Classical and Bayesian Estimation of Two-Parameter Power Function Distribution

The power function distribution is a flexible waiting time model that may provide better fit for some failure data. This paper presents the comparison of the maximum likelihood estimates and the Bayes estimates of two-parameter power function distribution. The Bayes estimates are obtained, using conjugate priors, under five loss functions consist of square error, precautionary, weighted, LINEX and DeGroot loss function. The Gibbs sampling algorithm is proposed to generate samples from posterior distributions and in result the Bayes estimates are computed. The comparison of the maximum likelihood estimates and the Bayes estimates are done through the root mean squared errors. One real-life data set is analyzed to illustrate the evaluation of proposed methods of estimation. Finally, results from the simulation are discussed to assess the performance behavior of the maximum likelihood estimates and the Bayes estimates.


Introduction
The power function distribution is widely used for semiconductor devices and electrical component reliability. Meniconi and Barry [1] verified that the power function distribution is the best model to test the reliability of an electrical component over exponential, lognormal and Weibull distribution. Zarrin et al. [2] used the power function distribution to estimate the component failure of a semiconductor device. The power function distribution is studied by many authors. For example, Kleiber and Kotz [3] showed that the power function distribution is a particular case of Pareto distribution, Bhatt [4] discussed the characterization of the power function distribution through expectation, Chang [5] considered the power function distribution and discussed its characterizations with the use of independence of record values, Lutful and Ahsanullah [6] used the linear function of the order statistics for estimation of the power function distribution, Malik [7] calculated expressions for the exact moments of order statistics for the power function distribution, Saran and Pandey [8] estimated the power function distribution and its characterizations by kth record value, Saleem et al. [9] derived Bayesian estimators for the finite mixture model of power function distribution with censored sample, Shahzad et al. [10] compared the L-moments method and Trim L-moments methods for the power function distribution and Shakeel et al. [11] used the probability weighted moments method and the generalized probability weighted method to estimate the power function distribution.
This paper presents the comparison of maximum likelihood estimation and Bayesian estimation using a complete sample from the power function distribution. The maximum likelihood and the Bayes estimates for unknown parameters and are derived and then compared through the root mean squared error. Finally, numeric illustration and comparisons are presented. In the Bayesian estimation problems, it is essential to specify a loss function. In this regard, five loss functions are selected, which consist of squared error loss function (SELF), precautionary loss function (PLF), weighted loss function (WLF), DeGroot loss function (DLF) and LINEX loss function (LLF). The SELF loss function is introduced by Legendre and Gauss in developing the least square theory. This loss function is symmetric and it assign equal weights to positive and negative errors. The PLF is proposed by Norstrom [12]. This loss function is asymmetric loss function and very useful when lower failure rate is under study. The DLF is proposed by DeGroot [13]. This loss function is asymmetric loss function. The LLF is proposed by Varian [14]. This loss function is also asymmetric loss function and preferred to use when there is under estimation is expected. If be the parameter of interest then a list of above-mentioned loss functions with their respective Bayes estimates are given in Table 1.
The rest of the paper is outlined as: Section 2 provides the introduction of the power function distribution. Section 3 consists of maximum likelihood estimation for the power function distribution. Section 5 describes the Bayesian estimation. Section 5 presents the Markov chain Monte Carlo (MCMC) technique. Simulation study for the maximum likelihood estimates and the Bayes estimates is conducted in Section 6. A real-life data analysis is performed in Section 7 for illustrative purposes, while a conclusion is given in Section 8.

Power Function Distribution
The two-parameter power function distribution is defined density function is a shape parameter, is a scale parameter. The two-parameter power function distribution is denoted by the notation ( , ). The distribution function of the ( , ) is given The survival function of the ( , ) is Similarly, the hazard function of the ( , ) is given by ( ; , ) = , , 0 < < , , > 0 (4) Figure 1. Pdf of power function distribution for different value of at = 5 The Figure 1 provides the graphical presentation of the ( , ) for different values of at = 5, which shows that the ( , ) is positively skewed and heavy-tailed distribution for < 1, uniform for = 1, right triangular for = 2 and negatively skewed for > 2.

Maximum Likelihood Estimation
This section presents maximum likelihood estimates (MLEs) of the ( , ). Here and are both assumed unknown. Suppose a random sample of size is = ( , . . . , ) taken from the PDF in (1) then likelihood function based on is given by The log-likelihood function is given by The normal equations are ℓ( , | ) = − ln( ) + ∑ ln( ) = 0 The MLE of is cannot obtained from the equation (8), However the value that maximizes the log-likelihood function for is the MLE of , given by where ( ) is the largest order statistic.
The MLE of is obtained by solving the equation (5) and (7), given by

Bayesian Estimation
This section introduces the Bayesian analysis of the ( , ). The Bayesian analysis incorporates the prior knowledge of the parameters through the informative prior densities. If the prior knowledge is not available, non-informative priors are used. Here, the Bayes estimates for unknown parameters and are derived under their respective independent conjugate priors. The conjugate prior for the unknown parameter is gamma prior, given by Similarly, the conjugate prior for unknown parameter is Pareto prior, given by Here , , , assumed known and non-negative constants, called hyperparameters. The joint prior for and , given by In Bayesian inference, the information about the model parameters is extracted through posterior distribution. According to the Bayes theorem the joint posterior density function of , given defined by The joint posteriors of and is obtained as follows, where = ( , ( ) ).
Once the posterior distribution is constructed, the next step is to compute the Bayes estimates for any function of and , say ( , ) under various loss functions. The Bayes estimate of under SELF given by The Bayes estimate of under PLF is given by The Bayes estimate of under WLF is given by The Bayes estimate of under DLF is given by The Bayes estimate of under LLF is given by  [20], Gearhart [21] and the references therein. To apply the Gibbs algorithm, the full conditional distributions of the parameters and are constructed from (14). The conditional density of given , defined by

Markov Chain Monte Carlo Technique
which is a density of gamma distribution with parameters + and + ∑ ln( ) − ln( ). Similarly, the conditional density of given , defined by which is a density of Pareto distribution with parameters + and = maximum( ( ) , ). The Gibbs algorithm consists of the following steps  Step 1: Setting some initial values for and , say and , respectively.
where is the burn-in period.

Simulation Study
This section presents a simulation study that is executed to compare the performance behavior of the MLE and the Bayes estimates. The simulation study is carried out for different sample sizes and with different hyperparameter values. In particular, sample of sizes = 15, 25, 50 and 75 are generated by the inverse transformation method from ( , ). The conjugate priors are used for the estimation of  Table 2 to 5. The RMSE are given in the parentheses below the estimates. All the results in Table 2 to 5 are based on 100000 replications.    The performance of the different estimators can be described as  All the estimates hold the consistency property, i.e., as the sample size increases, the RMSE decreases.
 The Bayes estimates are more efficient than MLE as their RMSE is smaller than the RMSE of MLE.
 The Bayes-I estimates are superior from Bayes-II estimates as the RMSE for Bayes-I is smaller than the RMSE of Bayes-II.

Real Life Data Analysis
This section provides a real-life data analysis to see how the ( , ) works in practice. The data set is considered here, used by Ghitany et al. [22], related to a clinical trial performed to study the effectiveness of an antibiotic ointment in relieving pain. This data set represents the 20 failure times (time for a patient to get relief from pain), as follows 0. 138. Before using this data set for the estimation, one natural question arises whether this data set fits the ( , ) or not. The Kolmogorov-Smirnov (K-S) test is performed for the goodness of fit of the ( , ). Here, the K-S distance ( ) is found 0.19912 with a corresponding p-value 0.3579 at = 2.02 and = 1.18. As the p-value is quite high, which suggests that the ( , ) provides good fit for the given failure times data. Thus, there are enough evidence to use this data set for the estimation of ( , ). The data set is used to compute the MLEs and the Bayes estimates for the ( , ).  [23], the comparison among the MLEs, and the Bayes estimates are performed on the basis of the K-S test. The K-S test is performed and the K-S distance along with their p-value for various estimates are given in Table 6. Here, the above real-life data is used to construct the graph of the marginal posterior densities for and .