Finite mixtures of distributions; the problem of estimating the mixing proportions

Constructing estimators for the parameters of a mixture of distributions has attracted many statisticians. Given that the distribution function G of a random variable X is a mixture of known distribution functions with mixing proportions respectively where estimation of the mixing proportions is considered. Different estimation techniques are studied in depth and the properties of the resulting estimators are discussed. The necessary background to mixtures of distributions is first given and an extension of the method of moments for estimating is then proposed. The generalized (weighted) least squares method, when the observations are grouped into (m+l) intervals, is considered and it is shown that the estimators possess certain desired asymptotic properties. The case when is also investigated. Since the set of equations leading to the generalized least squares estimators are not in general solvable, an iteration process is proposed and is shown to produce satisfactory results after even one cycle. Finally, when the problem of maximum likelihood estimation of 0 is considered and the Fisher's scoring method is suggested to solve the likelihood equation. Properties of the first and second cycle solutions are derived.