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

  • J. I. Udobi Department of Statistics, Federal Polytechnic Oko, P.M.B 021 Aguata, Anambra State, Nigeria
  • G. A. Osuji Department of Statistics, Nnamdi Azikiwe University Awka, PMB 5025 Awka, Anambra state, Nigeria
  • S. I. Onyeagu Department of Statistics, Nnamdi Azikiwe University Awka, PMB 5025 Awka, Anambra state, Nigeria
  • H. O. Obiora-Ilouno Department of Statistics, Nnamdi Azikiwe University Awka, PMB 5025 Awka, Anambra state, Nigeria
Keywords: influence function, bootstrap, mixture exponential, standard error

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.

References

W. Blischke and D. N. Prabhakar Murthy, Reliability: Modeling, Prediction, and Optimization, John Wiley and Sons, 2000. https://doi.org/10.1002/9781118150481

D. Murthy, M. Xie and R. Jiang, Weibull Models, John Wiley and Sons, 2004. DOI:10.002/047147326X

B. Epstein, The Exponential Distribution and Its Role in Life Testing, Industrial Quality Control 15 (1958), 4-9.

B. D. Ripley, Robust Statistics, M.Sc. in Applied Statistics MT2004.

F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, Robust Statistics: The Approach Based on Influence Functions, John Wiley and Sons, Inc., Canada, 1986.

R.G. Staudte and S. J. Sheather, Robust Estimation and Testing, John Wiley and Sons, Inc., New York, 1990. https://doi.org/10.1002/9781118165485

F.R. Hampel, Contributions to the theory of robust estimation, Ph.D. thesis, University of California, Berkeley, 1968.

F.R. Hampel, The influence curve and its role in robust estimation, J. Am. Statist. Assoc. 69 (1974), 383-393. https://doi.org/10.1002/9781118165485

P. J. Huber, Robustness: Where are We Now?, L1-Statistical Procedures and Related Topics, IMS Lecture Notes – Monograph Series, Volume 31, 1997.

H. Wainer, Robust statistics: A survey and some prescriptions, Journal of Educational Statistics 1(4) (1976), 285-312. https://doi.org/10.3102/10769986001004285

P. J. Huber and E. M. Ronchetti, Robust Statistics, 2nd ed., Wiley, New York, 2009.

R.R. Wilcox, Introduction to Robust Estimation and Hypothesis Testing, 2nd ed., Academic Press, 2005.

R.A. Maronna, R.D. Martin and V.J. Yohai, Robust Statistics: Theory and Methods, John Wiley and Sons, Ltd., England, 2006. https://doi.org/10.1002/0470010940

B. Efron, Bootstrap methods: Another look at the jackknife, Ann. Statist. 7 (1979), 1-26. https://doi.org/10.1214/aos/1176344552

M. R. Chernick and R. A. LaBudde, An Introduction to Bootstrap Methods with Applications to R, John Wiley & Sons, Inc., Hoboken, New Jersey and Canada, 2011.

B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans, SIAM, Philadelphia, 1982. https://doi.org/10.1137/1.9781611970319

Published
2021-06-21
How to Cite
Udobi, J. I., Osuji, G. A., Onyeagu, S. I., & Obiora-Ilouno, H. O. (2021). Influence Function and Bootstrap Methods of Estimating the Standard Errors of the Estimators of Mixture Exponential Distribution Parameter. Earthline Journal of Mathematical Sciences, 7(1), 113-136. https://doi.org/10.34198/ejms.7121.113136
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Articles