Properties of the Inverse of a Noncentral Wishart Matrix

Hillier, Grant; Kan, Raymond

doi:10.1017/s0266466621000190

Cited by 7 publications

(5 citation statements)

References 33 publications

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“…The matrix quadratic risk (10) of MEM at M = O is (p + 1)I p [33]. Here, we extend this result by using the recent result by [25] on the expectation of the inverse of a noncentral Wishart matrix.…”

Section: B Matrix Quadratic Losssupporting

confidence: 56%

Adapting to General Quadratic Loss via Singular Value Shrinkage

Matsuda

2024

IEEE Trans. Inform. Theory

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The Gaussian sequence model is a canonical model in nonparametric estimation. In this study, we introduce a multivariate version of the Gaussian sequence model and investigate adaptive estimation over the multivariate Sobolev ellipsoids, where adaptation is not only to unknown smoothness but also to arbitrary quadratic loss. First, we derive an oracle inequality for the singular value shrinkage estimator by Efron and Morris, which is a matrix generalization of the James-Stein estimator. Next, we develop an asymptotically minimax estimator on the multivariate Sobolev ellipsoid for each quadratic loss, which can be viewed as a generalization of Pinsker's theorem. Then, we show that the blockwise Efron-Morris estimator is exactly adaptive minimax over the multivariate Sobolev ellipsoids under the corresponding quadratic loss. It attains sharp adaptive estimation of any linear combination of the mean sequences simultaneously.

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Section: B Matrix Quadratic Losssupporting

confidence: 56%

Adapting to General Quadratic Loss via Singular Value Shrinkage

Matsuda

2024

IEEE Trans. Inform. Theory

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“…The matrix quadratic risk ( 9) of MEM at M = O is (p+1)I p [21]. Here, we extend this result by using the recent result by [15] on the expectation of the inverse of a noncentral Wishart matrix.…”

Section: Matrix Quadratic Losssupporting

confidence: 55%

“…Now, we derive an asymptotically minimax estimator on the multivariate Sobolev ellipsoid (15), which can be viewed as an extension of Pinsker's theorem (Proposition 2.1). Several technical lemmas are given in the Appendix.…”

Section: Asymptotically Minimax Estimator On Multivariate Sobolev Ell...mentioning

confidence: 99%

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Adapting to arbitrary quadratic loss via singular value shrinkage

Matsuda¹

2022

Preprint

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The Gaussian sequence model is a canonical model in nonparametric estimation. In this study, we introduce a multivariate version of the Gaussian sequence model and investigate adaptive estimation over the multivariate Sobolev ellipsoids, where adaptation is not only to unknown smoothness and scale but also to arbitrary quadratic loss. First, we derive an oracle inequality for the singular value shrinkage estimator by Efron and Morris, which is a matrix generalization of the James-Stein estimator. Next, we develop an asymptotically minimax estimator on the multivariate Sobolev ellipsoid for each quadratic loss, which can be viewed as a generalization of Pinsker's theorem. Then, we show that the blockwise Efron-Morris estimator is exactly adaptive minimax over the multivariate Sobolev ellipsoids under any quadratic loss. It attains sharp adaptive estimation of any linear combination of the mean sequences simultaneously.

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“…The complex Wishart distribution frequently arises in multivariate analysis as distributions of complex random matrices (for example, see Gupta and Nagar [21]) and hence plays a pivotal role in various branches of science and engineering. For properties and different variations of the Wishart distribution the reader is referred to James [2], Nagar, Gupta and Sánchez [22], Latec and Massam [23], Nagar, Roldán-Correa, and Gupta [24], Di Nardo [25], Dharmawansa and McKay [26], Hillier and Can [27], and Tralli and Conti [19].…”

Section: Introductionmentioning

confidence: 99%

Expected Values of Scalar-Valued Functions of a Complex Wishart Matrix

2023

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The complex Wishart distribution has ample applications in science and engineering. In this paper, we give explicit expressions for E(tr(Wh))g(tr(Wj))i and E(tr(W−h))g(tr(W−j))i, respectively, for particular values of g, h, i, j, g+h+i+j≤5, where W follows a complex Wishart distribution. For specific values of g, h, i, j, we first write (tr(Wh))g(tr(Wj))i and (tr(W−h))g(tr(W−j))i in terms of zonal polynomials and then by using results on integration evaluate resulting expressions. Several expected values of matrix-valued functions of a complex Wishart matrix have also been derived.

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Properties of the Inverse of a Noncentral Wishart Matrix

Cited by 7 publications

References 33 publications

Adapting to General Quadratic Loss via Singular Value Shrinkage

Adapting to General Quadratic Loss via Singular Value Shrinkage

Adapting to arbitrary quadratic loss via singular value shrinkage

Expected Values of Scalar-Valued Functions of a Complex Wishart Matrix

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