Exact Finite-Sample Inference for the Exponential Proportional Hazards Model via the Empirical Characteristic Function

Abstract We study estimation of the baseline rate $\theta$ in the exponential proportional hazards model (PHM) under random right censoring, the censoring intensity $\beta$ being an unknown nuisance parameter, using the empirical characteristic function (ECF). We obtain \emph{exact finite-sample formulas} for the bias and variance of the ECF-based estimator $\thetahat=\dbar/\zbar$: for every $n\ge 3$ the bias equals $\theta/(n-1)$ and the variance admits a closed form, both from the exact gamma distribution of the summed censored observations. Being exact for each fixed $n$ rather than to first asymptotic order, they yield exact confidence intervals, an exact bias correction, and an explicit sample-size rule. A single ECF/score-weighted estimating equation reproduces the maximum-likelihood equations on any support. Two extensions go beyond the closed-form regime: for the Weibull PHM with unknown shape we give a consistent, $\sqrt{n}$-asymptotically normal moment system nearly as efficient as maximum likelihood; and for dependent censoring we obtain the exact probability limit of $\thetahat$ under arbitrary dependence. The estimator's coincidence with maximum likelihood and its asymptotic normality are known; the finite-sample formulas, their consequences, and the extensions are new. A closed-form Fisher information computation shows $\thetahat$ attains the parametric Cram\'{e}r--Rao bound for the model with $\beta$ unknown. All formulas are validated by extensive simulation and a real-data application. MSC Classification: 62N02 , 62F10 , 62F12 , 62E15