Estimation of the Rayleigh Parameters Based On Interval Grouped Data - Solayman, Hatim (2008)
In this thesis the performance, the efficiency, the accuracy and the validity of the statistical estimation using the interval grouped data derived from the intermittent inspection life testing experiment are tested, improved and modified. To achieve these objectives several estimation methods are investigated employing Rayleigh as the underlying survival model. Based on the interval grouped data the likelihood functions of the unknown Rayleigh parameters are constructed using the unconditional probability and the conditional probability(in case of censoring) of failure in the corresponding intervals. The existence and the uniqueness of the MLE’s are proved. Using the equidistance case partitioning the MLE’s of the scale parameter are bounded and hence bisection and secant numerical methods can be applied to arrive at faster solution. The intervals end points and the cumulative number of failures at these ends are used to derive the mid interval and the compound grouped estimators .These estimators are in explicit forms and evaluated in terms of their bias and consistency. The results of applying the maximum likelihood estimation to real life time data relatively show better estimates of the survival and the hazard functions, as compared to the classical non parametric estimates. In the least square estimation and based on the multinomial distribution of failures the resulting estimators are compared to the corresponding estimators obtained by fitting regression models based on the nonparametric estimates of both the survival and the hazard functions at a pre given time. In the Bayesian estimation approach the conjugate priors are derived using both the complete and the interval grouped data. High posterior credible intervals are obtained and mathematical improvements of the Bayesian estimators obtained by the interval grouped are made to increase their relative efficiency and performance. Applying the modified Bayesian estimation procedures to a generated Rayleigh lifetimes data show a significance efficiency of the Bayesian estimation method. Despite the fact that there is a considerable loss of information in the exact unobservable lifetimes, simulation studies at different settings of the life testing experiment show a high relative efficiency of the estimators obtained using the interval grouped data in comparison with the estimators obtained using type I and right censored data. To measure the loss of information due to the intermittent inspection life testing experiment Shannon information and distance divergence measures are considered. Modifications in the Shannon information measure and derivation of a new information measure based on the sufficient statistics are investigated to reflect the actual loss of information. A criterion for minimizing the loss of information, selecting the suitable number of intervals, the inspection times and the sample size is extracted. The performance of the estimation procedures is also tested on some known survival analysis issues with application to a real lifetimes data. Hence, modifications in the conventional methods and formulating of alternative models are devoted to guarantee existence of the solution, improve the performance and reduce computations. Finally general conclusions on the overall thesis are given together with highlights for further researches.