Data from certain life test experiments exhibit such an unusually high concentration of failures near time 0 that the assumption of an over-all exponential density is unwarranted. One hypothesis for this phenomenon is that due to faulty construction or defective parts, certain test items fail prematurely; these items are termed “early failures.” To handle this type of data, an early failure model is postulated in which one failure rate is assumed to be in effect for an initial time interval [0,T0) and another, lower failure rate is operative thereafter. Estimators for the two failure rates are given in the case whereT0is known and in the case whereT0is not known exactly but can be assumed to be within a specified interval. Methods for obtaining approximate large sample confidence regions are outlined and procedures for handling small samples are described.