Finite-Sample Inference for Exponential Competing Risks Under Combined Type-II Generalized Progressive Hybrid Censoring

Abstract

Time-limited life tests may terminate before both competing failure causes are observed. We derive the finite-sample distributions of ordinary maximum likelihood estimators (MLEs) for two independent exponential risks under combined Type-II generalized progressive hybrid censoring, conditional on observing both causes. Count-indexed transforms of the total time on test yield matrix-exponential and finite signed shifted-gamma representations for the conditional distributions and moments. The analysis also gives the probability that both scale estimates are finite. A numerical study examines cause imbalance, withdrawal timing, and time thresholds, supplemented by comparisons with conditional-likelihood and Bayesian estimators. The comparisons show design-dependent changes in relative performance and frequent numerical boundary candidates for conditional likelihood under severe cause imbalance. Additional calculations distinguish sensitivity to the total rate at fixed time limits from scale equivariance. A retrospective appliance example illustrates likelihood fitting, model diagnostics, and screening of candidate test plans. The observed time–cause association limits its interpretation to working-model calculations. Together, the results show how exact finite-sample laws can assess estimator performance and identify designs that fail specified availability or precision criteria.