Estimation of the error density in a semiparametric transformation model

Consider the semiparametric transformation model $$\Lambda _{\theta _o}(Y)=m(X)+\varepsilon $$ Λ θ o ( Y ) = m ( X ) + ε , where $$\theta _o$$ θ o is an unknown finite dimensional parameter, the functions $$\Lambda _{\theta _o}$$ Λ θ o and $$m$$ m are smooth, $$\varepsilon $$ ε is independent of $$X$$ X , and $${\mathbb {E}}(\varepsilon )=0$$ E ( ε ) = 0 . We propose a kernel-type estimator of the density of the error $$\varepsilon $$ ε , and prove its asymptotic normality. The estimated errors, which lie at the basis of this estimator, are obtained from a profile likelihood estimator of $$\theta _o$$ θ o and a nonparametric kernel estimator of $$m$$ m . The practical performance of the proposed density estimator is evaluated in a simulation study. Copyright The Institute of Statistical Mathematics, Tokyo 2015