Semi‐Parametric Reliability Inference for a Piecewise‐Constant Proportional Hazards Model Based on Censored Competing‐Risks Data

ABSTRACT This paper considers semi‐parametric reliability inference for Type‐I censored competing‐risks data under an independent competing‐risks framework. Existing parametric approaches require the baseline hazard function to be specified as a given parametric family in advance, which may lead to model misspecification, make closed‐form solutions difficult to obtain, and render numerical computation sensitive to the choice of initial values. To alleviate these difficulties, a piecewise‐constant proportional hazards model that does not assume any specific parametric form for the baseline hazard function is adopted. Instead, the baseline hazard function is approximated by a piecewise‐constant hazard function with a given number of cut points. Based on the observed data, the likelihood function of the unknown parameters is derived. The maximum likelihood estimators of the unknown parameters can be obtained in closed form in the unordered case. Under order restrictions, the maximum likelihood estimators are obtained using the pool‐adjacent‐violators algorithm, and the associated Bootstrap confidence intervals are also constructed. In contrast, the Bayesian inferences under Dirichlet–Gamma and ordered Dirichlet–Gamma priors are developed. Since the corresponding Bayes estimators and associated credible intervals cannot be obtained in closed form, the unordered and order‐restricted Bayes estimators, together with the highest posterior density credible intervals, are computed using the importance sampling technique. Simulation results show that the order‐restricted Bayes estimators achieve the smallest MSEs, while the order‐restricted MLEs yield lower MSEs than their unordered counterparts. Order restrictions also reduce interval lengths, with average empirical coverage probabilities remaining close to the nominal 95% level. Finally, a real‐data analysis of solar lighting device lifetime data illustrates the effectiveness of the proposed method.