BIAS-REDUCED ESTIMATION OF MEAN ABSOLUTE DEVIATION AROUND THE MEDIAN Michele Lambardi di San Miniato

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BIAS-REDUCED ESTIMATION OF MEAN ABSOLUTE DEVIATION

AROUND THE MEDIAN

Michele Lambardi di San Miniato

Department of Economics and Statistics

University of Udine

Udine, Italy

michele.lambardi@uniud.it

ABSTRACT

A bias-reduced estimator is proposed for the mean absolute deviation parameter of a median regression

model. A workaround is devised for the lack of smoothness in the sense conventionally required

in general bias-reduced estimation. A local asymptotic normality property and a Bahadur–Kiefer

representation sufﬁce in proving the validity of the bias correction. The proposal is developed under

a classical asymptotic regime but, based on simulations, it seems to work also in high-dimensional

settings.

Keywords Asymptotic theory ·Bias reduction ·Mean absolute deviation ·Median regression

1 Introduction

The purpose of this note is to present the possibility of bias reduction for Z-estimators even in the absence of some

smoothness requirements that are typical in this ﬁeld (van der Vaart, 1998, Ch. 5). Bias can be corrected via bootstrap,

but this approach can be troubled in high-dimensions (Singh, 1998). Here, an interpretable bias-reduced estimator

is proposed, in the style of Firth (1993), for the case of a mean absolute deviation (MAD) parameter of a median

regression model, which lacks some conventional smoothness requirements. The proposal is tailored down to MAD

around the median, but it can be generalized to other dispersion parameters. In this case, bias reduction involves some

unknown model quantities that will be replaced with feasible estimates in the style of Severini (1999).

The issue of estimation bias in high-dimensions was ﬁrst pointed out by Neyman and Scott (1948), based on a many-

normal-means example. As a response, the classical asymptotic theory, henceforth addressed as ﬁrst-order asymptotics

(FOA), evolved into higher-order asymptotics (HOA). The behavior of square-root-consistent estimators has since been

assessed more precisely, leading to bias-reduced estimators (Firth, 1993). The style of proof in FOA stems from the

work by Cramér and relies heavily on certain local quadratic approximations to the logarithms of likelihood ratios (Le

Cam and Lo Yang, 2000, Sec. 6.1). HOA relies on even more detailed stochastic expansions and has thus been built

upon stronger smoothness requirements, after the success of Cramér’s research line.

A parallel research line can be found in the work by Le Cam, who aimed at developing a more parsimonious asymptotic

theory with weaker assumptions. This theory is based on concepts like contiguity, differentiability in quadratic mean,

and local asymptotic normality (LAN), which are less demanding than the smoothness assumptions of HOA (Le Cam

and Lo Yang, 2000, Ch. 6) (van der Vaart, 1998, Ch. 6–7) (Pollard, 1997). Both Cramér’s and Le Cam’s styles of proof

have their merits, as the former is useful at developing given some hypotheses, but the latter is needed for reﬁning

them. Concerns after Le Cam motivate the development of bias reduction in a famous non-smooth estimation problem,

namely, quantile regression (Koenker, 2005).

Here, we propose an improved estimator of MAD in median regression. A close resemblance with the famous degrees

of freedom correction for the variance estimator can be noticed. This correction arises naturally while assessing the

LAN property of the criterion that is minimized by the median regression estimator. This LAN condition must be

complemented with a Bahadur–Kiefer (BK) representation of the coefﬁcients estimator (Bahadur, 1966; Kiefer, 1967).

Both the LAN property and the BK representation are collected in a theorem by Koenker and its proof (Koenker, 2005).

arXiv:2210.03622v1 [stat.ME] 7 Oct 2022

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摘要：

BIAS-REDUCEDESTIMATIONOFMEANABSOLUTEDEVIATIONAROUNDTHEMEDIANMicheleLambardidiSanMiniatoDepartmentofEconomicsandStatisticsUniversityofUdineUdine,Italymichele.lambardi@uniud.itABSTRACTAbias-reducedestimatorisproposedforthemeanabsolutedeviationparameterofamedianregressionmodel.Aworkaroundisdevisedforth...

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BIAS-REDUCED ESTIMATION OF MEAN ABSOLUTE DEVIATION AROUND THE MEDIAN Michele Lambardi di San Miniato

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