1. Introduction
Survival analysis, a branch of statistics pertaining to death or failure, encompasses various types of statistical methods to draw conclusions. These methods include (1) nonparametric statistics, such as the Kaplan–Meier estimator and the log-rank test; (2) semi-parametric statistics, exemplified by the Cox proportional hazards model; and (3) parametric statistics, which focus on simulating survival time probabilities. Analysts may deduce that the survival function has a parametric distribution. For instance, if the survival time adheres to an exponential distribution, the hazard rate will be constant. Conversely, if the survival time conforms to a log-normal distribution, the hazard rate varies with time. Consequently, estimation of the survival function, calculation of the confidence interval, and assessment of the relative risk ensue. The utilization of a parametric survival function proves highly effective when appropriate distributions and parameter values are selected. The parametric survival distribution serves as a comprehensive representation of various types of survival data.
Hundreds of univariate continuous distributions exist. Mixture models play a crucial role in numerous applications, including survival analysis, such as Farewell [
1], Hunsberger et al. [
2], and Joudaki et al. [
3]. These models involve the combination of two or more statistical distributions to create a new distribution, thereby addressing various challenges encountered in the field. Recognizing the evident necessity for mixture distributions, extensive efforts have been devoted to integrating multiple well-established distributions and utilizing them to tackle relevant issues. In the context of complete samples, Niyomdecha and Srisuradetchai [
4] introduce a novel continuous three-parameter survival distribution referred to as the Complementary Gamma Zero-Truncated Poisson distribution. The traits of the maximum value in a series of independently identical gamma-distributed random variables are combined with those of zero-truncated Poisson random variables in this distribution. Abdullahi and Phaphan [
5] present a mixture of Nakagami distribution, accompanied by statistical properties and a comparative analysis of the efficacy of estimators utilizing the quasi-Newton method and simulated annealing. Nanuwong et al. [
6] proposed the mixture Pareto distribution by combining a Pareto distribution and a length-biased Pareto distribution. This distribution was formulated based on the concept of a weighted two-component distribution. Further investigation pertaining to the mixture models can be found in the references [
7,
8].
The Fréchet distribution, alternatively referred to as the inverse Weibull distribution, holds extensive application in the field of survival modeling. Fréchet [
9] initially introduced the Fréchet distribution, which subsequently underwent further exploration by Fisher and Tippett [
10] as well as Gumbel [
11]. Furthermore, Abbas and Yincai [
12] conducted a comparative analysis of the scale parameter estimation for the Fréchet distribution, employing maximum likelihood, probability-weighted moments, and Bayes estimations. Nasir and Aslam [
13] utilized a Bayesian technique to estimate the parameter of the Fréchet distribution. Reyad et al. [
14] established QE-Bayes and E-Bayes estimates for the scale parameters associated with the Fréchet distribution. Recent developments have introduced various extensions to the Fréchet distribution. Notably, Mead et al. [
15] proposed the beta exponential Fréchet distribution.
Consequently, this article paid special attention to developing a new survival distribution by employing the notion of a mixture distribution, which is based on the Fréchet distribution, to obtain a new alternative distribution with the value of the time-varying hazard rate and investigating the statistical properties of the new distribution, such as the probability density function, cumulative distribution function, ordinary moment, skewness, kurtosis, moment-generating function, mean, variance, mode, survival function, hazard function, asymptotic behavior, comparison of the estimators with several methods, and samples of applying to real data, which will be extremely useful in survival analysis.
2. The Fréchet Distribution
The Fréchet distribution, being a specific case of the generalized extreme value distribution [
16], finds extensive application in the field of hydrology [
17]. This distribution is commonly employed to model extreme events, including daily rainfall [
18] and river discharges [
19]. Moreover, the Fréchet distribution holds considerable significance in survival analysis utilizing experimental data from clinical research. Given its status as the inverse Weibull distribution, the Fréchet distribution exhibits properties akin to the Weibull distribution, such as time-varying hazard rates. As a result, the Fréchet distribution has been a subject of widespread discussion in the field of survival analysis.
Afify et al. [
20] provides the probability density function (PDF), cumulative distribution function (CDF), and mean of the Fréchet distribution. The PDF of the Fréchet distribution described by
Given that
represents a scale parameter and
represents a shape parameter, the CDF associated with these parameters can be expressed as follows:
Furthermore, the mean of the distribution can be determined as follows:
where
represents a gamma function:
.
5. Illustrative Example
The proposed distribution is applied to an actual dataset in this part. The dataset used in this analysis was collected from a clinical trial conducted by Freireich et al. [
22], where patients received a placebo to evaluate the efficacy of 6-mercaptopurine (6-MP) in maintaining remission. Following the completion of the trial after a year, the following remission times were recorded and are expressed in weeks: 1, 1, 2, 2, 3, 4, 4, 5, 5, 8, 8, 8, 8, 11, 11, 12, 12, 15, 17, 22, 23.
Based on the results shown in
Figure 3, the remission times of patients who received a placebo had a right-skewed distribution. In order to compare the goodness of fit, four right-skewed distributions—the Fréchet distribution, the length-biased Fréchet distribution, the mixture of Nakagami distribution [
5], and the proposed mixture Fréchet distribution—are chosen.
While the parameters of the other candidate distributions are determined using maximum likelihood estimation utilizing simulated annealing, the parameters of the novel mixture Fréchet (NMF) distribution are estimated using the EM algorithm. The best model is the one that provides the smallest Akaike information criterion (AIC) value, which is used as the evaluation criterion.
Based on the findings presented in
Table 9, it is evident that the NMF distribution yields the lowest value of the AIC. This indicates that the NMF distribution outperforms the other potential distributions when using an AIC statistic as a measure of goodness-of-fit for this example data. Therefore, as indicated by Equations (
20) and (
26), the mean and standard deviation of the remission times observed in a group of 21 patients who received a placebo are 3.091147 weeks and 2.792774 weeks, respectively.
6. Conclusions and Discussion
This article presents the introduction of a novel survival distribution known as the novel mixture Fréchet (NMF) distribution. This distribution is characterized by its right-skewed distribution. The study explores various statistical properties of this newly proposed distribution and estimates its two parameters using both EM algorithms and simulated annealing. To assess the performance of both methods, a simulation study is conducted, involving twenty-four different combination scenarios. The illustrative examples of the proposed distribution are implemented using patient remission times data. The results reveal that the EM estimators exhibit greater efficiency compared to the simulated annealing estimators. Additionally, the NMF distribution demonstrates a better fit when compared to other candidate distributions, as indicated by the Akaike information criterion (AIC). Consequently, this article presents a novel right-skewed distribution that holds potential application in diverse areas, including extreme value analysis, survival analysis, and reliability analysis.
In future research, it is advisable to investigate interval estimation using different methods, such as [
23,
24], to further enhance the accuracy of the estimations.