On maximum likelihood estimators of shape and scale parameters and their application in constructing confidence contours.
Titel:
On maximum likelihood estimators of shape and scale parameters and their application in constructing confidence contours.
Auteur:
Saunders, Sam C.
Verschenen in:
Communications in statistics
Paginering:
Jaargang 11 (1982) nr. 16 pagina's 1769-1791
Jaar:
1982
Inhoud:
The seminal result of a “Student”, which initiated exact statistical inference, stated that for the normal distribution the statistic [image omitted] where X¯ is the sample mean and s the sample standard deviation, has a t-distribution which depends upon the sample size but not upon the population mean μ or the standard deviation σ, see “Student” the Probable Error of the Mean Biometrika, 1908. This result can be generalized to other situat,ions by using maximum likelihood estimates, namely the statistic [image omitted] has a distribution independent of the location and scale parameters μ and σ, respectively, where [image omitted] denote the correspondi corresponding MLE's based on complete or type II censored samples, see Antle and Bain, “ A Property of Maximum Likelihood Estimators of Location and Scale Parameters” SIAM Review, 1969. In this report the MLE's of shape and scale parameters,a and fi respectively, for the survival distribution F¯of a r.v. X defined by [image omitted] with R a prescribed reliability function, are shown to have the property that [image omitted] have a joint distribution independent of α and β under a wider type of censoring than type II. The analogous “Student” result is that [image omitted] has a distribution depending upon R, and the nature of censoring, but not upon the parameters α and β. This result is applied to the problem of constructing upper and lower confidence contours for the reliability F¯ using the estimated function [image omitted] . This is accomplished by utilizing appropriate pseudo-random number generators, in conjunction with analytic results, in a computer program which enables the statistician to compute the desired percentiles of the requisite distributions for a wide class of sampling situations rather than having to rely upon tables published for a few. This technique is illustrated in a fatigue life application when the censoring distribution has parameters with a known relationship to the unknown parameters of life.
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