Asymptotic properties of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$M$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>M</mi> </math> </EquationSource> </InlineEquation>-estimators in linear and nonlinear multivariate regression models

We consider the (possibly nonlinear) regression model in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathbb{R }^q$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>q</mi> </msup> </math> </EquationSource> </InlineEquation> with shift parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation> in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\mathbb{R }^q$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>q</mi> </msup> </math> </EquationSource> </InlineEquation> and other parameters <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\beta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">β</mi> </math> </EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\mathbb{R }^p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>p</mi> </msup> </math> </EquationSource> </InlineEquation>. Residuals are assumed to be from an unknown distribution function (d.f.). Let <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\widehat{\phi }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover accent="true"> <mi mathvariant="italic">ϕ</mi> <mo stretchy="true">^</mo> </mover> </math> </EquationSource> </InlineEquation> be a smooth <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$M$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>M</mi> </math> </EquationSource> </InlineEquation>-estimator of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\phi={{\beta }\atopwithdelims (){\alpha }}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">ϕ</mi> <mo>=</mo> <mfenced close=")" open="(" separators=""> <mfrac linethickness="0pt"> <mi mathvariant="italic">β</mi> <mi mathvariant="italic">α</mi> </mfrac> </mfenced> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$T(\phi )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi mathvariant="italic">ϕ</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$$T(\widehat{\phi })$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mover accent="true"> <mi mathvariant="italic">ϕ</mi> <mo stretchy="true">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> and an estimator of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$$T(\phi )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi mathvariant="italic">ϕ</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> with bias <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$$\sim n^{-2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo>∼</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math> </EquationSource> </InlineEquation> requiring <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$$\sim n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo>∼</mo> <mi>n</mi> </mrow> </math> </EquationSource> </InlineEquation> calculations. (In contrast, the jackknife and bootstrap estimators require <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$$\sim n^2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo>∼</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> </math> </EquationSource> </InlineEquation> calculations.) For a linear regression with random covariates of low skewness, if <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$$T(\phi )=\nu \beta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi mathvariant="italic">ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="italic">ν</mi> <mi mathvariant="italic">β</mi> </mrow> </math> </EquationSource> </InlineEquation>, then <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$$T(\widehat{\phi })$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mover accent="true"> <mi mathvariant="italic">ϕ</mi> <mo stretchy="true">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> has bias <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$$\sim n^{-2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo>∼</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math> </EquationSource> </InlineEquation> (not <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$$n^{-1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math> </EquationSource> </InlineEquation>) and skewness <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$$\sim n^{-3}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo>∼</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </mrow> </math> </EquationSource> </InlineEquation> (not <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$$n^{-2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </math> </EquationSource> </InlineEquation>), and the usual approximate one-sided confidence interval (CI) for <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$$T(\phi )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi mathvariant="italic">ϕ</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> has error <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$$\sim n^{-1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo>∼</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math> </EquationSource> </InlineEquation> (not <InlineEquation ID="IEq26"> <EquationSource Format="TEX">$$n^{-1/2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math> </EquationSource> </InlineEquation>). These results extend to random covariates. Copyright Springer-Verlag Berlin Heidelberg 2014