Semi-parametric distributional inference for mean residual life with latent random inspection

In this paper, a semi-parametric distributional inference framework was developed for mean residual life analysis when the inspection age is latent, random, or only indirectly specified through a stochastic monitoring mechanism. Classical mean residual life methods evaluate the expected remaining lifetime at a fixed and fully observed age, an assumption that may be restrictive in reliability, survival, and predictive-maintenance studies where inspection times are irregular, missing, or governed by external monitoring policies. For a lifetime variable T and an independent random inspection age X, the proposed framework targeted the conditional residual-life functional $ (T-X|T\;>X) $, thereby incorporating uncertainty in the observation process directly into lifetime-distribution analysis. The lifetime distribution was estimated nonparametrically, whereas the latent inspection-age distribution was modeled parametrically, leading to a flexible and implementable inference procedure for complete and right-censored lifetime data. Under the exponential inspection model, explicit finite-sum representations were derived, avoiding numerical double integration and enabling direct computation from ordered lifetime observations and Kaplan-Meier survival estimates. The theoretical development established strong consistency and derived the asymptotic normality of the complete-data estimator. A plug-in estimator of the asymptotic variance was also introduced, yielding analytically justified standard errors and large-sample confidence intervals. Simulation studies under Weibull and gamma lifetime distributions demonstrated coherent finite-sample performance across aging regimes, inspection intensities, sample sizes, and censoring proportions. A fixed-age benchmark comparison further showed that replacing the random inspection age by its mean may conceal important distributional effects of monitoring uncertainty. Real-data analyses using kidney catheter survival data, aircraft air-conditioning failure times, and NASA C-MAPSS turbofan degradation lifetimes illustrated the practical value of the proposed methodology in biomedical, engineering, and predictive-maintenance applications. The proposed approach therefore contributes to statistical distribution theory and its applications by extending classical residual-life inference to modern lifetime-data settings in which observation ages are unavailable, irregularly recorded, or operationally uncertain.