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Asymptotic properties of $$M$$ M -estimators in linear and nonlinear multivariate regression models

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  • Christopher Withers
  • Saralees Nadarajah

Abstract

We consider the (possibly nonlinear) regression model in $$\mathbb{R }^q$$ R q with shift parameter $$\alpha $$ α in $$\mathbb{R }^q$$ R q and other parameters $$\beta $$ β in $$\mathbb{R }^p$$ R p . Residuals are assumed to be from an unknown distribution function (d.f.). Let $$\widehat{\phi }$$ ϕ ^ be a smooth $$M$$ M -estimator of $$\phi={{\beta }\atopwithdelims (){\alpha }}$$ ϕ = β α and $$T(\phi )$$ T ( ϕ ) a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of $$T(\widehat{\phi })$$ T ( ϕ ^ ) and an estimator of $$T(\phi )$$ T ( ϕ ) with bias $$\sim n^{-2}$$ ∼ n - 2 requiring $$\sim n$$ ∼ n calculations. (In contrast, the jackknife and bootstrap estimators require $$\sim n^2$$ ∼ n 2 calculations.) For a linear regression with random covariates of low skewness, if $$T(\phi )=\nu \beta $$ T ( ϕ ) = ν β , then $$T(\widehat{\phi })$$ T ( ϕ ^ ) has bias $$\sim n^{-2}$$ ∼ n - 2 (not $$n^{-1}$$ n - 1 ) and skewness $$\sim n^{-3}$$ ∼ n - 3 (not $$n^{-2}$$ n - 2 ), and the usual approximate one-sided confidence interval (CI) for $$T(\phi )$$ T ( ϕ ) has error $$\sim n^{-1}$$ ∼ n - 1 (not $$n^{-1/2}$$ n - 1 / 2 ). These results extend to random covariates. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Christopher Withers & Saralees Nadarajah, 2014. "Asymptotic properties of $$M$$ M -estimators in linear and nonlinear multivariate regression models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(5), pages 647-673, July.
  • Handle: RePEc:spr:metrik:v:77:y:2014:i:5:p:647-673
    DOI: 10.1007/s00184-013-0458-4
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    References listed on IDEAS

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    1. Powell, James L., 1984. "Least absolute deviations estimation for the censored regression model," Journal of Econometrics, Elsevier, vol. 25(3), pages 303-325, July.
    2. H.J. Bierens, 1981. "Robust Methods and Asymptotic Theory in Nonlinear Econometrics," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 35(3), pages 173-173, September.
    3. Chang, Xiao-Wen, 2006. "Computation of Huber's M-estimates for a block-angular regression problem," Computational Statistics & Data Analysis, Elsevier, vol. 50(1), pages 5-20, January.
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