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Register a new userThe exact distribution of the sample variance from bounded continuous random variables
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Added by
Thomas Royen
Publication date
2008
fields
Mathematical Statistics
and research's language is
English
Authors
T. Royen
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For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial or a trigonometrical polynomial the coefficients of this series are simple finite terms containing only the error function, the exponential function and powers. In more general cases - e.g. for all beta densities - the coefficients are given by some series expansions. The method is generalized to positive semi-definite quadratic forms of bounded independent but not necessarily identically distributed random variables if the form matrix differs from a diagonal matrix D > 0 only by a matrix of rank 1
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Several representations of the exact cdf of the sum of squares of n independent gamma-distributed random variables Xi are given, in particular by a series of gamma distribution functions. Using a characterization of the gamma distribution by Laha, an expansion of the exact distribution of the sample variance is derived by a Taylor series approach with the former distribution as its leading term. In particular for integer orders alpha some further series are provided, including a convex combination of gamma distributions for alpha = 1 and nearly of this type for alpha > 1. Furthermore, some representations of the distribution of the angle Phi between (X1,...,Xn) and (1,...,1) are given by orthogonal series. All these series are based on the same sequence of easily computed moments of cos(Phi).
From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for the cumulative distribution function of the sample variance of i.i.d. random variables with a gamma density, supplementing former formulas of the author. Furthermore, from the above Laplace transform Fourier series are obtained for the density and the distribution function of the sample variance of i.i.d. random variables with a uniform distribution. This distribution can be applied e.g. to a statistical test for a scale parameter.
The exact distribution of the square sum of Dirichlet random variables is given by two different univariate integral representations. Alternatively, three representations by orthogonal series with Jacobi or Legendre polynomials are derived. As a special case the distribution of the square sum of spacings - also called Greenwoods statistic - is obtained. Nine quantiles of this statistic are tabulated with eight digits where the number of squares ranges from 10 to 100.
We apply the holonomic gradient method to compute the distribution function of a weighted sum of independent noncentral chi-square random variables. It is the distribution function of the squared length of a multivariate normal random vector. We treat this distribution as an integral of the normalizing constant of the Fisher-Bingham distribution on the unit sphere and make use of the partial differential equations for the Fisher-Bingham distribution.
Many inference problems, such as sequential decision problems like A/B testing, adaptive sampling schemes like bandit selection, are often online in nature. The fundamental problem for online inference is to provide a sequence of confidence intervals that are valid uniformly over the growing-into-infinity sample sizes. To address this question, we provide a near-optimal confidence sequence for bounded random variables by utilizing Bentkus concentration results. We show that it improves on the existing approaches that use the Cram{e}r-Chernoff technique such as the Hoeffding, Bernstein, and Bennett inequalities. The resulting confidence sequence is confirmed to be favorable in both synthetic coverage problems and an application to adaptive stopping algorithms.
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