This Chapter considers SP estimation for tests whose distribution function of the generic test statistic, i.e. Gm,λm, is a Gaussian with mean λm and unitary variance: Gm,λm = Φλm,1.
Two tests are studied, whose actual test statistic is approximately Gaussian distributed. The first test concerns the comparison between two proportions from different populations, which is sometimes called the “large sample test for proportions”. Then, a problem in survival analysis is discussed, considering the so-called “log-rank test”, which compares two survival curves.
It is interesting to compare the proportions p1 and p2 of a certain feature in two different populations. The outcomes are of the yes/no type and if they are coded as 1/0, then the elements of the two samples Xij have a Bernoullian distribution with parameters pi, i = 1, 2. In other words, Xij have distributions tFi = Ber(pi), where P(Xij = 1) = pi. The null hypothesis considered here is that of no difference between proportions: H0 : p1 = p2, and the test for the one-sided alternative H1 : p1 > p2 is developed.
The sample frequencies form the basis when building the test statistic, which is the standardized difference between them. Since V ar(i,mi) = pi(1 − pi)/mi, Tm results:
Noting that the sample frequencies can be viewed as sample ...