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Register a new userExact distribution of the sample variance from a gamma parent distribution
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Added by
Thomas Royen
Publication date
2007
fields
Mathematical Statistics
and research's language is
English
Authors
Thomas Royen
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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).
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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.
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
A sum of observations derived by a simple random sampling design from a population of independent random variables is studied. A procedure finding a general term of Edgeworth asymptotic expansion is presented. The Lindeberg condition of asymptotic normality, Berry-Esseen bound, Edgeworth asymptotic expansions under weakened conditions and Cramer type large deviation results are derived.
Herein, we review the properties of the Amoroso distribution, the natural unification of the gamma and extreme value distribution families. Over 50 distinct, named distributions (and twice as many synonyms) occur as special cases or limiting forms. Consequently, this single simple functional form encapsulates and systematizes an extensive menagerie of interesting and common probability distributions.
Data observed at high sampling frequency are typically assumed to be an additive composite of a relatively slow-varying continuous-time component, a latent stochastic process or a smooth random function, and measurement error. Supposing that the latent component is an It^{o} diffusion process, we propose to estimate the measurement error density function by applying a deconvolution technique with appropriate localization. Our estimator, which does not require equally-spaced observed times, is consistent and minimax rate optimal. We also investigate estimators of the moments of the error distribution and their properties, propose a frequency domain estimator for the integrated volatility of the underlying stochastic process, and show that it achieves the optimal convergence rate. Simulations and a real data analysis validate our analysis.
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