MLE of parameters of location-scale distribution for complete and partially grouped data

This article studies the MLEs of parameters of location-scale distribution functions. It gives the necessary and sufficient conditions under which the MLEs of the location and scale parameters uniquely exist with completely grouped data. The results for the completely grouped data further imply that the Pearson–Fisher test is applicable to location-scale families. It also gives sufficient conditions under which the MLEs of the two parameters uniquely exist with partially grouped data. Here, the partially grouped data include complete data, Type-I censored data and others as special cases.