Introduction

Left censoring occurs in a duration model when a statistician observes only those spells which either are in the middle of continuation at the time of the first observation or start during the observation period. It is assumed that the statistician has no record of those spells which had ended by the time of the first observation. A special treatment of the problem is necessary because ignoring left censoring will overestimate the mean duration as longer spells tend to be observed more frequently than shorter spells. This is called selectivity bias.

Different cases of left censoring arise depending on the following considerations: (1) Spells in the middle of continuation at the time of the first observation are either completely or partially observed. Suppose such a spell started at s, continued on to 0 (the time of the first observation), and ended at t. The statistician may observe only s (by asking how long the spell had lasted), only t, or both. (2) Spells which start after the time of the first observation are either observed or not observed. (3) For a single individual we either observe a single spell or a sequence of spells in different states.

In each possible case we will consider how the selectivity bias is eliminated. If the model is fully specified, this is accomplished by the method of maximum likelihood estimation, which is fully efficient.