Location testing for Gaussian scale mixture distributions
Encyclopedia
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
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
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,
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 uniform
distributions.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, 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
Student's t-test
A t-test is any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is supported. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known...
is inapplicable. Specifically, these cases allow tests of location
Location test
A location test is a statistical hypothesis test that compares the location parameter of a statistical population to a given constant, or that compares the location parameters of two statistical populations to each other...
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
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
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
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
, 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
Cauchy distribution
The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...
. 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 uniform
distributions.
Open problem
The counterpart of the Theorem above for the class of all symmetric distributions, or equivalently, for the class of scale mixtures of coin flipping random variables, leads to the following problem:- How many vertices of an n-dimensional unit cubeUnit cubeA unit cube, sometimes called a cube of side 1, is a cube whose sides are 1 unit long. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.- Unit Hypercube :...
can be covered by a sphere with given radius r (and varying center)? Answer this question for all positive integers n and all positive real numbers r. (Certain special cases can be easy to compute.)