On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

Imori, Shinpei; Rosen, Dietrich von

doi:10.13001/ela.2020.5091

Cited by 5 publications

(6 citation statements)

References 10 publications

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“…Proof. In accordance with page 130 in [28] where Lemma A3 (i) has been applied in the last equality. On the other hand, if we instead apply the inequality in Lemma 2.4 (ii) of [28], we obtain E[(α S + x) 2 ] ≤ c(n, p) −1 (λ 1 ( −1 )) 4 (2c 1 + c 2 )(α α)(x x) g(L)dL = (λ 1 ( −1 )) 4 (2c 1 + c 2 )(α α)(x x)…”

Section: Lemma A2 the Elements Of E[s + ] Have The Bounds Formentioning

confidence: 82%

“…Further, define g(L) = n i=1 |L i | + and c(n, p) = (2π) np/2 2 n s(n, p), where |L i | + and s(n, p) are defined as on pages 128 and 129 in [28].…”

Section: Lemma A2 the Elements Of E[s + ] Have The Bounds Formentioning

confidence: 99%

“…Unfortunately, there exist no derivation of the expected value or variance of S + , when p > N. In [21] however, these quantities are presented in the special case of = I p . The authors also provided approximate results, using moments of standard normal random variables, and exact results for moments of the generalized reflexive inverse, another quantity that can be applied as an inverse of S. Further, in a recent paper [28], several bounds on the mean and variance of S + are provided, based on the Poincaré separation theorem. Our paper builds on the results presented in [21] and [28] to provide bounds and approximations for the moments of the TP weights, E[ wT P ] and V[ wT P ], where E[•] and V[•] denote the expected value and variance, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…The authors also provided approximate results, using moments of standard normal random variables, and exact results for moments of the generalized reflexive inverse, another quantity that can be applied as an inverse of S. Further, in a recent paper [28], several bounds on the mean and variance of S + are provided, based on the Poincaré separation theorem. Our paper builds on the results presented in [21] and [28] to provide bounds and approximations for the moments of the TP weights, E[ wT P ] and V[ wT P ], where E[•] and V[•] denote the expected value and variance, respectively. We also present a simulation study, where various measures compare the derived bounds with the equivalent sample quantities obtained from simulated data.…”

Section: Introductionmentioning

confidence: 99%

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On the mean and variance of the estimated tangency portfolio weights for small samples

Alfelt¹,

Mazur²

2022

Modern Stochastics: Theory and Applications

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In this paper, a sample estimator of the tangency portfolio (TP) weights is considered. The focus is on the situation where the number of observations is smaller than the number of assets in the portfolio and the returns are i.i.d. normally distributed. Under these assumptions, the sample covariance matrix follows a singular Wishart distribution and, therefore, the regular inverse cannot be taken. In the paper, bounds and approximations for the first two moments of the estimated TP weights are derived, as well as exact results are obtained when the population covariance matrix is equal to the identity matrix, employing the Moore–Penrose inverse. Moreover, exact moments based on the reflexive generalized inverse are provided. The properties of the bounds are investigated in a simulation study, where they are compared to the sample moments. The difference between the moments based on the reflexive generalized inverse and the sample moments based on the Moore–Penrose inverse is also studied.

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Section: Lemma A2 the Elements Of E[s + ] Have The Bounds Formentioning

confidence: 82%

“…Further, define g(L) = n i=1 |L i | + and c(n, p) = (2π) np/2 2 n s(n, p), where |L i | + and s(n, p) are defined as on pages 128 and 129 in [28].…”

Section: Lemma A2 the Elements Of E[s + ] Have The Bounds Formentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the mean and variance of the estimated tangency portfolio weights for small samples

Alfelt¹,

Mazur²

2022

Modern Stochastics: Theory and Applications

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show abstract

“…cf. e.g., [43] (and similarly for B * ). Inserting this into (22) and combining with ( 19) and ( 21) yields the result.…”

Section: B1 Proof Of Lemma 10mentioning

confidence: 97%

k-Sliced Mutual Information: A Quantitative Study of Scalability with Dimension

Goldfeld¹,

Greenewald²,

Nuradha³

et al. 2022

Preprint

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Sliced mutual information (SMI) is defined as an average of mutual information (MI) terms between one-dimensional random projections of the random variables. It serves as a surrogate measure of dependence to classic MI that preserves many of its properties but is more scalable to high dimensions. However, a quantitative characterization of how SMI itself and estimation rates thereof depend on the ambient dimension, which is crucial to the understanding of scalability, remain obscure. This works extends the original SMI definition to k-SMI, which considers projections to k-dimensional subspaces, and provides a multifaceted account on its dependence on dimension. Using a new result on the continuity of differential entropy in the 2-Wasserstein metric, we derive sharp bounds on the error of Monte Carlo (MC)-based estimates of k-SMI, with explicit dependence on k and the ambient dimension, revealing their interplay with the number of samples. We then combine the MC integrator with the neural estimation framework to provide an end-to-end k-SMI estimator, for which optimal convergence rates are established. We also explore asymptotics of the population k-SMI as dimension grows, providing Gaussian approximation results with a residual that decays under appropriate moment bounds. Our theory is validated with numerical experiments and is applied to sliced InfoGAN, which altogether provide a comprehensive quantitative account of the scalability question of k-SMI, including SMI as a special case when k = 1.

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Spectral Analysis of Large Reflexive Generalized Inverse and Moore-Penrose Inverse Matrices

Bodnar¹,

Parolya²

2020

Recent Developments in Multivariate and Random Matrix Analysis

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On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

Cited by 5 publications

References 10 publications

On the mean and variance of the estimated tangency portfolio weights for small samples

On the mean and variance of the estimated tangency portfolio weights for small samples

k-Sliced Mutual Information: A Quantitative Study of Scalability with Dimension

Spectral Analysis of Large Reflexive Generalized Inverse and Moore-Penrose Inverse Matrices

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