by,
where
and
. Then, the informative Bayesian estimation under the SELF and LINEX loss functions (see, [
31]) are given respectively by
where
is a shape parameter, in which its negative value provides more weight to underestimation compared with the overestimation, while for it’s small or large values, the LINEX loss [
] function is almost symmetric (see [
32]).
Meanwhile, for non-informative priors, the posterior PDFs
are given by
where
and
.
The Bayesianestimation of parameters for MO-GEVL using the SELF and LINEX for the informative prior function. by
The systems of equations are presented as Equations (
53), (
54), (
57) and (
58) do not have analytical solutions, so it is necessary to use numerical methods for their solution. Among these methods, Lindley’s approximation technique stands out as a popular choice for Bayesian estimation. We will explore this method in detail in the subsequent subsection.
Lindley’s Approximation Method
Lindley’s approximation is a simple technique for approximating the ratio of integrals in systems of equations (Equations (
53), (
54), (
57) and (
58)). Other methods, such as the Markov Chain Monte Carlo method and Gibbs sampler, are also used. the easily analytical approach of Lindley’s approximation makes it a popular choice for solving problems among the other analytical approach. The method typically involves expanding the log-likelihood
and the log-prior
around a suitable point, such as the mode of the posterior distribution, and then approximating the integral using this expansion. This approach is particularly useful when the exact computation of the integral is challenging. Lindley’s approximation provides a way to estimate the expected value of
under the posterior distribution, which is crucial in Bayesian analysis for making inferences about the parameters based on the observed data. Below is the Lindley approximation of
In this context,
is a vector representing a set of parameters. The function
is defined as a function of these parameters.
where
,
,
,
, and
is equal to the variance covariance matrix. All partial derivatives are evaluated at the maximum likelihood estimation of parameters. For more details, see [
33].
5. Simulation
A Monte Carlo simulation is performed in this section to compare the performance of various estimator parameters discussed in the preceding sections. This simulation offers a comprehensive evaluation of different estimation methods on their effectiveness in parameter estimation. The simulation involves generating 1000 samples for two different sample sizes 100 and 200 for a variable
X. In this analysis, the maximum likelihood estimator (MLE) for the parameters
is evaluated compared with the Bayesian estimators within the framework of two distinct loss functions: the squared error loss and the linear exponential (LINEX) loss function. The reason for this comparison is to provide insight into their relative performance and suitability for statistical analysis. Moreover, we explore Bayesian estimation using Lindley’s method for the informative and non-informative priors. For the informative priors, the hyperparameters are determined following the approach initiated by [
30] and subsequently employed by [
34]. Additionally, an approximate
confidence interval for the parameters is calculated, offering a comprehensive statistical analysis based on the established methodologies. The LINEX loss function is evaluated for three different values of
, as shown in
Table 1 and
Table 2.
Table 3 introduces the lower bound (LB), upper bound (UB), and length of the confidence interval (LC). All calculations are performed using the R programming language. The estimators are evaluated under three different scenarios of random removals, with censoring schemes involving a 10% elimination from the sample size. Moreover, fixed random removal is executed either at the beginning or the end of the sample. For details on the algorithm for generating progressive censoring of type-II, see [
18].
From
Table 1, the informative Bayesian estimation gives a small bias and MSE compared with the non-informative Bayesian estimation. Bayesian estimation utilizing the LINEX loss function tends to give better outcomes in terms of bias and mean squared error (MSE) when compared with the squared error loss function. This advantage is particularly noted when the parameter
holds a negative value, emphasizing a greater penalty for underestimation than overestimation. The LINEX loss function approaches symmetry for extremely small or large values of
, as detailed in the study by [
32]. This finding underscores the effectiveness of the LINEX loss function in various estimation scenarios.
Furthermore, in the first situation, eliminating at the beginning provides a more accurate estimate of bias and MSE compared to eliminating at the end. In terms of the sensitivity of GA for the sample size, we find that the behavior is consistent for both small and high sample sizes. Furthermore, MLE provides better bias than Bayesian estimate. In
Table 2, it is apparent that the
confidence interval demonstrates that as sample size grows, we obtain a better estimate in terms of shorter intervals encompassing the real value. The Lindley approximation for the second scenario fails to effectively estimate parameters for a sample size of 100. However, for
, the estimation improves in terms of bias and MSE.
6. Real Data Example
In this section, we apply the discussed algorithms using actual real data, with a focus on extreme weather events. Extreme weather phenomena, including heatwaves, droughts, and wildfires, are often associated with high temperatures. It is important to recognize that such excessively high temperatures can lead to significant health risks and heat-related illnesses.
Given the recent global crisis triggered by an unexpected rise in temperature levels, understanding historical temperature extremes for each country has become vital. This information is crucial for mitigating the impact of catastrophic events caused by these temperature increases. It also helps in assessing the frequency, severity, and consequences of such events, thereby aiding in the development of more effective strategies for preparedness and response.
For practical demonstration, we analyze real data sets representing the temperatures in Egypt from 1999 to 2023. These data was sourced from
https://www.ogimet.com/home.phtml.en, accessed on 12 May 2024.
Table 4 provides basic statistics for this data set, including mean, variance, median, and other relevant measures. This statistical analysis offers insights into the temperature trends in Egypt over the specified period, contributing to a broader understanding of the impacts of rising temperatures on a national scale.
In
Table 5, these data sets are fitted to both GEVL and MO-GEVL by using computationally intensive ways of comparing model fits like the Kolmogorov–Smirnov(K–S) goodness of fit test at the level of significance of
,
, Akaike information criterion (AIC) [
35], and Bayesian information criterion (BIC) [
36].
Moreover,
Figure 5 shows the empirical data and fitted cumulative distribution function plot for GEVL and MO-GEVL.
The results in
Table 5 for that data set show that both (GEVL–MO-GEVL) distributions give a good fit for these data, while MO-GEVL gives a better fit than GEVL in terms of lower indicators (KS,
, AIC, and BIC). Moreover, the
Figure 5 indicates that the MO-GEVL distribution provides a better reasonable fit than GEVL.
In
Table 6, we introduce the maximum likelihood estimate and Bayes estimates of MO-GEVL parameters considered in the above sections for the three cases of elimination, considered using a GA.
7. Conclusions
This study focuses on parameter estimation for variables following the Marshall–Olkin extended generalized extreme value distribution (MO-GEVL) within the framework of type-II progressive censoring. This methodological approach aims to enhance the accuracy of statistical inferences drawn from censored data sets, providing significant insights into the distribution’s characteristics and behavior under specified conditions.
We utilize artificial intelligence, specifically genetic algorithms (GAs), for estimating maximum likelihood estimation (MLE) parameters. Additionally, a Bayesian estimation of parameters is examined under two types of loss, functions-symmetric and asymmetric, considering both informative and non-informative priors.
We also use progressive censoring type-II under three different scenarios of elimination: fixed, discrete uniform, and binomial, with a elimination from the sample size. Within the fixed random removal censoring scheme, we consider two specific cases: elimination at the beginning and at the end of the sample.
A Monte Carlo simulation is conducted to compare the performance of various estimators for the GEVL distribution parameters using Lindley’s method. The simulation results show that informative Bayesian estimation yields results very close to non-informative Bayesian estimation in terms of bias and mean squared error (MSE), but it is still better than the non-informative case. Furthermore, Bayesian estimation under the LINEX loss function shows comparable results with the SELF-loss function regarding both Bias and MSE. It is noted that a negative value of
in the LINEX loss function places more emphasis on underestimation compared with overestimation. For extreme (small or large) values of
, the LINEX loss function tends to be almost symmetric, as illustrated in
Table 1.
For the first case of elimination, removing samples at the beginning provides better estimates in terms of bias and MSE than elimination at the end. In terms of the sensitivity of estimates to sample size, the GA’s performance is consistent across both smaller and larger sample sizes. Moreover, MLE tends to offer better Bias compared with Bayesian estimation. As illustrated in
Table 3, we noticed that as sample sizes increase, we receive more accurate estimates of the confidence interval in terms of a smaller interval length that contains both the true and estimated values of the parameters.