<i>M</i>-quantiles

Breckling, Jens; Chambers, Ray

doi:10.1093/biomet/75.4.761

Cited by 198 publications

(107 citation statements)

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“…The tilt statistic, Section 3, makes this connection explicit, as discussed for the asymmetric least squares case in remark F of Efron (1991). The regression percentile idea goes back to Koenker and Bassett (1978); see also Breckling and Chambers (1988).…”

Section: Aml Regression Percentilesmentioning

confidence: 99%

Poisson Overdispersion Estimates Based on the Method of Asymmetric Maximum Likelihood

Efron

1992

Journal of the American Statistical Association

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A common difficulty in regression problems with Poisson (or binomial or other exponential family) response variables is Overdispersion: the scatter around the fitted regression is too large by the standards of Poisson variability. This article concerns the description, estimation, and testing of various patterns of overdispersion, with particular emphasis on the Poisson case. Asymmetric maximum likelihood (AML) is a method of fitting regressions for the conditional percentiles of the response variable as a function of the predictors (e.g., the conditional 90th percentile of y given x). Distances between the various regression percentiles give a direct assessment of overdispersion. The discussion is carried through in terms of an archaeological data set, where we see that the counts are overdispersed by a factor of 1.35 in one part of the covariate space, but not at all in another. Moreover, the overdispersion is about 40% larger in the positive response direction than in the negative. The AML estimates are easy to compute and relate nicely to the usual maximum likelihood estimates for generalized linear regression.

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Section: Aml Regression Percentilesmentioning

confidence: 99%

Poisson Overdispersion Estimates Based on the Method of Asymmetric Maximum Likelihood

Efron

1992

Journal of the American Statistical Association

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“…An overview of M -quantile regression models and their application to small area estimation 3.1. Linear M-quantile regression MQR was first introduced by Breckling and Chambers (1988), aiming to extend the concept of M-estimation (Huber, 1964) to a wider family of location parameters that lie between quan- tiles and expectiles. M-quantiles combine the robustness properties of quantiles (Koenker and Bassett, 1978) with the efficiency of expectiles (Newey and Powell, 1987 where ρ τ = 2|τ − I.u 0/|ρ.u/ is a tilted version of a convex loss function ρ.u/, and σ.τ / is a nuisance scale parameter.…”

Section: Italian 2001 Census and 2006 European Union Survey On Incomementioning

confidence: 99%

Parametric Modelling ofM-Quantile Regression Coefficient Functions with Application to Small Area Estimation

Frumento

Salvati

2019

Journal of the Royal Statistical Society Series A: Statistics in Society

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Summary Small area estimation methods can be used to obtain reliable estimates of a parameter of interest within an unplanned domain or subgroup of the population for which only a limited sample size is available. A standard approach to small area estimation is to use a linear mixed model in which the heterogeneity between areas is accounted for by area level effects. An alternative solution, which has gained popularity in recent years, is to use M‐quantile regression models. This approach requires much weaker assumptions than the standard linear mixed model and enables computing outlier robust estimators of the area means. We introduce a new framework for M‐quantile regression, in which the model coefficients, β(τ), are described by (flexible) parametric functions of τ. We illustrate the advantages of this approach and its application to small area estimation. Using the European Union Survey on Income and Living Conditions data, we estimate the average equivalized household income in three Italian regions. The paper is accompanied by an R package Mqrcm that implements the necessary procedures for estimation, inference and prediction.

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“…(5) Downloaded by [Southern Illinois University] at 00:05 04 January 2015 from Equation (5), Q S (u, F ) is obtained by inverting the map x → E{S(x − X)}, from which it is seen that spatial quantiles are a special case of the 'M-quantiles' introduced by Breckling and Chambers (1988) and also treated by Koltchinskii (1997) and Breckling, Kokic, and Lübke (2001). Solving Equation (5) for u yields the spatial centred rank function,…”

Section: Examplesmentioning

confidence: 99%

Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation

Serfling

2010

Journal of Nonparametric Statistics

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Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardisation of multivariate data to produce equivariance or invariance of procedures, three important types of matrix-valued functional are studied: 'weak covariance' (or 'shape'), 'transformation-retransformation' (TR), and 'strong invariant coordinate system' (SICS). The clarification of TR affine equivariant versions of the sample spatial quantile function is obtained. It is seen that geometric artefacts of SICS-standardised data are invariant under affine transformation of the original data followed by standardisation using the same SICS functional, subject only to translation and homogeneous scale change. Some applications of SICS standardisation are described.

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M-quantiles

Cited by 198 publications

References 0 publications

Poisson Overdispersion Estimates Based on the Method of Asymmetric Maximum Likelihood

Poisson Overdispersion Estimates Based on the Method of Asymmetric Maximum Likelihood

Parametric Modelling ofM-Quantile Regression Coefficient Functions with Application to Small Area Estimation

Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation

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