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Location testing for Gaussian scale mixture distributions
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<p>In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the topic of location testing for Gaussian scale mixture distributions arises in some particular types of situations where the more standard <a href="/facts/Student%27s_t-test/dMH8uC28">Student's t-test</a> is inapplicable. Specifically, these cases allow <a href="/facts/Location_test/9botwshp">tests of location</a> to be made where the assumption that sample observations arise from populations having a <a href="/facts/Normal_distribution/UapjjPyQ">normal distribution</a> 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 <a href="/facts/Stable_distribution/rvtRmhlz">stable distributions</a>, <a href="/facts/Laplace_distribution/teMwaLry">Laplace distributions</a>, <a href="/facts/Logistic_distribution/QlQNt0uC">logistic distributions</a>, and exponential power distributions, etc. </p><p>Introduce </p> <i>t</i>G<i>n</i>(<i>x</i>), <p>the counterpart of <a href="/facts/Student%27s_t-distribution/DeT1SDqH">Student's t-distribution</a> 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 <i>t</i><i>n</i><i>G</i>(<i>x</i>) (<i>x</i> ≥ 0) is the <a href="/facts/Infimum/oXCKVopz">infimum</a> of all monotone nondecreasing functions <i>u</i>(<i>x</i>) ≥ 1/2, <i>x</i> ≥ 0 such that if the critical values of the test are <i>u</i>−1(1 − <i>α</i>), then the <a href="/facts/Significance_level/PMxjdlAu">significance level</a> is at most <i>α</i> ≥ 1/2 for all Gaussian scale mixture distributions [<i>t</i><i>G</i><i>n</i>(x) = 1 − <i>t</i><i>G</i><i>n</i>(−<i>x</i>),for <i>x</i> < 0]. An explicit formula for <i>t</i><i>G</i><i>n</i>(<i>x</i>), is given in the papers in the references in terms of <a href="/facts/Student%E2%80%99s_t-distribution/DeT1SDqH">Student’s t-distributions</a>, <i>t</i><i>k</i>, <i>k</i> = 1, 2, ..., <i>n</i>. Introduce </p> Φ<i>G</i>(<i>x</i>):= lim<i>n</i> → ∞ <i>t</i><i>G</i><i>n</i>(<i>x</i>), <p>the Gaussian scale mixture counterpart of the standard normal <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a>, Φ(x). </p><p>Theorem. Φ<i>G</i>(<i>x</i>) = 1/2 for 0 ≤ <i>x</i> < 1, Φ<i>G</i>(1) = 3/4, Φ<i>G</i>(<i>x</i>) = <i>C</i>(<i>x</i>/(2 − <i>x</i>2)1/2) for quantiles between 1/2 and 0.875, where <i>C</i>(<i>x</i>) is the standard <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy cumulative distribution function</a>. This is the convex part of the curve Φ<i>G</i>(<i>x</i>), <i>x</i> ≥ 0 which is followed by a linear section Φ<i>G</i>(<i>x</i>) = <i>x</i>/(2√3) + 1/2 for 1.3136... < <i>x</i> < 1.4282... Thus the 90% quantile is exactly 4√3/5. Most importantly, </p> ΦG(<i>x</i>) = Φ(<i>x</i>) for <i>x</i> ≥ √3. <p>Note that Φ(√3) = 0.958..., thus the classical 95% confidence interval for the unknown expected value of Gaussian distributions covers the center of symmetry with at least 95% probability for Gaussian scale mixture distributions. On the other hand, the 90% quantile of ΦG(<i>x</i>) is 4√3/5 = 1.385... > Φ−1(0.9) = 1.282... The following critical values are important in applications: 0.95 = Φ(1.645) = ΦG(1.651), and 0.9 = Φ(1.282) = ΦG(1.386). </p><p>For the extension of the Theorem to all symmetric <a href="/facts/Unimodal_distribution/Xo3bhgeE">unimodal distributions</a> one can start with a classical result of <a href="/facts/Aleksandr_Khinchin/QThgstum">Aleksandr Khinchin</a>: namely that all symmetric unimodal distributions are scale mixtures of symmetric uniform distributions. </p>