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.