Suppose that in a life-testing situation the failure of an individual can be classified into one of k( >1) mutually exclusive classes, usually causes of failure. It is assumed that associated with each cause of failure there is a characteristic life distribution belonging to a specific class of distributions. A general likelihood function is obtained which allows for dependence of the causes, for both censored and uncensored data. The method of maximum likelihood is used to estimate the parameters when the underlying life distributions are independent Weibulls with equal shape constants (the exponential being a special case) and independent Weibulls with unequal shape constants. A study involving 121 patients treated for cancer of the breast by surgery and/or X-ray therapy is analyzed, and from the estimates of the parameters of the underlying life distributions (assumed to be Weibull with unequal shape constants) the relevant probabilities in competing risk theory are obtained.
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Melvin L. Moeschberger (1971) studied this question.
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