Notes on Counting Process in Survival Analysis

Distribution of some statistics, especially rank based, with censored data are hard to get. Even if they can be worked out for small samples, they are likely to be prohibitively expensive for larger sample sizes. Therefore we need to work on the asymptotic distribution which often works well for moderate and large sample sizes. This notes is intended to be read along with Fleming and Harrington’s book Counting Processes and Survival Analysis. It is not intended as a rigorous treatment of the subject of counting process martingale. The aim is to (1) present intuitions to help visualize the counting process and (2) supply simplified proofs (in special cases, or with more assumptions, perhaps), make the book easier to read and (3) select the materials that can be realistically covered in one semester or even one quarter. The students need only to have had background in Poisson process and basic probability/statistics. Some prior knowledge of survival analysis will help understand the example/application of counting process in the survival analysis. There is an equally good book on the counting processes: by Andersen, Borgan, Keiding, Gill Statistical Models Based on Counting Processes, Springer 1993 This book contains many more examples of application. The book also has more materials then can be covered in one semester. At a math level similar to FH book. Chapter 5 of Kalbflisch and Prentice, second edition is similar to this note. Review of Master level Survival Analysis. Review of central limit theorem (Linderberg condition) and Poisson process (outlined below).