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In statistics, the topic of location testing for Gaussian scale mixture distributions arises in some particular types of situations where the more standard Student's t-test is inapplicable. Specifically, these cases allow tests of location to be made where the assumption that sample observations arise from populations having a normal distribution can be replaced by the assumption that they arise from a Gaussian scale mixture distribution. The class of Gaussian scale mixture distributions contains all symmetric stable distributions, Laplace distributions, logistic distributions, and exponential power distributions, etc.
Introduce
the counterpart of Student's t-distribution for Gaussian scale mixtures. This means that if we test the null hypothesis that the center of a Gaussian scale mixture distribution is 0, say, then tn is the infimum of all monotone nondecreasing functions u ≥ 1/2, x ≥ 0 such that if the critical values of the test are u, then the significance level is at most α ≥ 1/2 for all Gaussian scale mixture distributions = 1 − tn,for x < 0]. An explicit formula for tn, is given in the papers in the references in terms of Student’s t-distributions, tk, k = 1, 2, …, n. Introduce
the Gaussian scale mixture counterpart of the standard normal cumulative distribution function, Φ.