Maximum Likelihood Estimation of a Linear Regression Function with Grouped Data

When the data in a linear regression problem come in grouped form, finding the maximum likelihood estimates of the parameters usually calls for considerable computational effort. An alternative to the fully grouped solution is to place the observations at the mid‐points of their groups, and then regard the resultant mid‐points as the observed data. Calculations for this approximate mid‐point (a.m.p.) solution here are very simple, of course. A natural conjecture is that the two kinds of estimate will not differ greatly if there are a large number of groups. We confirm this conjecture here using some data that are jointly normally distributed, though it does seem that a very fine mesh indeed is needed if all parameter estimates are to be tolerably close. But this is not the main aim of the paper. In practice, the data will often be far too coarse to allow us to use the a.m.p. estimate itself and our primary aim, again for normal data, is to show how this simple a.m.p. estimate can be “corrected” with very little effort 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. Finally, we have calculated the grouped maximum likelihood, a.m.p. and corrected a.m.p. estimates and some associated estimated variances for a particular body of data to see how things work out in practice.