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 (Tn ) 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 Tn in order to find out which approach yields the more reliable, precise and efficient estimate of Tn. 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.
Highlights
The exponential model has many practical applications especially in life testing
An abridged version of the results of the analysis for the coefficient of variation and the confidence interval are included in the body of the work while the unabridged results are provide as appendix
Coefficient of variation (CV) of the standard error estimate «\ When coefficient of variation (CV) is used as a tool for analysis, interest is typically to find out which procedure returns the smaller CV
Summary
The exponential model has many practical applications especially in life testing. Blischke and Prabhakar Murthy [1]; and Murthy et al [2] in their different works opined that the exponential distribution is the most commonly employed model in reliability and life testing analysis. The maximum likelihood estimator (MLE) of the exponential distribution parameter ( ) is the arithmetic mean and the measure of its long run accuracy is the squared error which is based on the mean. Writing on the effect of mild deviations from a parametric model on classical estimates, Hampel et al [5] noted that the effect of contaminations on the squared error is even worse than that on the arithmetic mean
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