Estimation of Gumbel Distribution Parameters of m-th Maxima from Doubly Censored Sample

Abstract

Let X be a random variable following the first asymptotic distribution function of extreme values with location parameter u and scale parameter b, b > 0. Harter and Moore (1968) presented the MLEs of u and b from censored samples with lowest r₁ and largest r₂ sample values being censored in a sample of size n. In the present study these results are extended to the estimation of the parameters of type-I asymptotic distribution of m-th largest observation of several samples. The method of maximum likelihood with a sample of size n having r₁ smallest and r₂ largest sample values censored, is used for estimation. The asymptotic variance-covariance matrix of the MLEs is 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 of the likelihood function.