arXiv subjects

Akimichi Takemura

Publications and source records attributed to Akimichi Takemura.

At least 109 records · Page 6Linked to original sources

arXiv2006

Asymptotic Distribution of Wishart Matrix for Block-wise Dispersion of Population Eigenvalues

This paper deals with the asymptotic distribution of Wishart matrix and its application to the estimation of the population matrix parameter when the population eigenvalues are block-wise infinitely dispersed. We show that the appropriately normalized eigenvectors and eigenvalues asymptotically generate two Wishart matrices and one normally distributed random matrix, which are mutually independent. For a family of orthogonally equivariant estimators, we calculate the asymptotic risks with respect to the entropy or the quadratic loss function and derive the asymptotically best estimator among the family. We numerically show 1) the convergence in both the distributions and the risks are quick enough for a practical use, 2) the asymptotically best estimator is robust against the deviation of the population eigenvalues from the block-wise infinite dispersion.

math.ST↗

arXiv2006

Integral representations of one dimensional projections for multivariate stable densities

We consider the numerical evaluation of one dimensional projections of general multivariate stable densities introduced by Abdul-Hamid and Nolan (1998). In their approach higher order derivatives of one dimensional densities are used, which seem to be cumbersome in practice. Furthermore there are some difficulties for even dimensions. In order to overcome these difficulties we obtain the explicit finite-interval integral representation of one dimensional projections for all dimensions. For this purpose we utilize the imaginary part of complex integration, whose real part corresponds to the derivative of the one dimensional inversion formula. We also give summaries on relations between various parameterizations of stable multivariate density and its one dimensional projection.

math.ST↗

arXiv2006

Skewness and kurtosis as locally best invariant tests of normality

Consider testing normality against a one-parameter family of univariate distributions containing the normal distribution as the boundary, e.g., the family of $t$-distributions or an infinitely divisible family with finite variance. We prove that under mild regularity conditions, the sample skewness is the locally best invariant (LBI) test of normality against a wide class of asymmetric families and the kurtosis is the LBI test against symmetric families. We also discuss non-regular cases such as testing normality against the stable family and some related results in the multivariate cases.

math.ST↗

arXiv2006

Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph

We characterize clique trees of a chordal graph in their relation to simplicial vertices and perfect sequences of maximal cliques. We investigate boundary cliques defined by Shibata and clarify their relation to endpoints of clique trees. Next we define a symmetric binary relation between the set of clique trees and the set of perfect sequences of maximal cliques. We describe the relation as a bipartite graph and prove that the bipartite graph is always connected. Lastly we consider to characterize chordal graphs from the aspect of non-uniqueness of clique trees.

cs.DM↗

arXiv2006

Hierarchical orbital decompositions and extended decomposable distributions

Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. Kamiya, Takemura and Kuriki (2006) generalized the elliptically contoured distributions to star-shaped distributions, for which the contours are allowed to be arbitrary proportional star-shaped sets. This was achieved by considering the so-called orbital decomposition of the sample space in the general framework of group invariance. In the present paper, we extend their results by conducting the orbital decompositions in steps and obtaining a further, hierarchical decomposition of the sample space. This allows us to construct probability models and distributions with further independence structures. The general results are applied to the star-shaped distributions with a certain symmetric structure, the distributions related to the two-sample Wishart problem and the distributions of preference rankings.

math.ST↗

arXiv2006

Star-shaped distributions and their generalizations

Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. We generalize elliptically contoured densities to ``star-shaped distributions'' with concentric star-shaped contours and show that many results in the former case continue to hold in the more general case. We develop a general theory in the framework of abstract group invariance so that the results can be applied to other cases as well, especially those involving random matrices.

math.ST↗

arXiv2006

Strong consistency of MLE for finite mixtures of location-scale distributions when the scale parameters are exponentially small (full version)

In a finite mixture of location-scale distributions maximum likelihood estimator does not exist because of the unboundedness of the likelihood function when the scale parameter of some mixture component approaches zero. In order to study the strong consistency of maximum likelihood estimator, we consider the case that the scale parameters of the component distributions are restricted from below by $c_n$, where $c_n$ is a sequence of positive real numbers which tend to zero as the sample size $n$ increases. We prove that under mild regularity conditions maximum likelihood estimator is strongly consistent if the scale parameters are restricted from below by $c_{n} = \exp(-n^d)$, $0 < d < 1$.

math.ST↗

arXiv2006

Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family

We give a sufficient condition for admissibility of generalized Bayes estimators of the location vector of spherically symmetric distribution under squared error loss. Compared to the known results for the multivariate normal case, our sufficient condition is very tight and is close to being a necessary condition. In particular we establish the admissibility of generalized Bayes estimators with respect to the harmonic prior and priors with slightly heavier tail than the harmonic prior. We use the theory of regularly varying functions to construct a sequence of smooth proper priors approaching an improper prior fast enough for establishing the admissibility. We also discuss conditions of minimaxity of the generalized Bayes estimator with respect to the harmonic prior.

math.ST↗

arXiv2006

Evaluation of per-record identification risk and swappability of records in a microdata set via decomposable models

We propose a strategy for disclosure risk evaluation and disclosure control of a microdata set based on fitting decomposable models of a multiway contingency table corresponding to the microdata set. By fitting decomposable models, we can evaluate per-record identification (or re-identification) risk of a microdata set. Furthermore we can easily determine swappability of risky records which does not disturb the set of marginals of the decomposable model. Use of decomposable models has been already considered in the existing literature. The contribution of this paper is to propose a systematic strategy to the problem of finding a model with a good fit, identifying risky records under the model, and then applying the swapping procedure to these records.

math.ST↗

arXiv2006

Conditions for swappability of records in a microdata set when some marginals are fixed

We consider swapping of two records in a microdata set for the purpose of disclosure control. We give some necessary and sufficient conditions that some observations can be swapped between two records under the restriction that a given set of marginals are fixed. We also give an algorithm to find another record for swapping if one wants to swap out some observations from a particular record. Our result has a close connection to the construction of Markov bases for contingency tables with given marginals.

math.ST↗

arXiv2006

Game-theoretic versions of strong law of large numbers for unbounded variables

We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (2001). We prove several versions of SLLN for the case that Reality's moves are unbounded. Our game-theoretic versions of SLLN largely correspond to standard measure-theoretic results. However game-theoretic proofs are different from measure-theoretic ones in the explicit consideration of various hedges. In measure-theoretic proofs existence of moments are assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices.

math.PR↗

arXiv2006

Indispensable monomials of toric ideals and Markov bases

Extending the notion of indispensable binomials of a toric ideal, we define indispensable monomials of a toric ideal and establish some of their properties. They are useful for searching indispensable binomials of a toric ideal and for proving the existence or non-existence of a unique minimal system of binomials generators of a toric ideal. Some examples of indispensable monomials from statistical models for contingency tables are given.

math.ST↗

arXiv2006

Goodness-of-Fit Tests for Symmetric Stable Distributions -- Empirical Characteristic Function Approach

We consider goodness-of-fit tests of symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard symmetric stable distribution with the characteristic exponent $α$ estimated from the data. We treat $α$ as an unknown parameter, but for theoretical simplicity we also consider the case that $α$ is fixed. For estimation of parameters and the standardization of data we use maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE) which minimizes the weighted integral. We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE and EISE. For the case of MLE, the eigenvalues of the covariance function are numerically evaluated and asymptotic distribution of the test statistic is obtained using complex integration. Simulation studies show that the asymptotic distribution of the test statistics is very accurate. We also present a formula of the asymptotic covariance function of the characteristic function process with parameters estimated by an efficient estimator for general distributions.

math.ST↗

arXiv2005

On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game

In the framework of the game-theoretic probability of Shafer and Vovk (2001) it is of basic importance to construct an explicit strategy weakly forcing the strong law of large numbers (SLLN) in the bounded forecasting game. We present a simple finite-memory strategy based on the past average of Reality's moves, which weakly forces the strong law of large numbers with the convergence rate of $O(\sqrt{\log n/n})$. Our proof is very simple compared to a corresponding measure-theoretic result of Azuma (1967) on bounded martingale differences and this illustrates effectiveness of game-theoretic approach. We also discuss one-sided protocols and extension of results to linear protocols in general dimension.

math.PR↗

arXiv2005

Defensive forecasting for linear protocols

We consider a general class of forecasting protocols, called "linear protocols", and discuss several important special cases, including multi-class forecasting. Forecasting is formalized as a game between three players: Reality, whose role is to generate observations; Forecaster, whose goal is to predict the observations; and Skeptic, who tries to make money on any lack of agreement between Forecaster's predictions and the actual observations. Our main mathematical result is that for any continuous strategy for Skeptic in a linear protocol there exists a strategy for Forecaster that does not allow Skeptic's capital to grow. This result is a meta-theorem that allows one to transform any continuous law of probability in a linear protocol into a forecasting strategy whose predictions are guaranteed to satisfy this law. We apply this meta-theorem to a weak law of large numbers in Hilbert spaces to obtain a version of the K29 prediction algorithm for linear protocols and show that this version also satisfies the attractive properties of proper calibration and resolution under a suitable choice of its kernel parameter, with no assumptions about the way the data is generated.

cs.LG↗

arXiv2005

Game theoretic derivation of discrete distributions and discrete pricing formulas

In this expository paper we illustrate the generality of game theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe how discrete distributions with finite support and the discrete pricing formulas, such as the Cox-Ross-Rubinstein formula, are naturally derived from game-theoretic probability protocols. Corresponding to any discrete distribution with finite support, we construct a finite-horizon discrete game, a replicating strategy of Skeptic, and a neutral forecasting strategy of Forecaster, such that the discrete distribution is derived from the game. Construction of a replicating strategy is the same as in the standard arbitrage arguments of pricing European options in the binomial tree models. However the game theoretic framework is advantageous because no a priori probabilistic assumption is needed.

math.PR↗

arXiv2005

Validity of the expected Euler characteristic heuristic

We study the accuracy of the expected Euler characteristic approximation to the distribution of the maximum of a smooth, centered, unit variance Gaussian process f. Using a point process representation of the error, valid for arbitrary smooth processes, we show that the error is in general exponentially smaller than any of the terms in the approximation. We also give a lower bound on this exponential rate of decay in terms of the maximal variance of a family of Gaussian processes f^x, derived from the original process f.

math.PR↗

arXiv2005

Defensive forecasting

We consider how to make probability forecasts of binary labels. Our main mathematical result is that for any continuous gambling strategy used for detecting disagreement between the forecasts and the actual labels, there exists a forecasting strategy whose forecasts are ideal as far as this gambling strategy is concerned. A forecasting strategy obtained in this way from a gambling strategy demonstrating a strong law of large numbers is simplified and studied empirically.

cs.LG↗