Maximum likelihood estimation of parameters with constraints in normaland multinomial distributions

Abstract

Motivated by problems in medicine, biology, engineering and economics, con-

strained parameter problems arise in a wide variety of applications. Among them

the application to the dose-response of a certain drug in development has attracted

much interest. To investigate such a relationship, we often need to conduct a dose-

response experiment with multiple groups associated with multiple dose levels of

the drug. The dose-response relationship can be modeled by a shape-restricted

normal regression. We develop an iterative two-step ascent algorithm to estimate

normal means and variances subject to simultaneous constraints. Each iteration

consists of two parts: an expectation{maximization (EM) algorithm that is utilized

in Step 1 to compute the maximum likelihood estimates (MLEs) of the restricted

means when variances are given, and a newly developed restricted De Pierro algorithm that is used in Step 2 to find the MLEs of the restricted variances when

means are given. These constraints include the simple order, tree order, umbrella

order, and so on. A bootstrap approach is provided to calculate standard errors of

the restricted MLEs. Applications to the analysis of two real datasets on radioim-munological assay of cortisol and bioassay of peptides are presented to illustrate

the proposed methods.

Liu (2000) discussed the maximum likelihood estimation and Bayesian estimation in a multinomial model with simplex constraints by formulating this

constrained parameter problem into an unconstrained parameter problem in the

framework of missing data. To utilize the EM and data augmentation (DA) algorithms, he introduced latent variables {Zil;Yil} (to be defined later). However,

the proposed DA algorithm in his paper did not provide the necessary individual

conditional distributions of Yil given (the observed data and) the updated parameter estimates. Indeed, the EM algorithm developed in his paper is based on the

assumption that{ Yil} are fixed given values. Fortunately, the EM algorithm is

invariant under any choice of the value of Yil, so the final result is always correct.

We have derived the aforesaid conditional distributions and hence provide a valid

DA algorithm. A real data set is used for illustration.