Generalized weighted likelihood density estimators with application to finite mixture of exponential family distributions

The family of weighted likelihood estimators largely overlaps with minimum divergence estimators. They are robust to data contaminations compared to MLE. We define the class of generalized weighted likelihood estimators (GWLE), provide its influence function and discuss the efficiency requirements. We introduce a new truncated cubic-inverse weight, which is both first and second order efficient and more robust than previously reported weights. We also discuss new ways of selecting the smoothing bandwidth and weighted starting values for the iterative algorithm. The advantage of the truncated cubic-inverse weight is illustrated in a simulation study of three-component normal mixtures model with large overlaps and heavy contaminations. A real data example is also provided.