Sankhya: The Indian Journal of Statistics

1996, Volume 58, Series B, Pt. 1, pp. 28--37

APPROXIMATE VARIANCE AND COVARIANCE OF MAXIMUM LIKELIHOOD ESTIMATORS FOR THE PARAMETERS OF EXTREME VALUE REGRESSION MODELS FOR CENSORED DATA

By

S. R. PAUL and K. THIAGARAJHA ,University of Windsor

SUMMARY.  In this paper we obtain approximate variances and covariances of the maximum likelihood estimates (MLEs) of the parameters of the extreme value regression model for complete and censored (type I and type II) data by inverting the finite-$n$-expected Fisher information matrix. For type II censored data exact expressions for the elements involving the scale parameter are mathematically and computationally  messy. So we obtain approximate expressions for these elements by using standard Taylor series expansion. These approximate results are very simple to calculate and are shown to be very accurate. Harter and Moore (1968) obtain asymptotic variances and covariance of the MLEs for a two parameter model involving type II censored data, by inverting the expected Fisher information matrix in which the elements are obtained in the limit as $n{\rightarrow}{\infty}$. For the two parameter model, a comparison of the approximate variances and covariance using the exact and the approximate elements of the finite-$n$-expected Fisher information matrix and the asymptotic results by Harter and Moore (1968) is given. For complete and type II censored data parameter orthogonalization (Cox and Reid, 1987) simplifies calculation of the approximate variance covariance of the ML estimates. For type I censored data such parameter orthogonalization does not seem possible, so the results are obtained by inverting the expected Fisher information matrix.

AMS (1980) subject classification. 62J02,  62F12.

Key words and phrases. Approximate and exact results; Parameter orthogonalization; Two parameter extreme value distribution; Type I and type II censored samples; Weibull distribution.

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