Estimation of the error density in a semiparametric transformation model

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 smooth, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\varepsilon $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">ε</mi> </math> </EquationSource> </InlineEquation> is independent of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$X$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>X</mi> </math> </EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$${\mathbb {E}}(\varepsilon )=0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="double-struck">E</mi> <mo stretchy="false">(</mo> <mi mathvariant="italic">ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation>. We propose a kernel-type estimator of the density of the error <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\varepsilon $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">ε</mi> </math> </EquationSource> </InlineEquation>, and prove its asymptotic normality. The estimated errors, which lie at the basis of this estimator, are obtained from a profile likelihood estimator of <InlineEquation ID="IEq9"> <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> and a nonparametric kernel estimator of <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>. The practical performance of the proposed density estimator is evaluated in a simulation study. Copyright The Institute of Statistical Mathematics, Tokyo 2015