# Inference: Functions of Brownian motion

Part of the Statistics for Biology and Health book series (SBH)

Inference based on likelihood and counting processes is treated in the following two chapters and, here, we outline our recommended approach to inference based on functions of Brownian motion. Commonly used tests, in particular those based on partial likelihood, arise as special cases. The basic theory is made possible by virtue of the main theorem and its consequences, described in the previous chapter. The theory itself requires no particular extra effort and is quite classical, being based on a simple application of Donsker’s theorem. Rather than study a relevant process, such as the score process, and show it to look like Brownian motion, we construct such a process so that, by construction alone, we can almost immediately claim to have a key convergence in distribution result. A wide array of tests are possible. The well-known partial likelihood score test obtains as a special case and can even find its justification in this approach. Non-proportional hazards models, partially proportional hazards models, proportional hazards models with intercept and simple proportional hazards models all come under the same heading. Graphical representation of the tests provides a useful additional intuitive tool to inference. Operating characteristics indicating those situations in which certain tests may outperform others are discussed.

## Keywords

Brownian Motion Origin Test Partial Likelihood Brownian Bridge Curve Test