Estimation of the error density in a semiparametric transformation model.

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