Serial and Nonserial Sign-and-Rank Statistics: Asymptotic Representation and Asymptotic Normality

CentER Working Paper No. 2003-23

Posted: 26 May 2004

See all articles by Marc Hallin

Marc Hallin

ECARES, Universite Libre de Bruxelles

Catherine Vermandele

Université Libre de Bruxelles (ULB)

Bas J. M. Werker

Tilburg University - Center for Economic Research (CentER)

Date Written: 2003

Abstract

The classical theory of rank-based inference is entirely based either on ordinary ranks, which do not allow for considering location nor intercept parameters, or on signed ranks, which require an assumption of symmetry. If the median, in the absence of a symmetry assumption, is considered as a location parameter, the maximal invariance property of ordinary ranks is lost to the ranks and the signs. As shown in Hallin and Werker (2003), conditioning on a maximal invariant in such situations is essential if semiparametric efficiency is to be reached. This new maximal invariant thus suggests a new class of statistics, based on ordinary ranks and signs. An asymptotic representation theory a la Hajek is developed here for such statistics, both in the nonserial and in the serial case. The corresponding asymptotic normality results clearly show how the signs are adding a separate contribution to the asymptotic variance, hence, potentially, to asymptotic efficiency. Applications to semiparametric inference in regression and time series models with median restrictions are treated in detail in a companion paper (Hallin, Werker, and Vermandele 2003).

Keywords: Ranks, signs, Hajek representation, median regression, median restrictions, maximal invariant

JEL Classification: G10, M10

Suggested Citation

Hallin, Marc and Vermandele, Catherine and Werker, Bas J.M., Serial and Nonserial Sign-and-Rank Statistics: Asymptotic Representation and Asymptotic Normality (2003). CentER Working Paper No. 2003-23, Available at SSRN: https://ssrn.com/abstract=550185

Marc Hallin (Contact Author)

ECARES, Universite Libre de Bruxelles ( email )

Ave. Franklin D Roosevelt, 50 - C.P. 114
Brussels, B-1050
Belgium
+32 2 650 5886 (Phone)
+32 2 650 5899 (Fax)

Catherine Vermandele

Université Libre de Bruxelles (ULB) ( email )

CP 132 Av FD Roosevelt 50
Brussels, Brussels 1050
Belgium

Bas J.M. Werker

Tilburg University - Center for Economic Research (CentER) ( email )

Econometrics and Finance Group
5000 LE Tilburg
Netherlands

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