Bayesian Estimation of a Geometric Life Testing Model under Different Loss Functions Using a Doubly Type-1 Censoring Scheme

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Introduction
It is customary in clinical or biological investigations to use censoring schemes while assessing the worth of new procedures. Among others, the doubly censoring scheme is widely used in clinical and other lifetime investigations. Te worth of censoring schemes is not hidden in literature, and many authors have focused their research study based on diferent censoring schemes. Bravo and De Fuentes [1] derived maximum likelihood estimates by considering the doubly type-II-censored exponential scenario. Fauzy et al. [2] constructed intervals to estimate the parameters of an exponential distribution under the doubly type-II censoring scheme. Krishna and Malik [3] considered reliability estimation for the doubly type-II censored Maxwell distribution. Algarni et al. [4] considered type-I censoring while estimating the parameters of the Chen distribution. Feroze [5] discussed the application of doubly censored data from a 2component mixture of inverse Weibull distributions . Ghosh and Nadarajah [6] described the Bayesian inference of Kumaraswamy distributions based on censored samples. Long [7] estimated the parameters of the Rayleigh distribution based on double type-I hybrid censored data.
While investigating lifetime phenomenon, discrete life testing models got less attention in the literature because the mathematics required in dealing with discrete lifetime model is difcult to handle though their signifcance in many felds needs no depiction. Among others, the geometric distribution is specifcally important in many sectors of biological, social, and life-testing experiments. Te geometric distribution has been considered as a lifetime model in reliability theory by Yaqub and Khan [8], Bhattacharya and Kumar [9], Krishna and Jain [10], Sarhan and Kundu [11], and many others. Tese researchers developed Bayes estimators for reliability measures of the individual components in a multicomponent geometric lifetime model using disguised systems of life testing data.
Te literature of estimation lacks, to the best of our knowledge, analyzing through Bayesian approach the Geometric Lifetime Model (GLTM) while considering doubly type-I censoring scheme. Hence, the current study is devoted to provide and analyze doubly type I censored GLTM using Bayesian estimation tools. For the unknown parameter of GLTM, informative and uninformative priors are examined under the Square Error Loss Function (SELF), DeGroot Loss Function (DLF), Quadratic Loss Function (QLF), Precautionary Loss Function (PLF) and Simple Asymmetric Precautionary Loss Function (SAPLF). Bayes estimators and Bayes risks for the unknown parameter of GLTM under doubly type-I censoring scheme are derived for the aforementioned priors and loss functions. A simulation and real-lifedata-based analyzes are carried out to evaluate the suggested model's strength and utility..

Methodology
A random variable X is said to follow a discrete geometric distribution if its probability distrition funcion of X can be written as Geometric distribution having the parameter θ. Te cumulative distribution function for a random variable X, the geometric distribution function is given by:

Te Likelihood Function and Posterior Distributions.
Consider n items are placed in a life testing experiment and we begin studying these items after time T 1 � t 1 and continue to observing them until time T 2 � t 2 . Te observations are assumed to come from the geometric distribution having the parameter θ; it is assumed that left sided observations m 1 are censored in experiment with fxed time T 1 � t 1 . Let r be the number of observed failures from the observations and experiment will proceed from time T 1 � t 1 up to time T 2 � t 2 , and m 2 items will be censored after T 2 � t 2 . Hence, total m 1 + m 2 items are censored. Terefore, the likelihood function of the geometric distribution the under doubly type-I censoring scheme can be derived as At frst, it is assumed that the parameter follows Uniform prior i.e. θ ∼ U(0, 1) Combining likelihood function (2) and prior probability density (3), the posterior density of θ is where Uniform prior is vital in situations where no prior information is available and only the current/sample information is in hand. Tis prior is widely used by the data analysts while analyzing the data through the Bayesian approach (Yaqub and Khan [8], Bhattacharya and Kumar [9], Krishna and Jain [10], and references cited therein).
In situations where prior information is available, the Bayesian analysts suggest using informative priors, which enhance the efciency of the estimation techniques. Among many informative priors, the Beta prior is considered a better informative prior as it is a natural conjugate prior for proportion (probability success) (Van de Schoot [12]. Keeping in view the importance of Beta prior, it is assumed that the parameter of GLTM follows Beta prior distribution with hyperparameters ω 1 and ω 2 , i.e. θ ∼ β(ω 1 , ω 2 ).
where to be proper density, we must have ω 1 > 0 and ω 2 > 0. Combining the likelihood function (2) and prior probability density (6), the posterior density of θ becomes where and Another informative prior considered in this study is the well-known Kumaraswamy distribution, which has signifcant rule in distribution theory (Dey et al. [13]). Terefore, we also assume a special type of Kumaraswamy distribution as prior for the parameter of GLTM, i.e. θ ∼ K(1, ω 3 ).
Te posterior distribution for this prior is derived as where

Elicitations of Hyperparameters of Informative Priors.
To elicit the hyperparameters of the informative (Beta and Kumaraswamy) priors, according to Garthwaite et al. [14]; elicitation of hyperparameters as a method is used to convert an expert's prior knowledge and professional judgment about unknown quantities of interest. To elicit the hyperparameters of the informative (Beta and Kumaraswamy) priors, the method suggested by Aslam [15] is employed in this study. Let Y � X n+1 be future value of a random variable X, then the prior predictive distribution of Y is defned as Te prior predictive distributions of Y for Beta and Kumaraswamy priors are, respectively, derived in the following: For Beta and Kumaraswamy priors, the elicited values of the hyperparameters are obtained by solving the following equations simultaneously in "Mathematica 10." Te resultant values of the hyperparameters of Beta prior are given in the following: Te resultant value of the hyperparameter of Kumaraswamy prior is given in the following:

Loss Functions.
In this section, we derive the Bayes estimators (BEs) and Bayes risks (BRs) using uninformative prior (UP) and two informative priors (IP) for fve diferent loss functions (SELF, DLF, QLF, PLS, and SAPLF).

BEs and BRs Using UP and IPs under SELF.
For a parameter θ with a BE θ * , SELF is defned as Te BE and BR under SELF are Te BEs and BRs under SELF using diferent priors are shown in Table 1.

BEs and BRs Using UP and IPs under DLF.
For a parameter θ with a BE θ * , DLF is defned as Te BE and BR under DLF are Te BEs and BRs under DLF using diferent priors are shown in Table 2.

BEs and BRs Using UP and IPs under QLF.
For a parameter θ with a BE θ * , QLF is defned as Te BE and BR under QLF are Te BEs and BRs under QLF using diferent priors are shown in Table 3.

BEs and BRs Using UP and IPs under PLF.
For a parameter θ with a BE θ * , PLF is defned as Te BE and BR under PLF are Te BEs and BRs under PLF using diferent priors are shown in Table 4.

BEs and BRs Using UP and IPs under SAPLF.
For a parameter θ with a BE θ * , PLF is defned as Te BE and BR under PLF are Te BEs and BRs under SAPLF using diferent priors are shown in Table 5.

Simulations Study
In this section, a thorough simulation study is carried out to check the efciency of the Geometric Lifetime Model under the doubly type-I censoring scheme. Random samples of diferent sizes with various combinations of the test termination times (T 1 � t 1 , T 2 � t 2 ) and various parametric settings are drawn from GLTM under doubly type-II censoring. Te BEs and BRs are determined using the resulting mathematical expressions under various loss functions and priors, the simulation process is performed 10,000 times, and the average of the results are obtained and showcased in Tables 6-15. Te results displayed in Tables 6-15 depict that BEs approach to the true parametric values as the sample size increases. Increasing sample size has negative association with BR as it follows a decreasing trend with increasing sample size. Te decreased test termination time T 1 and increased test termination time T 2 result in a smaller BR, which is obvious. While comparing the performance of diferent priors, it is evident from the numerical results that Beta prior outperforms the rest of the priors as it yields smaller BR. On the other hand, SELF stands higher among its competitors on the shoulder of its minimum BR.   Te bold and italic values in this table shows that these are the best results as compared with counterparts.

Applications
To further strengthen the utility of the GLTM, a real-life data is analyzed. Te data originally discussed by Krishna and Goel [16] is about the remission periods in months of 137 lung cancer patients. Te numerical results for this data set are presented in Table 16.
Te numerical results displayed in Table 16 for the lung cancer patients cement the fndings of the simulation study.

Conclusion
Tis paper presents an estimation technique for the GLTM parameter under the doubly type-1 censoring scheme. Five diferent loss functions (SELF, DLF, QLF, PLF, and SAPLF) and three priors (Uniform, Beta, and Kumaraswamy priors) are considered for the estimation strategy. Te strength of the estimation technique is tested through the simulation study and a real-data analysis. Te numerical results, obtained for diferent settings, depict that BR tends to decrease when larger sample size is considered. Also, the lower termination time (t 1 ) and the BR are positively correlated, while the correlation between the upper termination time (t 2 ) and BR is negative. While overviewing the performance of diferent priors, Beta prior cement itself as better one among its competitor by yielding a smaller BR under all the loss functions. On the other side, SELF performs efciently in comparison to the rest of the loss functions as it gives lower BR under all the three priors. Hence, Beta prior and SELF are suggested for estimating the parameter of GLTM while modeling real-life phenomena that are based on doubly type-AI censoring.

Data Availability
Data are available upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.