Let N(t) be a counting process that satisfies the multiplicative intensity model λ(t) = α(t)Y(t), and consider the null hypothesis   where α0(t) is a known function. In this exercise we derive a test...

Let N(t) be a counting process that satisfies the multiplicative intensity model λ(t) = α(t)Y(t), and consider the null hypothesis

where α0(t) is a known function. In this exercise we derive a test for the null hypothesis that has good properties for alternatives of the form
  or

a) Let
  and introduce

  Show that
  is a mean zero martingale when the null hypothesis holds true.

b) Find an expression for the predictable variation process of
  when the null hypothesis holds true.

Consider the test statistic

where L(t) is a nonnegative predictable “weight process” that takes the value 0 whenever Y(t) = 0.

c) Show that Z(t₀) is a mean zero martingale when the null hypothesis holds true (when considered as a process in t₀), and explain why it is reasonable to use Z(t₀) as a test statistic.

d) Show that

when the null hypothesis holds true, and explain why this is an unbiased estimator for the variance of Z(t₀) under the null hypothesis.

e) Explain briefly why the standardized test statistic

is approximately standard normally distributed when the null hypothesis holds true.

One possible choice of weight process is L(t) = Y(t). The test then obtained is usually denoted the one-sample log-rank test.

f) Show that, for the one-sample log-rank test, we have Z(t0) = N(t₀)−E(t₀),  where
  Explain why E(t₀) may be interpreted as (an estimate of) the expected number of events under the null hypothesis.

g) Show that the standardized version of the one-sample log-rank statistic takes the form