Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series A, Pt. 2, 208-217

ON CONSISTENCY OF THE BEST-R-POINTS-AVERAGE ESTIMATOR FOR THE MAXIMIZER OF A NONPARAMETRIC REGRESSION FUNCTION

By

Z.D. BAI,

and

MONG-NA LO HUANG, National Sun Yat-sen University, Kaohsiung

SUMMARY. This paper gives a necessary and sufficient condition for the consistency of a mode estimator, the so-called ``BRPA'' (best-$r$-points-average), in a nonparametric regression model, under certain conditions posed on the regression function. The BRPA estimator was proposed in Changchien (1990), and studied in Chen {\it et al}. (1996) under certain specified classes of error distributions. An important application of the main results is the case where the error distribution belongs to the domain of attraction of extreme value distributions. For ease of reference, some asymptotics of the spacings induced by the top $k$ out of $n$ order statistics are presented, when the underlying distribution belongs to the domain of attraction of extreme value distributions. Some comments and simulation results on the convergence rates are also presented.

AMS (1991) subject classification. Primary 62G05, secondary 62G30.

Key words and phrases. Extreme-value distribution, mode estimation, order statistics, spacings.

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