A Parametric Estimation Procedure for Relapse Time Distributions

In a relapse clinical trial patients who have recovered from some recurrent disease (e.g.,ulcer or cancer) are examined at a number of predetermined times. A relapse can be detected either at one of these planned inspections or at a spontaneous visit initiated by the patient because of symptoms. In the first case the observations of the time to relapse, X, is interval-censored by two predetermined time-points. In the second case the upper endpoint of the interval is an observation of the time to symptoms,Y . To model the progression of the disease we use a partially observable Markov process. This approach results in a bivariate phase-type distribution for the joint distribution of (X,Y). It is a flexible model which contains several natural distributions for X, and allows the conditional distributions of the marginals to smoothly depend on each other. To estimate the distributions involved we develop an EM-algorithm. The estimation procedure is evaluated and compared with a non-parametric method in a couple of examples based on simulated data.     

Stochastic Monotonicity and Conditioning in the Limit

Suppose that {( X _n , Y _n )} is a sequence of pairs of cector-valued stochastic variables which converges weakly to ( X , Y ), and that { y _n } converges to y . Sufficient conditions for the conditional distribution of X _n given Y = y are given in terms of stochastic monotonicity. Conditions, which guarantee that also moments of the conditional distributions converge to the moments of the ones of the limit, are also derived.