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We calculate finite sample and asymptotic distributions for the largest censored and uncensored survival times, and some related statistics, from a sample of survival data generated according to an iid censoring model. These statistics are important for assessing whether there is sufficient followup in the sample to be confident of the presence of immune or cured individuals in the population. A key structural result obtained is that, conditional on the value of the largest uncensored survival time, and knowing the number of censored observations exceeding this time, the sample partitions into two independent subsamples, each subsample having the distribution of an iid sample of censored survival times, of reduced size, from truncated random variables. This result provides valuable insight into the construction of censored survival data, and facilitates the calculation of explicit finite sample formulae. We illustrate for distributions of statistics useful for testing for sufficient followup in a sample, and apply extreme value methods to derive asymptotic distributions for some of those., Comment: 34 pages, 9 figures

We derive the large-sample distribution of the number of species in a version of Kingman’s Poisson–Dirichlet model constructed from an $\alpha$ -stable subordinator but with an underlying negative binomial process instead of a Poisson process. Thus it depends on parameters $\alpha\in (0,1)$ from the subordinator and $r>0$ from the negative binomial process. The large-sample distribution of the number of species is derived as sample size $n\to\infty$ . An important component in the derivation is the introduction of a two-parameter version of the Dickman distribution, generalising the existing one-parameter version. Our analysis adds to the range of Poisson–Dirichlet-related distributions available for modeling purposes.

The Dickman function and associated distribution play a prominent role in probabilistic number theory and in the theory of Poisson–Dirichlet distributions. These distributions in turn are closely related to properties of ordered jumps of subordinators. In the present paper we derive some remarkable invariance properties and other identities for the Dickman subordinator, related to the large jumps of the process. The implications of this for size-biased sampling from the Dickman distribution are drawn out.

The $$\text {PD}_\alpha ^{(r)}$$ distribution, a two-parameter distribution for random vectors on the infinite simplex, generalises the $$\text {PD}_\alpha $$ distribution introduced by Kingman, to which it reduces when $$r=0$$ . The parameter $$\alpha \in (0,1)$$ arises from its construction based on ratios of ordered jumps of an $$\alpha $$ -stable subordinator, and the parameter $$r>0$$ signifies its connection with an underlying negative binomial process. Herein, it is shown that other distributions on the simplex, including the Poisson–Dirichlet distribution $$\text {PD}(\theta )$$ , occur as limiting cases of $$\text {PD}_\alpha ^{(r)}$$ , as $$r\rightarrow \infty $$ . As a result, a variety of connections with species and gene sampling models, and many other areas of probability and statistics, are made.

In this article, we consider the situation under a life test, in which the failure time of the test units are not related deterministically to an observable stochastic time varying covariate. In such a case, the joint distribution of failure time and a marker value would be useful for modeling the step stress life test. The problem of accelerating such an experiment is considered as the main aim of this article. We present a step stress accelerated model based on a bivariate Wiener process with one component as the latent (unobservable) degradation process, which determines the failure times and the other as a marker process, the degradation values of which are recorded at times of failure. Parametric inference based on the proposed model is discussed and the optimization procedure for obtaining the optimal time for changing the stress level is presented. The optimization criterion is to minimize the approximate variance of the maximum likelihood estimator of a percentile of the products’ lifetime distrib...

A model is presented in this article based on a bivariate gamma process in which, the first component is latent and determines the failure time and the second represents the marker. This process is a more realistic model for a degradation process. After introducing the model, we obtain failure and survival probability distributions and discuss parametric and predictive inferences based on the Maximum Likelihood method and in a Bayesian setup.