Semi-Parametric Mixture Models Through Log-Concave Density Estimation
This dissertation consists of two parts. The first part considers a semi-parametric two-component mixture model with one component completely known. Assuming the density of the unknown component to be log-concave, which contains a very broad family of densities, we develop a semi-parametric maximum likelihood estimator and propose an EM algorithm to compute it. Our new estimation method finds the mixing proportion and the distribution of the unknown component simultaneously. We establish the identifiability of the proposed semi-parametric mixture model and prove the existence and consistency of the proposed estimators. We further compare our estimator with several existing estimators through simulation studies and apply our method to two real data sets from biological sciences and astronomy.
The second part of this dissertation considers the model g(x) = (1 − p)f0(x; θ) + pf(x), where θ represents the unknown parameters of a known distribution f0 , and f represents the distribution of possible outliers. We propose two innovative algorithms to estimate θ nonparametrically. The first method is called Minimum Search, which is based on identifiability of the mixture model. A strong sufficient condition is proposed for the model to be identifiable and a weaker condition is given for the model to be locally identifiable. The second estimator is the maximum likelihood estimator, which is obtained by EM algorithm assuming f is log-concave. Extensive simulation studies show that our methods give very promising performances.