In the methods and approaches developed for estimating the operating characteristics of the discrete item responses, the maximum likelihood estimate of the examinee based upon the "Old Test" has an important role. When Old Test does not provide a sufficient amount of test information for the upper and lower part of the ability interval, however, it is likely that the response patterns obtained are either all lowest item scores or all highest item scores, and the resultant maximum likelihood estimates are negative or positive infinity. Although such a test is undesirable to use as the Old Test, to a certain extent the situation can be salvaged by providing some alternative estimator for these two extreme response patterns; in the present paper, such an estimator is proposed and discussed. (Author/GK)