Modern Stochastics: Theory and Applications

In mixture with varying concentrations model (MVC) one deals with a nonhomogeneous sample which consists of subjects belonging to a fixed number of different populations (mixture components). The population which a subject belongs to is unknown, but the probabilities to belong to a given component are known and vary from observation to observation. The distribution of subjects’ observed features depends on the component which it belongs to.

We consider a mixture with varying concentrations in which each component is described by a nonlinear regression model. A modified least squares estimator is used to estimate the regressions parameters. Asymptotic normality of the derived estimators is demonstrated. This result is applied to confidence sets construction. Performance of the confidence sets is assessed by simulations.

A general jackknife estimator for the asymptotic covariance of moment estimators is considered in the case when the sample is taken from a mixture with varying concentrations of components. Consistency of the estimator is demonstrated. A fast algorithm for its calculation is described. The estimator is applied to construction of confidence sets for regression parameters in the linear regression with errors in variables. An application to sociological data analysis is considered.

Confidence ellipsoids for linear regression coefficients are constructed by observations from a mixture with varying concentrations. Two approaches are discussed. The first one is the nonparametric approach based on the weighted least squares technique. The second one is an approximate maximum likelihood estimation with application of the EM-algorithm for the estimates calculation.