Improved confidence interval estimation of the population standard deviation using ranked set sampling: A simulation study
Background & Aim: The sample standard deviation S is the common point estimator of σ, but S is sensitive to the presence of outliers and may not be an efficient estimator of σ in skewed and leptokurtic distributions. Although S has good efficiency in platykurtic and moderately leptokurtic distributions, its classical inferential methods may perform poorly in non-normal distributions. The classical confidence interval for σ relies on the assumption of normality of the distribution. In this paper, a performance comparison of six confidence interval estimates of σ is performed under ten distributions that vary in skewness and kurtosis.
Methods and Material: A Monte Carlo simulation study is conducted under the following distributions: normal, two contaminated normal, t, Gamma, Uniform, Beta, Laplace, exponential and χ2 with specific parameters. Confidence interval estimates obtained using the more powerful ranked set sampling (RSS) are compared with the traditional simple random sampling (SRS) technique. Performance of the confidence intervals is assessed based on width and coverage probabilities. A real data example representing birth weight of 189 newborns is used for assessment.
Results: It is not surprising that for normal data most of the intervals were close tot he nominal value especially using RSS. Simulation results indicated generally better performance of RSS in terms of coverage probability and smaller interval width as sample size increases, especially for contaminated and heavy-tailed skewed distributions.
Conclusion: Simulation results revealed that the use of RSS improved greatly the coverage probability. Also, it was found that the interval labeled (III) due to Bonett (2006) had the best performance in terms of coverage probability over the wide range of distributions investigated in this paper and would be recommended for use by practitioners. There may be a need to develop nonparametric intervals that is robust against outliers and heavy-tailed distributions.
2. Lehmann EL, Romano JP. Testing statistical hypotheses. Springer Science & Business Media; 2006.
3. Albatineh AN, Kibria BMG, Wilcox ML, Zogheib B. Confidence interval estimation for the population coefficient of variation using ranked set sampling: a simulation study. J Appl Stat. 2013; 41(4): 733-751.
4. Arnold SF. Mathematical Statistics. Prentice Hall College Div; 1990.
5. Bonett DG. Approximate confidence interval for standard deviation of non-normal distributions. Comput Stat Data Anal.2006; 50(3): 775-782.
6. Burch BD. Estimating kurtosis and confidence intervals for the variance under non-normality. J Stat Comput Simul.2013; 84(12): 2710-2720.
7. Cojbasic V, Tomovic A. Nonparametric confidence intervals for population variance of one sample and the difference of variances of two samples. J Comput Stat Data Anal.2007; 51(12): 5562-5578
8. Hosmer D W, Lemeshow S. Applied Logistic Regression: Second edition. New York: John Wiley & Sons, 2000.
9. Hummel R, Banga S, Hettmansperger TP. Better confidence intervals for the variance in a random sample. Minitab Technical Report. (2005)Retrieved from: http://www.minitab.com/support/documentation/a nswers/OneVariance.pdf
10. Kittani H, Zghoul A. A robust confidence interval for the population standard deviation. Journal of Applied Statistical Science (2010);18(2): 121-130.
11. MacEachern S, Ozturk¨O,¨ Wolfe DA. A new ranked set sample estimator of variance. J Royal Stat Soc B.(2002); 64(2): 177-188.
12. McIntyre GA. A method for unbiased selective sampling using ranked sets. Australian J Agri Res.1952; 3(4): 385-390.
13. Mood AM, Graybill FA, Boes DC. Introduction to the Theory of Statistics (1974). McGrawM Hill Kogakusha; (1974).
14. Rousseeuw PJ, Croux C. Alternatives to the median absolute deviation. J Am Stat Assoc.1993; 88(424):1273-83.
15. Scheffe H. The analysis of variance. 1959. New York. (1959);331-67.
16. Shoemaker LH. Fixing the F test for equal variances. Am Stat. 2003;57(2):105- 14.
17. Stokes SL. Estimation of variance using judgment ordered ranked set samples. Biometrics. 1980; 35-42.
18. Takahasi K, Wakimoto K. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annal Inst Stat Math. 1968; 20(1):1-31.
19. Terpstra JT, Wang P. Confidence intervals for a population proportion based on a ranked set sample. J Stat Comput Simul. 2008; 78(4):351-66.