Influence Function and Bootstrap Methods of Estimating the Standard Errors of the Estimators of Mixture Exponential Distribution Parameter

Abstract

This work estimated the standard error of the maximum likelihood estimator (MLE) and the robust estimators of the exponential mixture parameter (θ) using the influence function and the bootstrap approaches. Mixture exponential random samples of sizes 10, 15, 20, 25, 50, and 100 were generated using 3 mixture exponential models at 2%, 5% and 10% contamination levels. The selected estimators namely: mean, median, alpha-trimmed mean, Huber M-estimate and their standard errors (T_n ) were estimated using the two approaches at the indicated sample sizes and contamination levels. The results were compared using the coefficient of variation, confidence interval and the asymptotic relative efficiency of T_n in order to find out which approach yields the more reliable, precise and efficient estimate of T_n. The results of the analysis show that the two approaches do not equally perform at all conditions. From the results, the bootstrap method was found to be more reliable and efficient method of estimating the standard error of the arithmetic mean at all sample sizes and contamination levels. In estimating the standard error of the median, the influence function method was found to be more effective especially when the sample size is small and yet contamination is high. The influence function based approach yielded more reliable, precise and efficient estimates of the standard errors of the alpha-trimmed mean and the Huber M-estimate for all sample sizes and levels of contamination although the reliability of the bootstrap method improved better as sample size increased to 50 and above. All simulations and analysis were carried out in R programming language.