Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Estimation for Mean and Standard Deviation of Normal Distribution under Type II Censoring

  • Received : 2014.09.01
  • Accepted : 2014.11.04
  • Published : 2014.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider maximum likelihood estimators of normal distribution based on type II censoring. Gupta (1952) and Cohen (1959, 1961) required a table for an auxiliary function to compute since they did not have an explicit form; however, we derive an explicit form for the estimators using a method to approximate the likelihood function. The derived estimators are a special case of Balakrishnan et al. (2003). We compare the estimators with the Gupta's linear estimators through simulation. Gupta's linear estimators are unbiased and easily calculated; subsequently, the proposed estimators have better performance for mean squared errors and variances, although they show bigger biases especially when the ratio of the complete data is small.

Keywords

References

  1. Asgharzadeh, A. (2006). Point and interval estimation for a generalized logistic distribution under progressive type II censoring, Communications in Statistics-Theory and Methods, 35, 1685-1702. https://doi.org/10.1080/03610920600683713
  2. Asgharzadeh, A. (2009). Approximate MLE for the scaled generalized exponential distribution under progressive type-II censoring, Journal of the Korean Statistical society, 38, 223-229. https://doi.org/10.1016/j.jkss.2008.09.004
  3. Balakrishnan, N. (1989). Approximate MLE of the scale parameter of the Rayleigh distribution with censoring, IEEE Transactions on Reliability, 38, 355-357. https://doi.org/10.1109/24.44181
  4. Balakrishnan, N. and Asgharzadeh, A. (2005). Inference for the scaled half-logistic distribution based on progressively type II censored samples, Communications in Statistics-Theory and Methods, 34, 73-87. https://doi.org/10.1081/STA-200045814
  5. Balakrishnan, N. and Kannan, N. (2001). Point and interval estimation for the logistic distribution base on progressive type-II censored samples, in Handbook of Statistics, Balakrishnan, N. and Rao, C. R. Eds., 20, 431-456.
  6. Balakrishnan, N., Kannan, N., Lin, C. T. and Ng, H. K. T. (2003). Point and interval estimation for gaussian distribution, Based on progressively type-II censored samples, IEEE Transactions on Reliability, 52, 90-95. https://doi.org/10.1109/TR.2002.805786
  7. Balakrishnan, N., Kannan, N., Lin, C. T. and Wu, S. J. S. (2004). Inference for the extreme value distribution under progressive type-II censoring, Journal of Statistical Computation and Simulation, 74, 25-45. https://doi.org/10.1080/0094965031000105881
  8. Balakrishnan, N. andWong, K. H. T. (1991). Approximate MLEs for the location and scale parameters of the half-logistic distribution with type-II right censoring, IEEE Transactions on reliability, 40, 140-145. https://doi.org/10.1109/24.87114
  9. Blom, G. (1958). Statistical Estimates and Transformed Beta Variates, Wiley, New York.
  10. Cohen, A. C. (1959). Simplified estimators for the normal distribution when samples are singly censored or truncated, Technometrics, 1, 217-237. https://doi.org/10.1080/00401706.1959.10489859
  11. Cohen, A. C. (1961). Tables for maximum likelihood estimates: Singly truncated and singly censored samples, Technometrics, 3, 535-541. https://doi.org/10.1080/00401706.1961.10489973
  12. Cohen, A. C. (1991). Truncated and Censored Samples, Marcel Dekker, Inc., New York.
  13. Gupta, A. K. (1952). Estimation of the mean and standard deviation of a normal population from a censored sample, Biometrika, 39, 260-273. https://doi.org/10.1093/biomet/39.3-4.260
  14. Harter, H. L. (1961). Expected values of normal order statistics, Biometrika, 48, 151-165. https://doi.org/10.1093/biomet/48.1-2.151
  15. Hastings, Jr. C., Mosteller, F., Tukey, J.W. andWinsor, C. P. (1947). Low moments for small samples: A comparative study of order statistics, Annals of Mathematical Statistics, 18, 413-426. https://doi.org/10.1214/aoms/1177730388
  16. Kang, S. B., Cho, Y. S. and Han, J. T. (2008). Estimation for the half logistic distribution under progressively type-II censoring, Communications of the Korean Statistical Society, 15, 367-378. https://doi.org/10.5351/CKSS.2008.15.3.367
  17. Kim, C. and Han, K. (2009). Estimation of the scale parameters of the Rayleigh distribution under general progressive censoring, Journal of the Korean Statistical Society, 38, 239-246. https://doi.org/10.1016/j.jkss.2008.10.005
  18. Kim, N. (2014). Approximate MLE for the scale parameter of the generalized exponential distribution under random censoring, Journal of the Korean Statistical Society, 43, 119-131. https://doi.org/10.1016/j.jkss.2013.03.006
  19. Sarhan, A. E. and Greenberg, B. G. (1956). Estimation of location and scale parameters by order statistics from singly and doubly censored sample: Part I. The normal distribution up to size 10, Annals of Mathematical Statistics, 27, 427-451. https://doi.org/10.1214/aoms/1177728267
  20. Sarhan, A. E. and Greenberg, B. G. (1958). Estimation of location and scale parameters by order statistics from singly and doubly censored sample: Part II. Tables for the normal distribution for samples of size 11 to 15, Annals of Mathematical Statistics, 29, 79-105. https://doi.org/10.1214/aoms/1177706707
  21. Sarhan, A. E. and Greenberg, B. G., eds. (1962). Contributions to Order Statistics, Wiley, New York.
  22. Seo, E. H. and Kang, S. B. (2007). AMLEs for Rayleigh distribution based on progressively type-II censored data, The Korean Communications in Statistics, 14, 329-344. https://doi.org/10.5351/CKSS.2007.14.2.329
  23. Sultan, K. S., Alsada, N. H. and Kundu, D. (2014). Bayesian and maximum likelihood estimation of the inverse Weibull parameters under progressive type-II censoring, Journal of Statistical Computation and Simulation, 84, 2248-265. https://doi.org/10.1080/00949655.2013.788652
  24. Weibull, W. (1939). The phenomenon of rupture in solids, Ingeniors Vetenskaps Akademien Handlingar, 153, 17.