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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

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.

Acknowledgement

Supported by : Hongik University

References

  1. 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
  2. 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
  3. 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
  4. Cohen, A. C. (1991). Truncated and Censored Samples, Marcel Dekker, Inc., New York.
  5. 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
  6. Harter, H. L. (1961). Expected values of normal order statistics, Biometrika, 48, 151-165. https://doi.org/10.1093/biomet/48.1-2.151
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. Sarhan, A. E. and Greenberg, B. G., eds. (1962). Contributions to Order Statistics, Wiley, New York.
  13. 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
  14. 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
  15. 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
  16. Weibull, W. (1939). The phenomenon of rupture in solids, Ingeniors Vetenskaps Akademien Handlingar, 153, 17.
  17. 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
  18. 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
  19. 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
  20. 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.
  21. 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
  22. 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
  23. 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
  24. Blom, G. (1958). Statistical Estimates and Transformed Beta Variates, Wiley, New York.