Identifiability and rates of estimation for scale parameters in location mixture models

Abstract

In this paper we consider the problem of identifiability and estimation for the scale parameter $\theta$ in the location mixture model $\theta (X + Y)$, where X has a known distribution independent of the Y, whose distribution is unknown. Identification of $\theta$ is ensured by constraining Y based on the tail behavior of the distribution for X. Rates for estimation are described for those X which can be written as a square summable series of exponential variables. As a special case, our analysis shows that the structural parameters in the Weibull semiparametric mixture (Heckman and Singer) are not estimable at the usual parametric $O_p(1/ \sqrt{n})$. The exact relationship between identifying constraints and achievable rates is explained.