Fitting a Straight Line to Data from a Truncated Population

Abstract

It is usually assumed in linear regression theory that for a given value of the regressor variable t, the dependent variable y is distributed normally with expectation a linear function of t, say y N(a + bt, c-) However, situations arise in practice where the distribution of y must necessarily be truncated at a value independent of t. The following example which illustrates this arose in connection with studies of the wearing away of the tooth crowns of Red Deer, carried out by Dr. B. Mitchell of the Nature Conservancy's Speyside Research Station, Aviemore, Scotland. The following simplified model of the wearing process is assumed. Suppose that for all the deer in a given herd, the crowns finish growing at the same age, and that thereafter the rate of wear is the same for all animals, and constant in time. The randomness in the model is due to the fact that crown weights at maturity are distributed normally about their mean with variance o-. When a crown wears away completely, the deer is no longer able to eat, and dies. The exact age at which the crowns mature need not be known, provided all those in the sample are above that age. Consider an individual whose tooth crown weighed ym at maturity, (age t..). At age t (>t7n) it would weigh