Efficient Estimation of Parameters of the Extreme Value Distribution

Abstract

The problem of efficient estimation of the parameters of the extreme value distribution has not been addressed in the literature. Our paper attempts to obtain efficient estimators of the parameters without solving the likelihood equations. The paper provides for the first time simple expressions for the elements of the information matrix for type II censoring. We consider the problem of estimation of θ and σ in the model (1/σ)f((x − θ)/σ) when f is the standard form of the pdf of type I (maximum) extreme value distribution. We construct efficient estimators of θ and σ using linear combinations of order statistics of a random sample drawn from the population. We derive explicit formulas for the information matrix for this problem for type II censoring and construct efficient estimators of the parameters using linear combinations of available order statistics with additional weights to the smallest and the largest order statistics. We consider numerical examples to illustrate the applications of the estimators. We also consider a Monte Carlo simulation study to examine the performance of the estimators for small samples. Using the asymptotic covariance matrix of the estimators in the above model we then construct an efficient estimator of θ in the reduced model (1/cθ)f((x − θ)/cθ) where θ ( > 0) is unknown and c ( > 0) is known for complete and censored samples.