Least squares sieve estimation of mixture distributions with boundary effects

In this study, we propose two types of sieve estimators, based on least squares (LS), for probability distributions that are mixtures of a finite number of discrete atoms and a continuous distribution under the framework of measurement error models. This research is motivated by the maximum likelihood (ML) sieve estimator developed in Lee et al. (2013). We obtain two types of LS sieve estimators through minimizing the distance between the empirical distribution/characteristic functions and the model distribution/characteristic functions. The LS estimators outperform the ML sieve estimator in several aspects: (1) they need much less computational time; (2) they give smaller integrated mean squared error; (3) the characteristic function based LS estimator is more robust against mis-specification of the error distribution. We also use roughness penalization to improve the smoothness of the resulting estimators and reduce the estimation variance. As an application of our proposed LS estimators, we use the Framingham Heart Study data to investigate the distribution of genetic effects on body mass index. Finally asymptotic properties of the LS estimators are investigated.