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

M. Hallin, C. Vermandele, B.J.M. Werker

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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 H ajek 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 e ciency.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).
Original languageEnglish
Place of PublicationTilburg
PublisherFinance
Number of pages26
Volume2003-23
Publication statusPublished - 2003

Publication series

NameCentER Discussion Paper
Volume2003-23

Fingerprint

Rank Statistics
Asymptotic Representation
Asymptotic Normality
Maximal Invariant
Semiparametric Efficiency
Semiparametric Inference
Statistics
Symmetry
Location Parameter
Intercept
Asymptotic Variance
Time Series Models
Asymptotic Theory
Signed
Representation Theory
Conditioning
Invariance
Regression
Restriction

Keywords

  • ranking
  • statistics
  • regression analysis

Cite this

Hallin, M., Vermandele, C., & Werker, B. J. M. (2003). Serial and Nonserial Sign-and-Rank Statistics: Asymptotic Representation and Asymptotic Normality. (CentER Discussion Paper; Vol. 2003-23). Tilburg: Finance.
Hallin, M. ; Vermandele, C. ; Werker, B.J.M. / Serial and Nonserial Sign-and-Rank Statistics : Asymptotic Representation and Asymptotic Normality. Tilburg : Finance, 2003. (CentER Discussion Paper).
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Hallin, M, Vermandele, C & Werker, BJM 2003 'Serial and Nonserial Sign-and-Rank Statistics: Asymptotic Representation and Asymptotic Normality' CentER Discussion Paper, vol. 2003-23, Finance, Tilburg.

Serial and Nonserial Sign-and-Rank Statistics : Asymptotic Representation and Asymptotic Normality. / Hallin, M.; Vermandele, C.; Werker, B.J.M.

Tilburg : Finance, 2003. (CentER Discussion Paper; Vol. 2003-23).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Serial and Nonserial Sign-and-Rank Statistics

T2 - Asymptotic Representation and Asymptotic Normality

AU - Hallin, M.

AU - Vermandele, C.

AU - Werker, B.J.M.

N1 - Pagination: 26

PY - 2003

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N2 - 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 H ajek 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 e ciency.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).

AB - 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 H ajek 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 e ciency.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).

KW - ranking

KW - statistics

KW - regression analysis

M3 - Discussion paper

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T3 - CentER Discussion Paper

BT - Serial and Nonserial Sign-and-Rank Statistics

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Hallin M, Vermandele C, Werker BJM. Serial and Nonserial Sign-and-Rank Statistics: Asymptotic Representation and Asymptotic Normality. Tilburg: Finance. 2003. (CentER Discussion Paper).