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Akimichi Takemura

Publications and source records attributed to Akimichi Takemura.

At least 19 recordsLinked to original sources
arXiv2025

A new perspective on dominating the James-Stein estimator

This paper presents a novel approach to constructing estimators that dominate the classical James-Stein estimator under the quadratic loss for multivariate normal means. Building on Stein's risk representation, we introduce a new sufficient condition involving a monotonicity property of a transformed shrinkage function. We derive a general class of shrinkage estimators that satisfy minimaxity and dominance over the James-Stein estimator, including cases with polynomial or logarithmic convergence to the optimal shrinkage factor. We also provide conditions for uniform dominance across dimensions and for improved asymptotic risk performance. We present several examples and numerical validations to illustrate the theoretical results.

math.ST↗
arXiv2023

Non-minimaxity of debiased shrinkage estimators

We consider the estimation of the $p$-variate normal mean of $X\sim N_p(θ,I)$ under the quadratic loss function. We investigate the decision theoretic properties of debiased shrinkage estimator, the estimator which shrinks towards the origin for smaller $\|x\|^2$ and which is exactly equal to the unbiased estimator $X$ for larger $\|x\|^2$. Such debiased shrinkage estimator seems superior to the unbiased estimator $X$, which implies minimaxity. However we show that it is not minimax under mild conditions.

math.ST↗
arXiv2021

The volume-of-tube method for Gaussian random fields with inhomogeneous variance

The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well as its Laplace approximation. In particular, the critical radius of the tube is generalized for evaluation of the asymptotic approximation error. As an example, we discuss the approximation of the largest eigenvalue distribution of the Wishart matrix with a non-identity matrix parameter. The Bonferroni method is the tube method when the index set is a finite set. We provide the formula for the asymptotic approximation error for the Bonferroni method when the variance is not constant.

math.PR↗
arXiv2018

Game-theoretic derivation of upper hedging prices of multivariate contingent claims and submodularity

We investigate upper and lower hedging prices of multivariate contingent claims from the viewpoint of game-theoretic probability and submodularity. By considering a game between "Market" and "Investor" in discrete time, the pricing problem is reduced to a backward induction of an optimization over simplexes. For European options with payoff functions satisfying a combinatorial property called submodularity or supermodularity, this optimization is solved in closed form by using the Lovász extension and the upper and lower hedging prices can be calculated efficiently. This class includes the options on the maximum or the minimum of several assets. We also study the asymptotic behavior as the number of game rounds goes to infinity. The upper and lower hedging prices of European options converge to the solutions of the Black-Scholes-Barenblatt equations. For European options with submodular or supermodular payoff functions, the Black-Scholes-Barenblatt equation is reduced to the linear Black-Scholes equation and it is solved in closed form. Numerical results show the validity of the theoretical results.

q-fin.PR↗
arXiv2016

Distribution of ratio of two Wishart matrices and evaluation of cumulative probability by holonomic gradient method

We study the distribution of the ratio of two central Wishart matrices with different covariance matrices. We first derive the density function of a particular matrix form of the ratio and show that its cumulative distribution function can be expressed in terms of the hypergeometric function 2F1 of a matrix argument. Then we apply the holonomic gradient method for numerical evaluation of the hypergeometric function. This approach enables us to compute the power function of Roy's maximum root test for testing the equality of two covariance matrices.

math.ST↗
arXiv2016

Exact ZF Analysis and Computer-Algebra-Aided Evaluation in Rank-1 LoS Rician Fading

We study zero-forcing detection (ZF) for multiple-input/multiple-output (MIMO) spatial multiplexing under transmit-correlated Rician fading for an N_R X N_T channel matrix with rank-1 line-of-sight (LoS) component. By using matrix transformations and multivariate statistics, our exact analysis yields the signal-to-noise ratio moment generating function (m.g.f.) as an infinite series of gamma distribution m.g.f.'s and analogous series for ZF performance measures, e.g., outage probability and ergodic capacity. However, their numerical convergence is inherently problematic with increasing Rician K-factor, N_R , and N_T. We circumvent this limitation as follows. First, we derive differential equations satisfied by the performance measures with a novel automated approach employing a computer-algebra tool which implements Groebner basis computation and creative telescoping. These differential equations are then solved with the holonomic gradient method (HGM) from initial conditions computed with the infinite series. We demonstrate that HGM yields more reliable performance evaluation than by infinite series alone and more expeditious than by simulation, for realistic values of K , and even for N_R and N_T relevant to large MIMO systems. We envision extending the proposed approaches for exact analysis and reliable evaluation to more general Rician fading and other transceiver methods.

cs.IT↗
arXiv2016

Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability

We study the behavior of the capital process of a continuous Bayesian mixture of fixed proportion betting strategies in the one-sided unbounded forecasting game in game-theoretic probability. We establish the relation between the rate of convergence of the strong law of large numbers in the self-normalized form and the rate of divergence to infinity of the prior density around the origin. In particular we present prior densities ensuring the validity of Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm.

math.PR↗
arXiv2015

$A$-Hypergeometric Distributions and Newton Polytopes

We give a bijection between a quotient space of the parameters and the space of moments for any $A$-hypergeometric distribution. An algorithmic method to compute the inverse image of the map is proposed utilizing the holonomic gradient method and an asymptotic equivalence of the map and the iterative proportional scaling. The algorithm gives a method to solve a conditional maximum likelihood estimation problem in statistics. Our interplay between the theory of hypergeometric functions and statistics gives some new formulas of $A$-hypergeometric polynomials.

math.CA↗
arXiv2015

Holonomic gradient method for distribution function of a weighted sum of noncentral chi-square random variables

We apply the holonomic gradient method to compute the distribution function of a weighted sum of independent noncentral chi-square random variables. It is the distribution function of the squared length of a multivariate normal random vector. We treat this distribution as an integral of the normalizing constant of the Fisher-Bingham distribution on the unit sphere and make use of the partial differential equations for the Fisher-Bingham distribution.

math.ST↗
arXiv2015

Standard imsets for undirected and chain graphical models

We derive standard imsets for undirected graphical models and chain graphical models. Standard imsets for undirected graphical models are described in terms of minimal triangulations for maximal prime subgraphs of the undirected graphs. For describing standard imsets for chain graphical models, we first define a triangulation of a chain graph. We then use the triangulation to generalize our results for the undirected graphs to chain graphs.

math.ST↗
arXiv2015

MIMO Zero-Forcing Performance Evaluation Using the Holonomic Gradient Method

For multiple-input multiple-output (MIMO) spatial-multiplexing transmission, zero-forcing detection (ZF) is appealing because of its low complexity. Our recent MIMO ZF performance analysis for Rician--Rayleigh fading, which is relevant in heterogeneous networks, has yielded for the ZF outage probability and ergodic capacity infinite-series expressions. Because they arose from expanding the confluent hypergeometric function $ {_1\! F_1} (\cdot, \cdot, σ) $ around 0, they do not converge numerically at realistically-high Rician $ K $-factor values. Therefore, herein, we seek to take advantage of the fact that $ {_1\! F_1} (\cdot, \cdot, σ) $ satisfies a differential equation, i.e., it is a \textit{holonomic} function. Holonomic functions can be computed by the \textit{holonomic gradient method} (HGM), i.e., by numerically solving the satisfied differential equation. Thus, we first reveal that the moment generating function (m.g.f.) and probability density function (p.d.f.) of the ZF signal-to-noise ratio (SNR) are holonomic. Then, from the differential equation for $ {_1\! F_1} (\cdot, \cdot, σ) $, we deduce those satisfied by the SNR m.g.f. and p.d.f., and demonstrate that the HGM helps compute the p.d.f. accurately at practically-relevant values of $ K $. Finally, numerical integration of the SNR p.d.f. produced by HGM yields accurate ZF outage probability and ergodic capacity results.

cs.IT↗
arXiv2014

A power-law decay model with autocorrelation for posting data to social networking services

We propose a power-law decay model with autocorrelation for posting data to social networking services concerning particular events such as national holidays or major sport events. In these kinds of events we observe people's interest both before and after the events. In our model the number of postings has a Poisson distribution whose expected value decays as a power law. Our model also incorporates autocorrelations by autoregressive specification of the expected value. We show that our proposed model well fits the data from social networking services.

math.ST↗
arXiv2014

Schur Complement Based Analysis of MIMO Zero-Forcing for Rician Fading

For multiple-input/multiple-output (MIMO) spatial multiplexing with zero-forcing detection (ZF), signal-to-noise ratio (SNR) analysis for Rician fading involves the cumbersome noncentral-Wishart distribution (NCWD) of the transmit sample-correlation (Gramian) matrix. An \textsl{approximation} with a \textsl{virtual} CWD previously yielded for the ZF SNR an approximate (virtual) Gamma distribution. However, analytical conditions qualifying the accuracy of the SNR-distribution approximation were unknown. Therefore, we have been attempting to exactly characterize ZF SNR for Rician fading. Our previous attempts succeeded only for the sole Rician-fading stream under Rician--Rayleigh fading, by writing it as scalar Schur complement (SC) in the Gramian. Herein, we pursue a more general, matrix-SC-based analysis to characterize SNRs when several streams may undergo Rician fading. On one hand, for full-Rician fading, the SC distribution is found to be exactly a CWD if and only if a channel-mean--correlation \textsl{condition} holds. Interestingly, this CWD then coincides with the \textsl{virtual} CWD ensuing from the \textsl{approximation}. Thus, under the \textsl{condition}, the actual and virtual SNR-distributions coincide. On the other hand, for Rician--Rayleigh fading, the matrix-SC distribution is characterized in terms of determinant of matrix with elementary-function entries, which also yields a new characterization of the ZF SNR. Average error probability results validate our analysis vs.~simulation.

cs.IT↗