Asymptotic Variances and Covariances of Maximum-Likelihood Estimators, from Censored Samples, of the Parameters of Weibull and Gamma Populations

Abstract

For ordered samples of size $n$, with proportions $q_1$ and $q_2$ of the sample values censored from below and from above, respectively, from three-parameter Weibull and gamma populations, expressions are given for the elements of the maximum-likelihood information matrices, each element being the negative of the expected value of one of the second partial derivatives of the likelihood function with respect to the parameters. An important result is that, while one or more of the elements become infinite for values of the shape parameter less than or equal to 2 when $q_1 = 0$, this does not happen for $q_1 > 0$. If one lets $n \rightarrow \infty$ while holding $q_1$ and $q_2$ fixed and then inverts the information matrix and its submatrices, the results are the asymptotic variance-covariance matrices whose elements are the asymptotic variances and covariances of the joint maximum-likelihood estimators of all three parameters and of any one or two parameters, the other(s) being known. Tables are given of the coefficients of $(1/n)$ times powers of the scale parameter $\theta$ in the asymptotic variances and covariances for the cases $q_1 = 0.000(0.005)0.25, q_2 = 0.00(0.25)0.75$ for both Weibull and gamma populations with shape parameters 1, 2, and 3, omitting cases for which $q_1 = 0$ when the shape parameter is 1 or 2 and the location parameter is one of those being estimated. Results of a limited Monte Carlo study indicate that when at least one of the parameters is known, the variances and covariances of samples of size as small as 50 agree closely with the results given by the asymptotic formulas, but that when all three parameters are unknown, the variances and the absolute values of the covariances, even for samples of size as large as 100, are substantially in excess of the asymptotic values.