Multivariate generalized Laplace distribution and related random fields

In this paper, we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix‐variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large‐dimensional asymptotic regime, where the dimension p and the sample size n approach infinity such that p/n→c ∈ [0, + ∞) when the sample covariance matrix does not need to be invertible and p/n→c ∈ [0,1) otherwise.

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“…In the second case, we put

boldν \sim {scriptG scriptA scriptL}_{q} false({boldI}_{q}, {bold1}_{q}, 10 false)

, that is, ν has a q ‐variate generalized asymmetric Laplace distribution (cf. Kozubowski, Podgórski, & Rychlik, ). Moreover, we put q = 10.…”

Section: Numerical Studymentioning

confidence: 99%

Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

Bodnar

Mazur

Parolya

2019

Scandinavian J Statistics

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“…This distribution was introduced by Kotz et al (2001, p. 257). For a more detailed overview, see Kozubowski, Podgórski, and Rychlik (2013).…”

Section: Derivation Of the Marginal Likelihoodmentioning

confidence: 99%

Exact dimensionality selection for Bayesian PCA

Bouveyron

Latouche

Mattei

2019

Scandinavian J Statistics

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We present a Bayesian model selection approach to estimate the intrinsic dimensionality of a high‐dimensional dataset. To this end, we introduce a novel formulation of the probabilisitic principal component analysis model based on a normal‐gamma prior distribution. In this context, we exhibit a closed‐form expression of the marginal likelihood which allows to infer an optimal number of components. We also propose a heuristic based on the expected shape of the marginal likelihood curve in order to choose the hyperparameters. In nonasymptotic frameworks, we show on simulated data that this exact dimensionality selection approach is competitive with both Bayesian and frequentist state‐of‐the‐art methods.

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“…No statistical model known to us captures these behaviors of Y and DY in a unified and consistent manner. A step in that direction is represented by the fractional Laplace motion model of Meerschaert et al [2004] and Kozubowski et al [2006Kozubowski et al [ , 2013, which transitions automatically from heavy-tailed to Gaussian with increasing lag.…”

Section: Introductionmentioning

confidence: 99%

New scaling model for variables and increments with heavy‐tailed distributions

Riva

Neuman

Guadagnini

2015

Water Resources Research

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Many hydrological (as well as diverse earth, environmental, ecological, biological, physical, social, financial and other) variables, Y, exhibit frequency distributions that are difficult to reconcile with those of their spatial or temporal increments, DY. Whereas distributions of Y (or its logarithm) are at times slightly asymmetric with relatively mild peaks and tails, those of DY tend to be symmetric with peaks that grow sharper, and tails that become heavier, as the separation distance (lag) between pairs of Y values decreases. No statistical model known to us captures these behaviors of Y and DY in a unified and consistent manner. We propose a new, generalized sub-Gaussian model that does so. We derive analytical expressions for probability distribution functions (pdfs) of Y and DY as well as corresponding lead statistical moments. In our model the peak and tails of the DY pdf scale with lag in line with observed behavior. The model allows one to estimate, accurately and efficiently, all relevant parameters by analyzing jointly sample moments of Y and DY. We illustrate key features of our new model and method of inference on synthetically generated samples and neutron porosity data from a deep borehole.

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Multivariate generalized Laplace distribution and related random fields

Cited by 67 publications

References 21 publications

Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

Exact dimensionality selection for Bayesian PCA

New scaling model for variables and increments with heavy‐tailed distributions

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