| One of the most powerful algorithms for obtaining maximum likelihood estimates for many incomplete data problems is the EM algorithm. It is an iterative method and there are two steps in every iteration of the EM algorithm: one is called E-step which is to find a conditional expectation based on a conditional distribution and the other is called M-step which is to find a maximum likelihood estimator (MLE) based on the complete data problem.The restricted EM algorithm for maximum likelihood estimation under linear equalities restrictions on the parameters has been solved. However, when the parameters satisfy a set of linear or nonlinear inequalities restrictions, the EM algorithm may be too complicated to apply directly. This paper is devoted to give the restricted EM algorithm for regression coefficients of the linear model with missing data, here the constraints are given by linear and nonlinear inequalities, respectively. For the linear restrictions on the parameters, we give two algorithms, the first is modified projection algorithm of Kudo and Dykstra's, and this algorithm is applied to M-step of EM algorithm for two-dimensional normal distribution. The other optimal algorithm of M-step is the Hildreth-D'Esopo algorithm, some convergence properties of the EM sequence are discussed. And for the nonlinear restrictions on the parameters, by using the asymptotic normality of the maximum likelihood estimators, we change this kind of estimation problem to a stochastic optimization problem in the M-step, and give the limit problem of the stochastic optimization problem. That is, we get the optimal solution of the stochastic optimization problem by using the limit problem whose optimal solution is easily computed, and prove that the optimal solution of the stochastic optimization problem is converge to the optimal solution of the limit problem in probability. Finally, the suggested algorithms are illustrated by numerical examples.
|
PDF Full Text Request