On an Application of the Student Distribution in the Theory of Probability and Mathematical Statistics

Bening, V. E.; Korolev, V. Yu.

doi:10.1137/s0040585x97981159

Cited by 43 publications

(42 citation statements)

References 18 publications

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“…We also note here that other mechanisms leading to the production of infinite support q-Gaussian distributions, more commonly referred to as the Student t, have been known for some time, see for example [12] and [13]. In particular, if samples are obtained from a normal population via a sampling procedure governed by a random process which yields sample sizes which are themselves random, then the resulting sampling distribution of sample means may be pulled away from the Gaussian distribution towards an infinite support q-Gaussian.…”

Section: Discussionmentioning

confidence: 99%

q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations

et al. 2009

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Abstract:We study a strictly scale-invariant probabilistic N-body model with symmetric, uniform, identically distributed random variables. Correlations are induced through a transformation of a multivariate Gaussian distribution with covariance matrix decaying out from the unit diagonal, as ρ/r α for r =1, 2, …, N-1, where r indicates displacement from the diagonal and where 0 ρ 1 and α 0. We show numerically that the sum of the N dependent random variables is well modeled by a compact support q-Gaussian distribution. In the particular case of α = 0 we obtain q = (1-5/3 ρ) / (1-ρ), a result validated analytically in a recent paper by Hilhorst and Schehr. Our present results with these q-Gaussian approximants precisely mimic the behavior expected in the frame of non-extensive statistical mechanics. The fact that the N → ∞ limiting distributions are not exactly, but only approximately, q-Gaussians suggests that the present system is not exactly, but only approximately, q-independent in the sense of the q-generalized central limit theorem of Umarov, Steinberg and Tsallis. Short range interaction (α > 1) and long range interactions (α < 1) are discussed. Fitted parameters are obtained via a Method of Moments approach. Simple mechanisms which lead to the production of q-Gaussians, such as mixing, are discussed.

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Section: Discussionmentioning

confidence: 99%

q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations

et al. 2009

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“…In asymptotic settings, statistics constructed from samples with random sizes are special cases of random sequences with random indices. The randomness of indices usually leads to that the limit distributions for the corresponding random sequences are heavy-tailed even in the situations where the distributions of non-randomly indexed random sequences are asymptotically normal see, e. g., [3,4,9].…”

Section: Convergence Of the Distributions Of Statistics Constructed Fmentioning

confidence: 99%

On Mixture Representations for the Generalized Linnik Distribution and Their Applications in Limit Theorems

2020

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We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, exponential and stable laws and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e. g., random sums of independent random variables with finite variances and maximum random sums, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of geometric (or, in general, negative binomial) summation is far not the only asymptotic setting (even for sums of independent random variables) in which the generalized Linnik law appears as the limit distribution.

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“…If α = 1, then S α,1 ≡ 1 and according to lemma 1 the limit law in theorem 4 turns into that of the r.v. X Z r,1 W −1 1 d = XG −1/2 r,1 , that is, the Student distribution with 2r degrees of freedom (see [2,26]).…”

Section: Limit Theorems For Statistics Constructed From Samples With mentioning

confidence: 99%

Generalized negative binomial distributions as mixed geometric laws and related limit theorems*

Korolev

Zeifman

2019

Lith Math J

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In this paper we study a wide and flexible family of discrete distributions, the socalled generalized negative binomial (GNB) distributions that are mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma (GG) distributions. The latter was introduced by E. W. Stacy as a special family of lifetime distributions containing gamma, exponential power and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena being very convenient and almost universal models for the description of statistical regularities in discrete data. Analytic properties of GNB distributions are studied. A GG distribution is proved to be a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. The mixing distribution is written out explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative halfline. As a corollary, the representation is obtained for the GNB distribution as a mixed geometric distribution. The corresponding scheme of Bernoulli trials with random probability of success is considered. Within this scheme, a random analog of the Poisson theorem is proved establishing the convergence of mixed binomial distributions to mixed Poisson laws. Limit theorems are proved for random sums of independent random variables in which the number of summands has the GNB distribution and the summands have both light-and heavy-tailed distributions. The class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. Various representations for the limit laws are obtained in terms of mixtures of Mittag-Leffler, Linnik or Laplace distributions. Limit theorems are proved establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the GNB distribution to generalized variance gamma distributions. Some applications of GNB distributions in meteorology are discussed.

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On an Application of the Student Distribution in the Theory of Probability and Mathematical Statistics

Cited by 43 publications

References 18 publications

q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations

q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations

On Mixture Representations for the Generalized Linnik Distribution and Their Applications in Limit Theorems

Generalized negative binomial distributions as mixed geometric laws and related limit theorems*

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