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Azmoodeh E, Gasbarra D, Gaunt RE. On algebraic Stein operators for Gaussian polynomials. BERNOULLI 2023. [DOI: 10.3150/22-bej1460] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Affiliation(s)
- Ehsan Azmoodeh
- Department of Mathematical Sciences, Mathematical Sciences Bldg, The University of Liverpool, Liverpool L69 7ZL, UK
| | - Dario Gasbarra
- Department of Mathematics and Statistics, University of Helsinki, P.O.Box 68 00014, Finland
| | - Robert E. Gaunt
- Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
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2
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An asymptotic approach to proving sufficiency of Stein characterisations. LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS 2023. [DOI: 10.30757/alea.v20-06] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/17/2023]
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3
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Gong F, Sun X. On the characterization of Brownian bridge measure on the pinned path space over a compact Riemannian manifold. BERNOULLI 2022. [DOI: 10.3150/21-bej1420] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Fuzhou Gong
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
| | - Xiaoxia Sun
- School of Data Science and Artificial Intelligence, Dongbei University of Finance and Economics, Dalian, China
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4
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Barman K, Upadhye NS. On Brascamp–Lieb and Poincaré type inequalities for generalized tempered stable distribution. Stat Probab Lett 2022. [DOI: 10.1016/j.spl.2022.109600] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
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5
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Bounds for the chi-square approximation of the power divergence family of statistics. J Appl Probab 2022. [DOI: 10.1017/jpr.2022.7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
Abstract
Abstract
It is well known that each statistic in the family of power divergence statistics, across n trials and r classifications with index parameter
$\lambda\in\mathbb{R}$
(the Pearson, likelihood ratio, and Freeman–Tukey statistics correspond to
$\lambda=1,0,-1/2$
, respectively), is asymptotically chi-square distributed as the sample size tends to infinity. We obtain explicit bounds on this distributional approximation, measured using smooth test functions, that hold for a given finite sample n, and all index parameters (
$\lambda>-1$
) for which such finite-sample bounds are meaningful. We obtain bounds that are of the optimal order
$n^{-1}$
. The dependence of our bounds on the index parameter
$\lambda$
and the cell classification probabilities is also optimal, and the dependence on the number of cells is also respectable. Our bounds generalise, complement, and improve on recent results from the literature.
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6
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Asymptotic Normality in Linear Regression with Approximately Sparse Structure. MATHEMATICS 2022. [DOI: 10.3390/math10101657] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
Abstract
In this paper, we study the asymptotic normality in high-dimensional linear regression. We focus on the case where the covariance matrix of the regression variables has a KMS structure, in asymptotic settings where the number of predictors, p, is proportional to the number of observations, n. The main result of the paper is the derivation of the exact asymptotic distribution for the suitably centered and normalized squared norm of the product between predictor matrix, X, and outcome variable, Y, i.e., the statistic ∥X′Y∥22, under rather unrestrictive assumptions for the model parameters βj. We employ variance-gamma distribution in order to derive the results, which, along with the asymptotic results, allows us to easily define the exact distribution of the statistic. Additionally, we consider a specific case of approximate sparsity of the model parameter vector β and perform a Monte Carlo simulation study. The simulation results suggest that the statistic approaches the limiting distribution fairly quickly even under high variable multi-correlation and relatively small number of observations, suggesting possible applications to the construction of statistical testing procedures for the real-world data and related problems.
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Mukhopadhyay N. On Rereading Stein’s Lemma: Its Intrinsic Connection with Cramér-Rao Identity and Some New Identities. Methodol Comput Appl Probab 2021. [DOI: 10.1007/s11009-020-09830-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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8
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Gaunt RE, Mijoule G, Swan Y. Some new Stein operators for product distributions. BRAZ J PROBAB STAT 2020. [DOI: 10.1214/19-bjps460] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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9
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Arras B, Azmoodeh E, Poly G, Swan Y. Stein characterizations for linear combinations of gamma random variables. BRAZ J PROBAB STAT 2020. [DOI: 10.1214/18-bjps420] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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10
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Gaunt RE. Stein’s method for functions of multivariate normal random variables. ANNALES DE L'INSTITUT HENRI POINCARÉ, PROBABILITÉS ET STATISTIQUES 2020. [DOI: 10.1214/19-aihp1011] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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11
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Gaunt RE. Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I. J THEOR PROBAB 2018. [DOI: 10.1007/s10959-018-0867-4] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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13
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Gaunt RE. Products of normal, beta and gamma random variables: Stein operators and distributional theory. BRAZ J PROBAB STAT 2018. [DOI: 10.1214/16-bjps349] [Citation(s) in RCA: 20] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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14
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Gaunt RE. On Stein’s method for products of normal random variables and zero bias couplings. BERNOULLI 2017. [DOI: 10.3150/16-bej848] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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16
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Gaunt RE, Pickett AM, Reinert G. Chi-square approximation by Stein’s method with application to Pearson’s statistic. ANN APPL PROBAB 2017. [DOI: 10.1214/16-aap1213] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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17
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Bai S, Taqqu MS. Behavior of the generalized Rosenblatt process at extreme critical exponent values. ANN PROBAB 2017. [DOI: 10.1214/15-aop1087] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Ley C, Reinert G, Swan Y. Distances between nested densities and a measure of the impact of the prior in Bayesian statistics. ANN APPL PROBAB 2017. [DOI: 10.1214/16-aap1202] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Döbler C, Gaunt RE, Vollmer SJ. An iterative technique for bounding derivatives of solutions of Stein equations. ELECTRON J PROBAB 2017. [DOI: 10.1214/17-ejp118] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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20
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Ley C, Reinert G, Swan Y. Stein’s method for comparison of univariate distributions. PROBABILITY SURVEYS 2017. [DOI: 10.1214/16-ps278] [Citation(s) in RCA: 53] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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21
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Döbler C. Stein's method of exchangeable pairs for the Beta distribution and generalizations. ELECTRON J PROBAB 2015. [DOI: 10.1214/ejp.v20-3933] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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22
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