Maximum Likelihood Estimation of a Polynomial Regression Function with Grouped Data

Abstract

When the data in a polynomial regression problem come in grouped form, finding the maximum likelihood estimates of the parameters usually calls for considerable computational effort. One alternative to the fully grouped solution is to place the observations at the mid‐points of their groups, and then treat the resultant mid‐points as the observed data. Evaluation of this approximate mid‐point (a.m.p.) regression is straightforward. The two kinds of estimates (i.e. a.m.p. and grouped) should not differ greatly if there are a large number of groups. In practice, the data will often be coarsely grouped and it is advisable to regard the a.m.p. estimate as unsatisfactory. Simple corrections can be made to this estimate to bring it back into line with the fully grouped maximum likelihood estimate. Consideration is also given to the adequate approximation of the estimated variance of the grouped estimate. Calculations have been made to determine the grouped maximum likelihood, a.m.p. and “corrected” a.m.p. estimates and some associated estimated variances for a particular set of data.