Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 4, 635--647

A Unified Approach to Efficient Estimation in Simple Linear Regression

S. Mandal and M. Samanta, University of Manitoba, Canada

SUMMARY. This paper develops a unified approach to efficient estimation in simple linear regression without solving the likelihood equations, when the error distribution is completely known and multiple measurements on the response variable $y$ for each value of the explanatory variable $x$ are available. We construct efficient estimators of the parameters in the model using linear combinations of order statistics of random samples drawn from the population. We also construct efficient estimators of the parameters for symmetric type-II censoring and symmetric error distribution using linear combinations of available order statistics with additional weights to the smallest and the largest order statistics. It is shown that our estimators are asymptotically normally distributed with mean vector equal to the vector of the parameters in the model and covariance matrix equal to the inverse of the Fisher information matrix for the corresponding problem. Examples illustrating the applications of our methods for several error distributions including normal, logistic, Cauchy and the t with 2 degrees of freedom for complete and symmetrically censored samples are given.

AMS (2000) subject classification. Primary 62J05, 62F10.

Key words and phrases. Censored samples, complete samples, Cramer-Rao lower bound, efficient estimation, order statistics.

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