In this paper we study the problem of estimating the density of the error distribution in a random design regression model, where the error is assumed to be independent of the design variable. Our main result is that the L1 error of the kernel density estimate applied to residuals of a...

Consider the semiparametric transformation model <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\Lambda _{\theta _o}(Y)=m(X)+\varepsilon $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi mathvariant="normal">Λ</mi> <msub> <mi mathvariant="italic">θ</mi> <mi>o</mi> </msub> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi mathvariant="italic">ε</mi> </mrow> </math> </EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\theta _o$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="italic">θ</mi> <mi>o</mi> </msub> </math> </EquationSource> </InlineEquation> is an unknown finite dimensional parameter, the functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\Lambda _{\theta _o}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="normal">Λ</mi> <msub> <mi mathvariant="italic">θ</mi> <mi>o</mi> </msub> </msub> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$m$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>m</mi> </math> </EquationSource> </InlineEquation> are...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>