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In competing risk problems, an identifiable quantity of interest is cumulative incidence function which is the probability of occurrence of a cause-specific event on or before a given point in time in presence of other competing events. For parametric Bayesian estimation of cumulative incidence function, a prior for the concerned parameter may be selected by seeking expert advice or can be elicited using information available in the form of past data. However, if there is a possibility of error in elicitation of an adequate prior for the parameter of concerned risk, one can obtain robust results by selecting a prior in a data-dependent fashion. In this paper, we discuss the elicitation of data-dependent prior in the presence of competing risk data with missing causes and obtain robust Bayes estimates of cumulative incidence function by employing Weibull cause-specific hazard. We provide procedure to construct Weibull cause-specific hazard plot for considered data. We present numerical illustrations using simulated as well as real data sets.

Robust Bayesian estimation of cumulative incidence function for competing risk data with missing causes