Maximum-Likelihood Estimation, from Doubly Censored Samples, of the Parameters of the First Asymptotic Distribution of Extreme Values

Let X be a random variable having the first asymptotic distribution of smallest (largest) values, with location parameter u and scale parameter b, b > 0. The natural logarithm of the likelihood function of a sample of size n from such a distribution, the lowest r1 and the highest r2 sample values having been censored, is written down and its first and second partial derivatives with respect to the parameters are worked out. The likelihood equations, obtained by equating to zero the first partial derivatives, do not have explicit solutions, but an iterative procedure for solving them on an electronic computer is described. The asymptotic variances and covariances of the maximum-likelihood estimators of the parameters are obtained by inverting the information matrix, whose elements are the negatives of the limits, as n → ∞, of the expected values of the second partial derivatives, and tabulated for censoring proportions q1 = 0.0(0.1)0.9 from below and q2 = 0.0(0.1) (0.9 – q1) from above. The asymptotic variances and covariances are compared with corresponding sample values obtained from a Monte Carlo study of 2000 samples each of sizes n = 10 and n = 20 from the distribution of smallest values. For single censoring from above, the mean square errors of the sample estimates are also compared with the variances of the best linear unbiased estimators and the mean square errors of the best linear invariant estimators.