On the Behavior of Certain Maximum Likelihood Estimators from Large, Randomly Censored Samples

In the case the survival distribution of a life length X is of the form ln[1 - Fx (x)] = -αQ(βx) for x > 0, where the hazard function Q is known but α or β or both are unknown, the asymptotic distribution of the maximum likelihood estimators of the shape and scale parameters is found for large randomly censored samples. The covariance matrix of is expressed in terms of the distribution of the random variable that can be observed, namely, Y = min(X, T); here T is the random censoring time independent of X. Sufficient conditions are found (such as T having only finite support) to ensure that the estimates are strongly consistent and asymptotically normal. Application is made to decreasing hazard rate distributions.