Note on closed-form MLEs of failure rates in a fully parametric random censorship model with incomplete data

Closed-form maximum likelihood estimators (MLEs) of the failure rates in the competing risk model with masked data and an arbitrary number, say r, of exponentially distributed competing causes of failure are unknown at the present. Miyakawa (1984) gives closed-form MLEs for r = 2 and Usher and Hodgson (1988) find closed-form MLEs for r = 3 under certain assumptions on the data. In every other case, the MLEs have to be computed through numerical methods. The object of the present note consists of: (a) giving closed-form MLEs under assumptions that are either weaker or different as compared with the assumptions under which closed-form MLEs are currently known, (b) showing that general closed-form expressions of such estimators for arbitrary data sets do not exist. It is also indicated that the particular conditions under which closed-form estimators are obtained are easily met, at least approximately, in common frameworks.