Abstract
In this article, we consider the doubly type-1 censoring scheme that researchers frequently use in clinical trials and lifetime experiments. The Bayesian paradigm will be used to estimate the parameters of the Geometric Lifetime Model (GLTM) using a doubly type-I censoring scheme. Bayes estimators and their associated Bayes risks are examined in terms of closed-form algebraic expressions. This research also includes a strategy for eliciting hyperparameters based on prior prediction distributions. To evaluate the strength and effectiveness of the suggested estimating approach, thorough simulation studies as well as real-life data analysis are presented. The results depict that Squared Error Loss Function (SELF) is more efficient, and the Beta prior is suitable while estimating the parameter of GLTM.
1. 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. The worth of censoring schemes is not hidden in literature, and many authors have focused their research study based on different 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 2-component 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 difficult to handle though their significance in many fields needs no depiction. Among others, the geometric distribution is specifically important in many sectors of biological, social, and life-testing experiments. The 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. These researchers developed Bayes estimators for reliability measures of the individual components in a multicomponent geometric lifetime model using disguised systems of life testing data.
The 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..
2. 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 . The cumulative distribution function for a random variable X, the geometric distribution function is given by:
2.1. The Likelihood Function and Posterior Distributions
Consider n items are placed in a life testing experiment and we begin studying these items after time and continue to observing them until time . The observations are assumed to come from the geometric distribution having the parameter ; it is assumed that left sided observations are censored in experiment with fixed time . Let be the number of observed failures from the observations and experiment will proceed from time up to time , and items will be censored after . Hence, total items are censored. Therefore, the likelihood function of the geometric distribution the under doubly type-I censoring scheme can be derived as
At first, it is assumed that the parameter follows Uniform prior i.e.
Combining likelihood function (2) and prior probability density (3), the posterior density of iswhere
Uniform prior is vital in situations where no prior information is available and only the current/sample information is in hand. This 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 efficiency 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 , i.e. where to be proper density, we must have and . Combining the likelihood function (2) and prior probability density (6), the posterior density of becomeswhere
and
Another informative prior considered in this study is the well-known Kumaraswamy distribution, which has significant rule in distribution theory (Dey et al. [13]). Therefore, we also assume a special type of Kumaraswamy distribution as prior for the parameter of GLTM, i.e. .
The posterior distribution for this prior is derived aswhere
2.2. 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 be future value of a random variable X, then the prior predictive distribution of Y is defined as
The 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.”
The resultant values of the hyperparameters of Beta prior are given in the following:
The resultant value of the hyperparameter of Kumaraswamy prior is given in the following:
2.3. 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 five different loss functions (SELF, DLF, QLF, PLS, and SAPLF).
2.4. BEs and BRs Using UP and IPs under SELF
For a parameter with a BE , SELF is defined as
The BE and BR under SELF are
The BEs and BRs under SELF using different priors are shown in Table 1.
2.5. BEs and BRs Using UP and IPs under DLF
For a parameter with a BE , DLF is defined as
The BE and BR under DLF are
The BEs and BRs under DLF using different priors are shown in Table 2.
2.6. BEs and BRs Using UP and IPs under QLF
For a parameter with a BE , QLF is defined as
The BE and BR under QLF are
The BEs and BRs under QLF using different priors are shown in Table 3.
2.7. BEs and BRs Using UP and IPs under PLF
For a parameter with a BE , PLF is defined as
The BE and BR under PLF are
The BEs and BRs under PLF using different priors are shown in Table 4.
2.8. BEs and BRs Using UP and IPs under SAPLF
For a parameter with a BE , PLF is defined as
The BE and BR under PLF are
The BEs and BRs under SAPLF using different priors are shown in Table 5.
3. Simulations Study
In this section, a thorough simulation study is carried out to check the efficiency of the Geometric Lifetime Model under the doubly type-I censoring scheme. Random samples of different sizes with various combinations of the test termination times (, ) and various parametric settings are drawn from GLTM under doubly type-II censoring. The 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.
The 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. The decreased test termination time and increased test termination time result in a smaller BR, which is obvious. While comparing the performance of different 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.
4. Applications
To further strengthen the utility of the GLTM, a real-life data is analyzed. The data originally discussed by Krishna and Goel [16] is about the remission periods in months of 137 lung cancer patients. The numerical results for this data set are presented in Table 16.
The numerical results displayed in Table 16 for the lung cancer patients cement the findings of the simulation study.
5. Conclusion
This paper presents an estimation technique for the GLTM parameter under the doubly type-1 censoring scheme. Five different loss functions (SELF, DLF, QLF, PLF, and SAPLF) and three priors (Uniform, Beta, and Kumaraswamy priors) are considered for the estimation strategy. The strength of the estimation technique is tested through the simulation study and a real-data analysis. The numerical results, obtained for different settings, depict that BR tends to decrease when larger sample size is considered. Also, the lower termination time and the BR are positively correlated, while the correlation between the upper termination time and BR is negative. While overviewing the performance of different 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 efficiently 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
The authors declare that they have no conflicts of interest.