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Linear estimation of location and scale parameters using partial maxima

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Abstract

Consider an i.i.d. sample \({X^*_{1},X^*_{2},\ldots,X^*_{n}}\) from a location-scale family, and assume that the only available observations consist of the partial maxima (or minima) sequence, \({X^*_{1:1},X^*_{2:2},\ldots,X^*_{n:n}}\), where \({X^*_{j:j}=\max\{ X^*_1, \ldots,X^*_j \}}\). 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 (in Biometrica 39:88–95, 1952) 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.

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Authors and Affiliations

  1. Department of Mathematics, Section of Statistics and O.R., University of Athens, Panepistemiopolis, 157 84, Athens, Greece

    Nickos Papadatos

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  1. Nickos Papadatos
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Correspondence to Nickos Papadatos.

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Devoted to the memory of my six-years-old, little daughter, Dionyssia, who leaved us on August 25, 2010, at Cephalonia island.

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Papadatos, N. Linear estimation of location and scale parameters using partial maxima. Metrika 75, 243–270 (2012). https://doi.org/10.1007/s00184-010-0325-5

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