Bayesian Estimation of Gumbel Type-II Distribution under Type-II Censoring with Medical Applications

Department of Statistics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan Research Centre for Modeling and Simulation, National University of Sciences and Technology, H-12 Campus, Islamabad, Pakistan Department of Statistics, Allama Iqbal Open University, Islamabad, Pakistan Department of Statistics, Islamia College University, Peshawar, Khyber Pakhtunkhwa, Pakistan Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan Department of Statistics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan


Introduction
In medical research, data supporting the time until the occurrence of a particular event, such as the death of a patient, are frequently encountered. Such data are referred to as survival time data which has generally right-skewed distribution, and Gumbel type-II distribution can be used for this purpose. It was introduced by the German mathematician Gumbel in [1] and is useful to model "extreme values" such as floods, earthquakes, and natural disasters and also used in life expectancy tables, hydrology, and rainfall. e probability density function (PDF) of Gumbel type-II distribution is (1) where "α" is the shape and "β" is the scale parameter of the distribution. e corresponding cumulative distribution function (CDF) is studies, experiments are generally windup before failure times of all items are observed. erefore, adequate information and results on failure times of all objects cannot be obtained. During experimentation, these situations occur due to loss or removal of objects before they fail. erefore, generally, such experiments are preplanned and purposeful to save time and cost of these testing. Data obtained from such experiments are called censored. e type-I and type-II censoring are two well-known censoring schemes. In type-II censoring scheme, the number of failure units are fixed in advanced. For example, the investigator may decide to terminate the study after four of the six rats have developed tumors. ere is an enormous literature accessible on estimation of parameters of distributions using type-II censoring, for example, Abbas and Tang [2] considered ML and least square estimators of Frechet distribution using type-II censored samples. Okasha [3] estimated the unknown parameters, reliability, and hazard functions of Lomax distribution under type-II censoring using Bayesian and E-Bayesian estimation. Abu-Zinadah [4] studied on exponentiated Gompertz distribution based on type-II and complete censored data. El-Sagheer [5] studied the generalized pareto distribution under the different censoring schemes.
Recently, many authors have worked on Gumbel type-II distribution and Bayesian estimation using different loss functions. Abbas et al. [6] worked on Gumbel type-II distribution and obtained the Bayes estimators under different loss functions. Feroze and Aslam [7] obtained Bayes estimators of two components of Gumbel type-II distribution. Malinowska and Szynal [8] also derived Bayes estimators for Gumbel type-II distribution on kth lower record values. Sultana et al. [9] worked on a threecomponent mixture of Gumbel type-II distribution using Bayesian estimation under different priors such as informative and noninformative. Moreover, Metiri et al. [10] worked on the properties of the Lindley distribution.
e Bayes estimates were derived under LINEX loss function using informative and noninformative priors (Reyad and Ahmed [11]). Preda et al. [12] developed Bayes estimators of modified Weibull distribution under squared error loss function (SELF) and LINEX loss function.
However, Bayesian estimation of Gumbel type-II distribution based on type-II censoring is not frequently discussed; therefore, we are interested in estimating the unknown parameters of Gumbel type-II distribution under type-II censored data. Including this introduction section, the rest of the paper is arranged 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. e proposed estimators are compared in terms of their mean squared error (MSE) in Section 4. Section 5 illustrates the applications of proposed estimators with two examples, namely, data set of remission times for bladder cancer and survival times of inoperable adenocarcinoma of the lung. Finally, conclusions and recommendations are presented in Section 6.

Maximum Likelihood Estimation
Suppose that X 1 < X 2 < , . . ., < X r is a type-II censored sample of size "r" obtained from a life test on "n" items whose life times have the Gumbel type-II distribution with parameters "α" and "β." e likelihood function of "r" failures and (n − r) censored values may be written as It is more convenient to work with log-likelihood. e log-likelihood function is To get the ML estimator of α and β, differentiate equation (5) with respect to α and β and the resulting equations are Equations (6) and (7) cannot be written in closed form. erefore, here, we the use the Laplace approximation to get the point estimates of α and β.

Bayesian Estimation
In Bayesian estimation, we consider different loss functions such as squared error loss function (SELF) proposed by Legendre [13] and Gauss [14], LINEX (Varian [15]), and general entropy loss function (GELF) introduced by Calabria and Pulcini [16]. As both parameters are unknown, independent noninformative form of priors can be used. Supposed that α and β have independent Gamma (a, b) and Gamma (c, d) priors, respectively, for a, b, c, d > 0, i.e., e joint prior distribution of parameters is 2 Computational and Mathematical Methods in Medicine where K is the normalizing constant that makes Φ′(α, β | x) a proper PDF. us, erefore, the joint posterior density under any loss function is Posterior distribution (12) takes a ratio form that cannot be reduced to a closed form. erefore, we use Lindley approximation [17] to get the Bayesian estimates, which can be written as where (13) is provided in Appendix. e approximate Bayesian estimators of "α" and "β" based on SELF are Computational and Mathematical Methods in Medicine Similarly, the Bayesian estimators of "α" and "β" under LINEX loss function are e Bayesian estimators of "α" and "β" under GELF are Computational and Mathematical Methods in Medicine 5 where α and β are the ML estimators of α and β which can be obtained from equations (6) and (7), respectively.

Data Analysis
In this section, we consider two examples for illustration purposes.

Example 1.
e real data about remission times (in months) of a random sample of 128 bladder cancer patients presented in Table 5 were reported by Lee and Wang [20]. A total of 128 patients with different prespecified percentages of events, i.e., 40%, 50%, 60%, and 80%, represented patients whose treatment was terminated and rest of the percentages are censored. Clearly, Figure 1 confirms that the histogram is slightly skewed to the right and is leptokurtic. Moreover, ML and Bayesian estimates can also be envisioned in Figure 1, in which the x-axis represents the remission times (in months) of bladder cancer patients, while the Gumbel type-II density function is taken on the yaxis. erefore, it would be appropriate to select positively skewed distributions for describing the behavior of remission times of bladder cancer patients. Amongst the skewed distributions, Gumbel type-II distribution is fitted and the parameter estimates using ML and Bayesian methods are presented in Table 6 for comparison purposes.

Computational and Mathematical Methods in Medicine
It is concluded that the proposed estimators of Gumbel type-II distribution fit the data well. erefore, it is recommended that the Bayesian estimators can be more beneficial to address the uncertainty in medical-related censored data.

Example 2.
e survival times, in weeks, of 61 patients with unoperable lung cancer treated with cyclophosphamide considered in Lagakos and Williams ( [18]) and in Lee and Wolfe ([19]) are presented in Table 7.
ere are 33 uncensored observations and 28 censored observations, representing the patients whose treatment was terminated because of a devolving condition. e point estimates of α and β obtained by all the methods are summarized in Table 8. Figure 2 shows the results of different estimation methods and depicts that Gumbel type-II distribution fits the data better, in which x-axis comprises the survival times in weeks of 61 patients with inoperable adenocarcinoma of     the lung as the Gumbel type-II density function is taken on the y-axis.

Conclusion and Recommendations
In medical decision-making, Bayesian tools incorporate the state of uncertainty and provide a rational framework for studying such problems. Usually, medical data are generally skewed to the right, and positively skewed distributions can be most suitable for describing unimodal medical data. In this study, an attempt has been made to develop the Bayesian estimators for Gumbel type-II distribution based on type-II censored data using squared error loss, GELF, and LINEX loss functions via Lindley's approximation. It is concluded that ML and Bayesian estimators become closer by increasing the sample sizes and prespecified percentages of failures. Based on the outcomes of this research study, we may suggest that this study can be further extended by using other skewed distributions considering the Bayesian framework with other loss functions using medical data.