Estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data

Bayes and frequentist estimators are obtained for the two-parameter Gompertz distribution (GD), as well as the reliability and hazard rate functions, using progressive first-failure censoring plan. We have examined Bayes estimates under symmetric and asymmetric loss functions. We show that the Bayes estimates relative to asymmetric loss function includes the maximum likelihood estimate (MLE) and other Bayes estimates as special cases. This is done using the conjugate prior for the scale parameter and discrete prior for the shape parameter. It has been seen that the Bayes estimators are obtained in closed form. Also, based on this new censoring scheme, exact and approximate confidence intervals as well as exact confidence region for the parameters of GD are developed. A practical example using simulated data set was used for illustration. Finally, to assess the performance of the proposed estimators, numerical results using Monte Carlo simulation study were reported.

[1]  N. Gordon.  Maximum likelihood estimation for mixtures of two gompertz distributions when censoring occurs , 1990 .

[2]  Veeresh Gadag,et al.  Progressively Censored Reliability Sampling Plans for the Weibull Distribution , 2000, Technometrics.

[3]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[4]  Ahmed A. Soliman,et al.  Comparison of estimates using record statistics from Weibull model: Bayesian and non-Bayesian approaches , 2006, Comput. Stat. Data Anal..

[5]  R. Soland.  Bayesian Analysis of the Weibull Process With Unknown Scale and Shape Parameters , 1969 .

[6]  Benjamin Gompertz,et al.  XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. F. R. S. &c , 1825, Philosophical Transactions of the Royal Society of London.

[8]  Gannat R. Al-Dayian,et al.  On finite mixture of two-component gompertz lifetime model , 2000 .

[9]  R. Makany.  A Theoretical Basis for Gompertz'S Curve , 1991 .

[10]  N. Balakrishnan,et al.  Progressive Censoring: Theory, Methods, and Applications , 2000 .

[11]  Ahmed A. Soliman,et al.  Estimations for Pareto Model Using General Progressive Censored Data and Asymmetric Loss , 2008 .

[12]  Nancy R. Mann,et al.  Best Linear Invariant Estimation for Weibull Parameters Under Progressive Censoring , 1971 .

[13]  R. Calabria,et al.  Point estimation under asymmetric loss functions for left-truncated exponential samples , 1996 .

[14]  Uditha Balasooriya,et al.  Failure–censored reliability sampling plans for the exponential distribution , 1995 .

[15]  Lonnie C. Vance,et al.  Estimators for the 2-Parameter Weibull Distribution with Progressively Censored Samples , 1983, IEEE Transactions on Reliability.

[16]  Tzong-Ru Tsai,et al.  Limited failure-censored life test for the Weibull distribution , 2001, IEEE Trans. Reliab..

[17]  P. Franses.  Fitting a Gompertz Curve , 1994 .

[18]  Narayanaswamy Balakrishnan,et al.  Progressive censoring methodology: an appraisal , 2007 .

[19]  DavidR . Thomas,et al.  Linear Order Statistic Estimation for the Two-Parameter Weibull and Extreme-Value Distributions from Type II Progressively Censored Samples , 1972 .

[20]  B. Gompertz,et al.  On the Nature of the Function Expressive of the Law of Human Mortality , 1825 .

[21]  Siu-Keung Tse,et al.  Parameters estimation for weibull distributed lifetimes under progressive censoring with random removals , 1996 .

[22]  Narayanaswamy Balakrishnan,et al.  Interval Estimation of Parameters of Life From Progressively Censored Data , 1994 .

[23]  H. Martz.  Bayesian reliability analysis , 1982 .

[24]  M. Uschold,et al.  Methods and applications , 1953 .

[25]  Shuo-Jye Wu,et al.  On estimation based on progressive first-failure-censored sampling , 2009, Comput. Stat. Data Anal..

[26]  Jong-Wuu Wu,et al.  Statistical inference about the shape parameter of the Burr type XII distribution under the failure-censored sampling plan , 2005, Appl. Math. Comput..

[27]  Jong-Wuu Wu,et al.  Estimation of the parameters of the Gompertz distribution under the first failure-censored sampling plan , 2003 .

[28]  E. Cramer.  Balakrishnan, Narayanaswamy ; Aggarwala, Rita: Progressive censoring : theory, methods, and applications / N. Balakrishnan ; Rita Aggarwala. - Boston ; Basel ; Berlin, 2000 , 2000 .

[29]  A. Cohen,et al.  Progressively Censored Samples in Life Testing , 1963 .

[30]  Wen-Chuan Lee,et al.  Characterization of the Mixtures of Gompertz Distributions by Conditional Expectation of Order Statistics , 1999 .

[31]  B. R. Rao,et al.  New Better than Used and Other Concepts for a Class of Life Distributions , 1992 .

[32]  C. Redmond,et al.  Maximum‐Likelihood Estimation of the Parameters of the Gompertz Survival Function , 1970 .

[33]  Narayanaswamy Balakrishnan,et al.  Estimation of parameters from progressively censored data using EM algorithm , 2002 .

[34]  Zhenmin Chen.  Parameter Estimation of the Gompertz Population , 1997 .

[35]  A. A. Soliman,et al.  Estimation of parameters of life from progressively censored data using Burr-XII model , 2005, IEEE Transactions on Reliability.