Strong consistency of pseudo-likelihood parameter estimator for univariate Gaussian mixture models

We consider a new method for estimating the parameters of univariate Gaussian mixture models. The method relies on a nonparametric density estimator $\hat{f}_n$ (typically a kernel estimator). For every set of Gaussian mixture components, $\hat{f}_n$ is used to find the best set of mixture weights. That set is obtained by minimizing the $L_2$ distance between $\hat{f}_n$ and the Gaussian mixture density with the given component parameters. The densities together with the obtained weights are then plugged in to the likelihood function, resulting in the so-called pseudo-likelihood function. The final parameter estimators are the parameter values that maximize the pseudo-likelihood function together with the corresponding weights. The advantages of the pseudo-likelihood over the full likelihood are: 1) its arguments are the means and variances only, mixture weights are also functions of the means and variances; 2) unlike the likelihood function, it is always bounded above. Thus, the maximizer of the pseudo-likelihood function -- referred to as the pseudo-likelihood estimator -- always exists. In this article, we prove that the pseudo-likelihood estimator is strongly consistent.