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Estimators of location based on Kolmogorov‐Smirnov‐type statistics
Author(s) -
Boulanger Alain,
Eeden Constance Van
Publication year - 1983
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314887
Subject(s) - estimator , mathematics , statistics , statistic , m estimator , rank (graph theory) , extremum estimator , asymptotic distribution , distribution (mathematics) , kolmogorov–smirnov test , sample (material) , combinatorics , statistical hypothesis testing , mathematical analysis , physics , thermodynamics
Two classes of estimators of a location parameter ø 0 are proposed, based on a nonnegative functional H 1 * of the pair ( D 1 ø N , G ø N ), where\documentclass{article}\pagestyle{empty}\begin{document}$$ {D_{1N}^\theta (x)} & { = \sqrt N \left\{ {1 - F_N (( - x + \theta)^ -) - F_N (x + \theta)} \right\},} $$\end{document}\documentclass{article}\pagestyle{empty}\begin{document}$$ {G_N^\theta (x)} & { = \frac{1}{2}\left\{ {1 - F_N (( - x + \theta)^ -)F_N (x + \theta)} \right\},} $$\end{document}and where F N is the sample distribution function. The estimators of the first class are defined as a value of ø minimizing H 1 * ; the estimators of the second class are linearized versions of those of the first. The asymptotic distribution of the estimators is derived, and it is shown that the Kolmogorov‐Smirnov statistic, the signed linear rank statistics, and the Cramérvon Mises statistics are special cases of such functionals H 1 * ;. These estimators are closely related to the estimators of a shift in the two‐sample case, proposed and studied by Boulanger in B2 (pp. 271–284).