Prevalent cohort studies are commonly conducted in many areas of research when incident cohort studies are deemed infeasible due to logistic or other constraints. While such studies are cost effective, it is known that survival data collected on prevalent cases do not form a representative sample from the target population. When the incidence (e.g. onset of disease) arise from a stationary Poisson process, it allows developing a more efficient methodology. While the stationarity assumption holds in many applications, to the best of our knowledge, the problem of establishing uniform confidence bands using data arisen in such settings has not been addressed in the current literature. We devise a method for obtaining uniform confidence bands for the cumulative hazard and the survival function built on their nonparametric maximum likelihood estimators (NPMLEs). To attain this objective, we first present results on uniform strong consistency, weak convergence and asymptotic efficiency of the NPMLE of the cumulative hazard function. Given the intractable forms of the limiting processes in this case, the idea is to numerically approximate the functionals of the asymptotic processes of the normalized NPMLEs. Our simulation studies reveal the efficiency of the estimators for finite samples. The proposed procedures are illustrated using a set of real data on patients with dementia from the Canadian Study of Health and Aging.

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