Location testing for Gaussian scale mixture distributions

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 t_n^G(x) (x ≥ 0) is the infimum of all monotone nondecreasing functions u(x) ≥ 1/2, x ≥ 0 such that if the critical values of the test are u⁻¹(1 − α), then the significance level is at most α ≥ 1/2 for all Gaussian scale mixture distributions [t^G_n(x) = 1 − t^G_n(−x),for x < 0]. An explicit formula for t^G_n(x), is given in the papers in the references in terms of Student’s t-distributions, t_k, k = 1, 2, …, n. Introduce

the Gaussian scale mixture counterpart of the standard normal cumulative distribution function, Φ(x).

Theorem. Φ^G(x) = 1/2 for 0 ≤ x < 1, Φ^G(1) = 3/4, Φ^G(x) = C(x/(2 − x²)^1/2) for quantiles between 1/2 and 0.875, where C(x) is the standard Cauchy cumulative distribution function. This is the convex part of the curve Φ^G(x), x ≥ 0 which is followed by a linear section Φ^G(x) = x/(2√3) + 1/2 for 1.3136… < x < 1.4282…. Thus the 90% quantile is exactly 4√3/5. Most importantly,

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