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• Location_testing_for_Gaussian_scale_mixture_distributions abstract "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 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 tGn(x),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 tnG(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 [tGn(x) = 1 − tGn(−x),for x < 0]. An explicit formula for tGn(x), is given in the papers in the references in terms of Student’s t-distributions, tk, k = 1, 2, …, n. Introduce ΦG(x):= limn → ∞ tGn(x),the Gaussian scale mixture counterpart of the standard normal cumulative distribution function, Φ(x).Theorem. ΦG(x) = 1/2 for 0 ≤ x < 1, ΦG(1) = 3/4, ΦG(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 ΦG(x), x ≥ 0 which is followed by a linear section ΦG(x) = x/(2√3) + 1/2 for 1.3136… < x < 1.4282…. Thus the 90% quantile is exactly 4√3/5. Most importantly, ΦG(x) = Φ(x) for x ≥ √3.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(x) 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).For the extension of the Theorem to all symmetric unimodal distributions one can start with a classical result of Aleksandr Khinchin: namely that all symmetric unimodal distributions are scale mixtures of symmetric uniformdistributions.".
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• Location_testing_for_Gaussian_scale_mixture_distributions subject Category:Statistical_tests.
• Location_testing_for_Gaussian_scale_mixture_distributions type Abstraction100002137.
• Location_testing_for_Gaussian_scale_mixture_distributions type Cognition100023271.
• Location_testing_for_Gaussian_scale_mixture_distributions type Experiment105798043.
• Location_testing_for_Gaussian_scale_mixture_distributions type HigherCognitiveProcess105770664.
• Location_testing_for_Gaussian_scale_mixture_distributions type Inquiry105797597.
• Location_testing_for_Gaussian_scale_mixture_distributions type ProblemSolving105796750.
• Location_testing_for_Gaussian_scale_mixture_distributions type Process105701363.
• Location_testing_for_Gaussian_scale_mixture_distributions type PsychologicalFeature100023100.
• Location_testing_for_Gaussian_scale_mixture_distributions type StatisticalTests.
• Location_testing_for_Gaussian_scale_mixture_distributions type Thinking105770926.
• Location_testing_for_Gaussian_scale_mixture_distributions type Trial105799212.
• Location_testing_for_Gaussian_scale_mixture_distributions comment "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 sample observations arise from populations having a normal distribution can be replaced by the assumption that they arise from a Gaussian scale mixture distribution.".
• Location_testing_for_Gaussian_scale_mixture_distributions label "Location testing for Gaussian scale mixture distributions".
• Location_testing_for_Gaussian_scale_mixture_distributions sameAs m.0cnx8gv.
• Location_testing_for_Gaussian_scale_mixture_distributions sameAs Q6664822.
• Location_testing_for_Gaussian_scale_mixture_distributions sameAs Q6664822.
• Location_testing_for_Gaussian_scale_mixture_distributions sameAs Location_testing_for_Gaussian_scale_mixture_distributions.
• Location_testing_for_Gaussian_scale_mixture_distributions wasDerivedFrom Location_testing_for_Gaussian_scale_mixture_distributions?oldid=597644143.
• Location_testing_for_Gaussian_scale_mixture_distributions isPrimaryTopicOf Location_testing_for_Gaussian_scale_mixture_distributions.