Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 9089856, 8 pages

https://doi.org/10.1155/2019/9089856

## Bayesian Analysis of Three-Parameter Frechet Distribution with Medical Applications

^{1}Department of Statistics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan^{2}Department of Statistics, Allama Iqbal Open University, Islamabad, Pakistan^{3}Department of Statistics, Islamia College, Peshawar, Khyber Pakhtunkhwa, Pakistan^{4}Department of Statistics, University of Peshawar, Khyber Pakhtunkhwa, Pakistan^{5}Department of Statistics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan^{6}Research Centre for Modeling and Simulation, National University of Sciences and Technology, Islamabad, Pakistan^{7}Faculty of Basic Sciences and Humanities, University of Engineering and Technology, Taxila, Pakistan

Correspondence should be addressed to Amjad Ali; kp.ude.pci@dajma

Received 28 October 2018; Revised 24 December 2018; Accepted 6 February 2019; Published 12 March 2019

Academic Editor: Martti Juhola

Copyright © 2019 Kamran Abbas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The medical data are often filed for each patient in clinical studies in order to inform decision-making. Usually, medical data are generally skewed to the right, and skewed distributions can be the appropriate candidates in making inferences using Bayesian framework. Furthermore, the Bayesian estimators of skewed distribution can be used to tackle the problem of decision-making in medicine and health management under uncertainty. For medical diagnosis, physician can use the Bayesian estimators to quantify the effects of the evidence in increasing the probability that the patient has the particular disease considering the prior information. The present study focuses the development of Bayesian estimators for three-parameter Frechet distribution using noninformative prior and gamma prior under LINEX (linear exponential) and general entropy (GE) loss functions. Since the Bayesian estimators cannot be expressed in closed forms, approximate Bayesian estimates are discussed via Lindley’s approximation. These results are compared with their maximum likelihood counterpart using Monte Carlo simulations. Our results indicate that Bayesian estimators under general entropy loss function with noninformative prior (BGENP) provide the smallest mean square error for all sample sizes and different values of parameters. Furthermore, a data set about the survival times of a group of patients suffering from head and neck cancer is analyzed for illustration purposes.

#### 1. Introduction

Frechet distribution (FD) was introduced by Maurice Frechet (1878–1973) for largest extremes [1]. It had been derived with nonnegative initial variates. The FD deals with extreme events and also recognized as extreme value Type-II distribution. The cumulative distribution function of three-parameter FD iswhere is the shape, is the scale, and is the location parameter. If then it becomes two-parameter FD. The corresponding probability density function is

A number of authors have studied the estimation of its parameters, namely, Gumbel [2], Mann [3], Singh [4], and Hooda et al. [5]. Moreover, Afify [6] estimated the parameters of FD using principal components and least median of squares. Mubarak [7, 8] derived the best linear unbiased estimators and the best linear invariant estimators of location and scale parameters of FD under progressive Type-II censoring, respectively. Abbas and Tang [9] discussed classical as well as the Bayesian estimators of FD assuming that the shape parameter was known. Abbas and Tang [10] developed maximum likelihood and least squares estimators for FD with Type-II censored samples. Furthermore, Abbas and Tang [11, 12] derived the reference and matching priors for the Frechet stress-strength model and developed Bayesian estimators for FD under reference prior, respectively. Nasir and Aslam [13] obtained Bayes estimators of FD and their risks by using four loss functions under Gumbel Type-II prior and Levy prior. Yet, the Bayesian analysis of three-parameter FD is not conducted.

The aim of this paper is to develop Bayesian estimators for three-parameter FD using noninformative prior and gamma prior under two loss functions for the case of complete samples. Including this introduction section, the rest of the paper unfolds as follows: in Section 2, maximum likelihood estimators (MLEs) for the parameters are obtained. In Section 3, Bayesian estimators based on different loss functions by taking noninformative and gamma priors are derived. The proposed estimators are compared in terms of their mean squared error (MSE) in Section 4. Section 5 illustrates the applications of proposed estimators using head and neck cancer data set. Finally, conclusions and recommendations are presented in Section 6.

#### 2. Maximum Likelihood Estimation

Let be random samples of size from a three-parameter FD, then the likelihood function of (2) is

The corresponding log-likelihood function isfrom equation (4), we have

Clearly, the above equations cannot be written in a closed form. Therefore, BFGS quasi-Newton optimization method (Broyden Fletcher GoldFarb Shanno, Battiti, and Masulli [14]) is applied to compute the MLEs.

#### 3. Bayesian Estimation

In Bayesian estimation, we consider the two types of loss functions. The first is LINEX loss function, introduced by Varian [15]. This loss function was widely used by several authors, for example, Rojo [16], Basu and Ebrahimi [17], Pandey [18], Soliman [19], and Soliman et al. [20]. The second is general entropy loss function (GELF), defined by Calabria and Pulcini [21]. For Bayesian analysis, we need prior distribution. When prior information about the parameters is unavailable, then the noninformative prior can be considered for the Bayesian study. So, we supposed the noninformative form of priors for the all unknown parameters , , and of three-parameter FD as

If someone has a few information about parameters, then informative priors may be used for Bayesian analysis. It is noted that FD converts an inverse exponential distribution if its shape parameter equal to 1 and takes the form of the inverse Rayleigh distribution for shape parameter equal to 2, and when shape equal to 0.5, it approximates the inverse gamma distribution. So, we consider gamma prior for the scale parameter by assuming that shape parameter is known and independent priors for the shape and location parameters. Thus the proposed prior is

The joint prior distribution of parameters , , and is

The joint posterior density can be written as

Posterior distribution (9) takes a ratio form that cannot be reduced to a closed form. Therefore, we use Lindley’s approximation [22] to get the Bayesian estimate, which can be written as

The detail of equation (10) is given in Appendix. Therefore, the approximate Bayesian estimators of parameters , , and by using noninformative prior under LINEX loss function are

Bayesian estimators of parameters , , and with gamma prior under LINEX loss function are

Similarly, Bayesian estimators of , , and using noninformative prior under GELF are

Bayesian estimators of parameters , , and with gamma prior under GELF arewhere , , and are the ML estimates of parameters , , and , respectively. Further, the observed Fisher information matrix is obtained by taking the second and mixed partial derivatives of equation (4) with respect to parameters , , and , respectively, provided in Appendix.

#### 4. Simulation Study

To demonstrate the performance of the proposed Bayesian estimators with their ML counterpart in terms of biases and MSE (within parenthesis), different sample sizes and different values of parameters are considered using Monte Carlo simulation. Monte Carlo simulation is conducted as follows:(1)Take the initial values of , , and , respectively. Samples are generated from the FD using inverse transformation technique, i.e., , where *U* is uniformly distributed random variable over the interval of [0, 1] and considering .(2)Calculate the ML and Bayesian estimators of by , where is the function of , , and using informative and noninformative priors and is the number of iterations.(3)The process is replicated 3000 times for each sample size and averages of these estimates and the corresponding MSEs (within parenthesis) were calculated for each method.

The results are listed in Tables 1–4 for comparison purposes. Table 1 contains simulation results for the case where , , and , and Table 2 presents the simulation results when , , and . Moreover, Tables 3 and 4 comprise the results for the case where , , and and , , and , respectively. From the results of the simulation study, conclusions are drawn regarding the behavior of the estimators, which are summarized below:(i)MSE decreases for both ML and Bayesian method when the sample sizes increases.(ii)In terms of MSE, the BGENP estimator provides the smallest MSE for all samples sizes and different values of parameters.(iii)Apparently, Bayesian and MLEs become better when the sample size increases. However, similar performance can be observed for large sample sizes.(iv)Based on simulation study and real data analysis, we suggest that the BGENP estimators in each scenario execute considerably, because the MSE is significantly smaller.