Linear Estimation of Location and Scale Parameters Using Partial Maxima

TL;DR

This paper develops linear estimators for location and scale parameters using only partial maxima data from a location-scale family, demonstrating their properties and efficiency, especially for the scale parameter, under various distributions.

Contribution

It introduces BLUEs based on partial maxima for location-scale models and analyzes their consistency and variance properties, extending classical methods to truncated data.

Findings

Variance of scale parameter estimator is at most O(1/log n)

Estimators are consistent under broad distribution classes

Partial maxima BLUEs resemble classical order statistic estimators

Abstract

Consider an i.i.d. sample X^*_1,X^*_2,...,X^*_n from a location-scale family, and assume that the only available observations consist of the partial maxima (or minima)sequence, X^*_{1:1},X^*_{2:2},...,X^*_{n:n}, where X^*_{j:j}=max{X^*_1,...,X^*_j}. This kind of truncation appears in several circumstances, including best performances in athletics events. In the case of partial maxima, the form of the BLUEs (best linear unbiased estimators) is quite similar to the form of the well-known Lloyd's (1952, Least-squares estimation of location and scale parameters using order statistics, Biometrika, vol. 39, pp. 88-95) BLUEs, based on (the sufficient sample of) order statistics, but, in contrast to the classical case, their consistency is no longer obvious. The present paper is mainly concerned with the scale parameter, showing that the variance of the partial maxima BLUE is at most of order O(1/log n), for a wide class of distributions.