This Chapter considers SP estimation for tests whose distribution function of the generic test statistic, i.e. *G*_{m,λm}, is a Gaussian with mean λ_{m} and unitary variance: *G*_{m,λm} = Φ_{λm,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 *p*_{1} and *p*_{2} 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 *X*_{ij} have a Bernoullian distribution with parameters *p*_{i}, *i* = 1, 2. In other words, *X*_{ij} have distributions _{t}*F*_{i} = *Ber*(*p*_{i}), where *P*(*X*_{ij} = 1) = *p*_{i}. The null hypothesis considered here is that of no difference between proportions: *H*_{0} : *p*_{1} = *p*_{2}, and the test for the one-sided alternative *H*_{1} : *p*_{1} > *p*_{2} 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}) = *p*_{i}(1 − *p*_{i})/*m*_{i}, *T*_{m} results:

Noting that the sample frequencies can be viewed as sample ...

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