Closed-form estimation of the exponential scale under two-sided informative random censorship

RESUMO

We estimate the scale of an exponential lifetime from observations censored randomly from both sides, under an informative model in which each censoring law is a power of the lifetime survival function -- a two-sided generalization of the Koziol--Green proportional hazards model. Substituting the empirical distribution function into the likelihood equation yields a closed-form pseudo-maximum-likelihood estimator of the scale that is computed in one pass over the order statistics and needs neither iteration nor numerical optimization. A digamma identity shows the estimating equation to be \emph{exactly}, not merely asymptotically, Fisher-consistent. Strong consistency and asymptotic normality are established with \emph{no restriction on the censoring depth}; the key is an identity that rewrites the underlying $L$-statistic without its unbounded score function and so removes the condition that classical limit theorems would impose. The influence function and the asymptotic variance are explicit in polygamma functions, so confidence intervals require no numerical integration. Against the information bound of the observed-data model the efficiency exceeds $98.6\%$ in the designs considered and stays above $96.2\%$ over a wide sweep. Only the entry half of the model is needed: the estimator remains exactly Fisher-consistent and strongly consistent whatever the right-censoring law, and exponentiality is shown to be the precise price of that freedom. The construction extends verbatim to a proportional hazards class with known baseline. Three data sets illustrate the procedure, including a cohort in which left censoring is genuine.