Bayesian Approach for Confidence Intervals of Variance on the Normal Distribution

Abstract

This research aims to compare estimating the confidence intervals of variance based on the normal distribution with the primary method and the Bayesian approach. The maximum likelihood is the well-known method to approximate variance, and the Chi-squared distribution performs the confidence interval. The central Bayesian approach forms the posterior distribution that makes the variance estimator, which depends on the probability and prior distributions. Most introductory prior information looks for the availability of the prior distribution, informative prior distribution, and noninformative prior distribution. The gamma, Chi-squared, and exponential distributions are defined in the prior distribution. The informative prior distribution uses the Markov Chain Monte Carlo (MCMC) method to draw the random sample from the posterior distribution. The Fisher information performs the Wald confidence interval as the noninformative prior distribution. The interval estimation of the Bayesian approach is obtained from the central limit theorem. The performance of these methods considers the coverage probability and minimum value of the average width. The Monte Carlo process simulates the data from a normal distribution with the true parameter of mean and several variances and the sample sizes. The R program generates the simulated data repeated 10,000 times in each situation. The results showed that the maximum likelihood method employed on the small sample sizes. The best confidence interval estimation was when sample sizes increased the Bayesian approach with an available prior distribution. Overall, the Wald confidence interval tended to outperform the large sample sizes. For application in real data, we expressed the reported airborne particulate matter of 2.5 in Bangkok, Thailand. We used the 10–1000 records to estimate the confidence interval of variance and evaluated the interval width. The results are similar to those of the simulation study.