Establishment of Preliminary Test Estimators and Preliminary Test Confidence Intervals for Measures of Reliability of an Exponentiated Distribution Based on Type-II Censoring
DOI:
https://doi.org/10.6092/issn.1973-2201/8543Keywords:
Exponentiated distributions, Preliminary test estimator, Type-II censoring, Uniformly minimum variance unbiased estimator, Maximum likelihood estimatorAbstract
The present paper has developed the preliminary test estimators (PTEs) of the model parameter raised to certain power, σp, and the two measures of reliability, namely, the reliability function, R(t ) and the reliability of an item or a system, P of an exponentiated distribution, under Type- II censoring, based on their uniformly minimum variance unbiased estimators (UMVUEs) and maximum likelihood estimators (MLEs). The preliminary test confidence intervals (PTCIs) are also developed for σ, R(t ) and P based on their UMVUEs and MLEs. Further, the paper has derived expression for coverage probability of the PTCI of the model parameter, σ. Merits of the proposed PTEs are also established through analysis of simulated numerical data.
References
A. H. ABDEL-HAMID, E. K. AL-HUSSAINI (2009). Estimation in step-stress accelerated life tests for the exponentiated exponential distribution with Type I censoring. Computational Statistics and Data Analysis, 53, pp. 1328–1338.
I. B. ABDUL-MONIEM, H. F. ABDEL-HAMEED (2012). On exponentiated Lomax distribution. International Journal of Mathematical Archive, 3, no. 5, pp. 2144–2150.
E. K. AL-HUSSAINI (2010). On exponentiated class of distributions. Journal of Statistical Theory and Applications, 8, pp. 41–63.
E. K. AL-HUSSAINI, M. HUSSEIN (2011). Bayes prediction of future observables from exponentiated populations with fixed and random sample size. Open Journal of Statistics, 1, pp. 24–32.
A. M. AWAD, M. K. GHARRAF (1986). Estimation of P(Y < X) in the Burr case: A comparative study. Communications in Statistics - Simulation and Computation, 15, no. 2, pp. 389–403.
T. A. BANCROFT (1944). On biases in estimation due to the use of preliminary tests of significance. The Annals of Mathematical Statistics, 15, no. 2, pp. 190–204.
D. J. BARTHOLOMEW (1957). A problem in life testing. Journal of the American Statistical Association, 52, pp. 350–355.
D. J. BARTHOLOMEW (1963). The sampling distribution of an estimate arising in life testing. Technometrics, 5, pp. 361–374.
A. P. BASU (1964). Estimates of reliability for some distributions useful in life testing. Technometrics, 6, pp. 215–219.
R. A. BELAGHI, M. ARASHI, S. M. M. TABATABAEY (2014). Improved confidence intervals for the scale parameter of Burr XII model based on record values. Computational Statistics, 29, no. 5, pp. 1153–1173.
R. A. BELAGHI, M. ARASHI, S. M. M. TABATABAEY (2015). On the construction of preliminary test estimator based on record values for the Burr XII model. Communications in Statistics - Theory and Methods, 44, no. 1, pp. 1–23.
A. CHAO (1982). On comparing estimators of pr{X > Y} in the exponential case. IEEE Transactions on Reliability, R-26, pp. 389–392.
A. CHATURVEDI, A. MALHOTRA (2016). Estimation and testing procedures for the reliability functions of a family of lifetime distributions based on records. International Journal of System Assurance Engineering and Management, 8, no. 2, pp. 836–848.
A. CHATURVEDI, A. MALHOTRA (2017). Inference on the parameters and reliability characteristics of three parameter Burr distribution based on records. Applied Mathematics and Information Sciences, 11, no. 3, pp. 837–849.
A. CHATURVEDI, A. MALHOTRA (2018). On the construction of preliminary test estimators of the reliability characteristics for the exponential distribution based on records.
American Journal of Mathematical and Management Sciences, 37, no. 2, pp. 168–187.
A. CHATURVEDI, A. PATHAK (2012). Estimation of the reliability functions for exponentiated Weibull distribution. Journal of Statistics and Applications, 7, pp. 1–8.
A. CHATURVEDI, A. PATHAK (2013). Bayesian estimation procedures for three parameter exponentiated Weibull distribution under entropy loss function and Type II censoring. URL interstat.statjournals.net/YEAR/2013/abstracts/1306001.php.
A. CHATURVEDI, A. PATHAK (2014). Estimation of the reliability function for four parameter exponentiated generalized Lomax distribution. International Journal of Scientific & Engineering Research, 5, no. 1, pp. 1171–1180.
A. CHATURVEDI, U. RANI (1997). Estimation procedures for a family of density functions representing various life-testing models. Metrika, 46, pp. 213–219.
A. CHATURVEDI, U. RANI (1998). Classical and Bayesian reliability estimation of the generalized Maxwell failure distribution. Journal of Statistical Research, 32, pp. 113–120.
A. CHATURVEDI, K. G. SINGH (2006). Bayesian estimation procedures for a family of lifetime distributions under squared-error and entropy losses. Metron, 64, no. 2, pp.179–198.
A. CHATURVEDI, K. G. SINGH (2008). A family of lifetime distributions and related estimation and testing procedures for the reliability function. Journal of Applied Statistical Science, 16, no. 2, pp. 35–50.
A. CHATURVEDI, K. SURINDER (1999). Further remarks on estimating the reliability function of exponential distribution under Type-I and Type II censorings. Brazilian Journal of Probability and Statistics, 13, pp. 29–39.
A. CHATURVEDI, S. K. TOMER (2002). Classical and Bayesian reliability estimation of the negative binomial distribution. Journal of Applied Statistical Science, 11, pp. 33–43.
A. CHATURVEDI, S. K. TOMER (2003). UMVU estimation of the reliability function of the generalized life distributions. Statistical Papers, 44, no. 3, pp. 301–313.
A. CHATURVEDI, S. VYAS (2017). Estimation and testing procedures for the reliability functions of exponentiated distributions under censorings. Statistica, 77, no. 3, pp. 207–
R. C. GUPTA, R. D.GUPTA, P. L.GUPTA (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics - Theory and Methods, 27, pp. 887–904.
R. D. GUPTA, D. KUNDU (1999). Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41, pp. 173–188.
R. D. GUPTA, D. KUNDU (2001a). Exponentiated exponential family: An alternative to gamma and Weibull distributions. Biometrical Journal, 43, pp. 117–130.
R. D.GUPTA, D. KUNDU (2001b). Generalized exponential distribution: Different methods of estimation. Journal of Statistical Computation and Simulation, 69, pp. 315–337.
R. D. GUPTA, D. KUNDU (2002). Generalized exponential distributions: statistical inferences. Journal of Statistical Theory and Applications, 1, pp. 101–118.
R. D. GUPTA, D. KUNDU (2003a). Closeness of gamma and generalized exponential distribution. Communications in Statistics - Theory and Methods, 32, pp. 705–721.
R. D. GUPTA, D. KUNDU (2003b). Discriminating between the Weibull and the generalized exponential distributions. Computational Statistics and Data Analysis, 43, pp. 179–196.
R. D. GUPTA, D. KUNDU, A. MANGLICK (2002). Probability of correct selection of gamma versus GE or Weibull versus GE models based on likelihood ratio test. In Y. P. CHAUBEY (ed.), Recent Advances In Statistical Methods, Imperial College Press, London, pp. 147–156.
R. JIANG, D. N. P. MURTHY (1999). The exponentiated Weibull family: A graphical approach. IEEE Transactions on Reliability, 48, pp. 68–72.
N. L. JOHNSON (1975). Letter to the editor. Technometrics, 17, p. 393.
G. D. KELLY, J. A. KELLY, W. R. SCHUCANY (1976). Efficient estimation of p(y < x) in the exponential case. Technometrics, 18, pp. 359–360.
B. M. G. KIBRIA (2004). Performance of the shrinkage preliminary tests ridge regression estimators based on the conflicting of W, LR and LM tests. Journal of Statistical Computation and Simulation, 74, no. 11, pp. 793–810.
B. M. G. KIBRIA, A. K. M. E. SALEH (1993). Performance of shrinkage preliminary test
estimator in regression analysis. Jahangirnagar Review, A17, pp. 133–148.
B. M. G. KIBRIA, A. K. M. E. SALEH (2004). Preliminary test ridge regression estimators with Student's t errors and conflicting test-statistics. Metrika, 59, no. 2, pp. 105–124.
B. M. G. KIBRIA, A. K. M. E. SALEH (2005). Comparison between Han-Bancroft and Brook method to determine the optimum significance level for pre-test estimator. Journal of Probability and Statistical Science, 3, pp. 293–303.
B. M. G. KIBRIA, A. K. M. E. SALEH (2006). Optimum critical value for pre-test estimators. Communications in Statistics-Theory and Methods, 35, no. 2, pp. 309–320.
B. M. G. KIBRIA, A. K. M. E. SALEH (2010). Preliminary test estimation of the parameters of exponential and Pareto distributions for censored samples. Statistical Papers, 51, pp.
–773.
D. KUNDU, R. GUPTA (2005). Estimation of P(Y < X) for generalized exponential distribution. Metrika, 61, no. 3, pp. 291–308.
D. KUNDU, R. D. GUPTA, A. MANGLICK (2005). Discriminating between the lognormal and generalized exponential distribution. Journal of Statistical Planning and Inference, 127, pp. 213–227.
D. KUNDU, M. Z. RAQAB (2005). Generalized Rayleigh distribution: different methods of estimation. Computational Statistics and Data Analysis, 49, pp. 187–200.
C. LAI, M. XIE, D. MURTHY (2003). Modified Weibull model. IEEE Transactions on Reliability, 52, pp. 33–37.
M. LJUBO (1965). Curves and concentration indices for certain generalized Pareto distributions. Statistical Review, 15, pp. 257–260.
G. S. MUDHOLKAR, A. D. HUTSON (1996). The exponentiated Weibull family: Some properties and a flood data application. Communications in Statistics - Theory and Methods, 25, no. 12, pp. 3059–3083.
G. S. MUDHOLKAR, D. K. SRIVASTAVA (1993). Exponentiated Weibull family for analyzing bathtub failure-real data. IEEE Transaction on Reliability, 42, pp. 299–302.
G. S. MUDHOLKAR, D. K. SRIVASTAVA, M. FREIMER (1995). The exponentiated Weibull family: A reanalysis of the bus-motor-failure data. Technometrics, 37, pp. 436–445.
M. M. NASSAR, F. H. EISSA (2003). On the exponentiated Weibull distributions. Communications in Statistics - Theory and Methods, 32, pp. 1317–1333.
M. M. NASSAR, F. H. EISSA (2004). Bayesian estimation for the exponentiated Weibull model. Communications in Statistics - Theory and Methods, 33, pp. 2343–2362.
M. PAL, M. M. ALI, J. WOO (2006). Exponentiated Weibull distribution. Statistica, 66, no. 2.
M. PAL, M. M. ALI, J. WOO (2007). Some exponentiated distributions. The Korean Communications in Statistics, 14, pp. 93–109.
E. L. PUGH (1963). The best estimate of reliability in the exponential case. Operations Research, 11, pp. 57–61.
M. Z. RAQAB (2002). Inferences for generalized exponential distribution based on record statistics. Journal of Statistical Planning and Inference, 104, pp. 339–350.
A. K. M. E. SALEH (2006). Theory of Preliminary Test and Stein-Type Estimation with Applications. JohnWiley & Sons, Inc., Hoboken.
A. K. M. E. SALEH, B. M. G. KIBRIA (1993). Performance of some new preliminary test ridge regression estimators and their properties. Communications in Statistics - Theory and Methods, 22, no.10, pp. 2747–2764.
Y. S. SATHE, S. P. SHAH (1981). On estimating P(X > Y) for the exponential distribution. Communications in Statistics - Theory and Methods, 10, no. 1, pp. 39–47.
P. K. SEN, A. K. M. E. SALEH (1978). Nonparametric estimation of location parameter after a preliminary test on regression in the multivariate case. Journal of Multivariate Analysis, 9, no. 2, pp. 322–331.
A. I. SHAWKY, H. H. ABU-ZINADAH (2009). Exponentiated Pareto distribution: Different method of estimations. International Journal of Contemporary Mathematical Sciences, 14, pp. 677–693.
P. TADIKAMALLA (1980). A look at the Burr and related distributions. International Statistical Review, 48, pp. 337–344.
H. TONG (1974). A note on the estimation of P(Y < X) in the exponential case. Technometrics, 16, p. 625.
H. TONG (1975). Letter to the editor. Technometrics, 17, p. 393.
R. K. TYAGI, S. K. BHATTACHARYA (1989). A note on the MVU estimation of reliability for the Maxwell failure distribution. Estadistica, 41, pp. 73–79.
M. XIE, Y. TANG, T. GOH (2002). A modified Weibull extension with bathtub shape failure rate function. Reliability Engineering and System Safety, 76, pp. 279–285.
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