A Note on Bayesian Analysis of Error Function Distribution under Different Loss Functions

The Bayesian analysis of the scale parameter of error function distribution has been considered in this paper. A class of informative and non-informative priors has been assumed to derive the corresponding posterior distributions. The Bayes estimators and associated risks have calculated under different loss functions. The Bayesian credible intervals have been constructed under each prior. The performance of the Bayes estimators have been evaluated and compared under a comprehensive simulation study. The purpose is to find out the combination of a loss function and a prior having the minimum Bayes risk and hence producing the best results. The study depicts that in order to estimate the said parameter use of entropy loss function under informat ive priors can be preferred.


Introduction
The error function distribution is one of the most widely used distributions in statistics. Estimating its parameter using Bayesian inference is extremely useful. Eberly and Casella [1] discussed the construction of Bayesian cred ible intervals using Rao-Blackwellized construction which offers smallest standard error of estimate. Korsgaard et al. [2] considered the mu ltivariate normal d istribution and concentrated on the model where residuals associated with liabilit ies of the binary traits have been assumed to be independent. A Bayesian analysis using Gibbs sampling has been outlined for the model where this assumption has been relaxed. Wang [3] proposed a criterion to choose a loss function in Bayesian analysis. Liang [4] introduced and derived De mpster EM -Algorithm for the two-co mponent normal mixtu re models to obtain the iterat ive co mputation estimates, also used data augmentation and general Gibbs sampler to get the sample fro m posterior distribution under conjugate prior. Wang [5]developed the new method, called matrix-variate graphical models (M GGMs), wh ich involves simu ltaneously modeling variable and sample dependencies with the matrix-normal distribution. Khan and Islam [6] evaluated the maintenance performance of the system when time is continuous and consider half-normal failure lifetime model as well as repair t ime model. However, error function distribution has rarely received the attention of the analysts. But it is always of interest to study the behaviour and properties of the estimators for the parameters of the new/deprived distributions. So, the problem of estimation of the parameter of the error function distribution under a Bayesian framework has been addressed in this paper. A class of priors have been assumed under various loss functions to estimate the parameter o f the distribution.

Model and Likelihood Function
The probability density function of error function distribution is:

Bayesian Analysis under the Assumption of Uniform Prior
The uniform prior is assumed to be: ( ) 1 p ω ∝ The Bayes estimators and risks under SELF, QLF, ELF and PLF are respectively presented in the following. x

Bayesian Analysis under the Assumption of Jeffreys Prior
The Jeffreys prior is defined as: The Bayes estimators and risks under SELF, QLF, ELF and PLF are respectively given in the following.

Bayesian Analysis under the Assumption of Maxwell Prior
The Maxwell prior is assumed to be: The posterior distribution under Maxwell prior is: The Bayes estimators and risks under SELF, QLF, ELF and PLF are respectively shown in the follo wing.

Bayesian Analysis under the Assumption of Rayleigh Prior
The Rayleigh prio r is assumed to be: The posterior distribution under Ray leigh prior is: The Bayes estimators and risks under SELF, QLF, ELF and PLF are respectively presented in the following.

Bayesian Analysis under the Assumption of Chi Prior
The chi prior is assumed to be: The Bayes estimators and risks under SELF, QLF, ELF and PLF are respectively derived in the fo llowing.

Bayesian Analysis under the Assumption of Normal Prior Considering Location Parameter to be Zero
The normal prior is assumed to be: The Bayes estimators and risks under SELF, QLF, ELF and PLF are respectively presented in the following.

Bayesian Credible Intervals under Different Priors
The Bayesian cred ible intervals, as d iscussed by Saleem and Raza [7], under uniform, Jeffreys, Maxwell, Rayleigh, chi and normal priors are respectively constructed in the following. 2

Simulation Study
Simu lation study has been carried out using n = 50, 100, 200 and 300 for . Different values of the hyper-parameters have been used and the results for the values giving better convergence and the minimu m risks have been presented. In order to have more precise estimates, the results have been replicated sufficiently. The risks associated with Bayes estimates have been underlined in the tables. Similarly, the differences between lower and upper limits of credib le intervals have been underlined. Fro m the above study it can be seen that by increasing the sample size the estimated value of the parameter converges to the true value of the parameter and magnitude of risk associated with each estimate decreases. The increasing values of the parameter impose a negative impact on rate of convergence under each prior; similarly, the performance of squared error loss function and precautionary loss function is badly affected. However, the performance of quadratic loss function and entropy loss function is independent of choice of parametric value. In co mparison of non-informative priors the uniform prio r g ives the better estimates as the corresponding risks are smaller for each loss function. While in case of informat ive priors the Maxwell prior for QLF and ELF, Chi prior for SELF and Rayleigh prior fo r PLF provide the best results. Similarly, estimates under entropy loss function give the min imu m risks among all loss functions for each prior. It can also be assessed that the performance of estimates under informat ive priors is better than those under non-informative p riors. So me prior elicitation technique may further strengthen this argument. Hence, the use of Maxwell prior under entropy loss function can be preferred to estimate the parameter of the error function distribution using a Bayesian framework.
In case of interval estimat ion, the credib le intervals under uniform prior are again narrower than those under Jeffreys prior. Using info rmative priors, the intervals under chi prior are having the min imu m width. So for Bayesian interval estimation of the parameter of error function distribution, the use of chi prior can be preferred.

Conclusions and Recommendations
The study has been conducted to estimate the parameter of the error function distribution using four different loss functions and under six informat ive and non-informat ive priors. The study indicates that for Bayesian point estimation, the use of entropy loss function under Maxwell prior can be preferred. While for interval estimation, the ch i prior can affectively be emp loyed.
The study can be extended by using more priors and loss functions. Some censoring procedures and finite mixture of components of error function distribution can also be used.