2017年度秋季総合分科会(於:山形大学)

SUMMARY: In this talk we will state several properties for random walks on graphs. We will mainly focus on simple random walks on infinite connected graphs such as fractal graphs and percolation clusters. This talk will consist of two large parts. First, we will state the range of random walk on graphs satisfying a uniform condition, which includes several fractal graphs. Second, we will state a level-2 quenched large deviation principle for simple random walk on a class of percolation clusters including long-range correlations. The second part is based on a joint work with Noam Berger and Chiranjib Mukherjee.

SUMMARY: In this talk, I will review a class of most important results, chosen based on my personal preference, on phase transitions and critical behavior for the Ising model, a model of ferromagnetism in classical equilibrium statistical mechanics. First, I will overview the results obtained by the end of the previous century and recall the problems left unsolved then. Next, I will review how some of those problems have been solved since 2001. The common tool to solve most of those problems is the random-current representation. I will explain the derivation of the representation and its implication by showing some examples, such as exponential decay of the subcritical two-point function, uniqueness of the critical point, the mean-field bound on the 1-arm exponent, and the lace expansion for the two-point function. Finally, I will summarize the remaining open problems for future researches.

SUMMARY: We discuss a scaling problem of continuous probability distributions on the Euclidean space, where scaling means a coordinate-wise transformation in order that some functional identity is satisfied. In linear algebra, it is known that any non-negative definite matrix has a unique diagonal scaling such that the transformed matrix is quasi-doubly stochastic, whenever it is strictly copositive. We generalize the result to the space of probability distributions, where the set of matrices is identified with the Gaussian family. It is shown that, under a strictly copositive condition for distributions, there exists a unique coordinate-wise transformation such that the transformed distribution satisfies a Stein-type identity. The result is interpreted as an alternative representation of copulas. The proof is based on an energy minimization problem over a subset of the Wasserstein space. Some open problems will be discussed.

SUMMARY: This talk introduces robust statistical inference for various time series models under non-standard settings. In these few decades, non-standard aspects of real data in some sense are frequently observed in practical situations. First, a prominent example of long-range dependence was found by Hurst (1951, Trans. Amer. Soc. Civil Eng.) via the analysis of records of water flows through the Nile and though other rivers. Second, Mandelbrot (1963, J. Polit. Econ.) and Fama (1965, J. Bus.) found heavy-tailed economic and financial data which were poorly captured by the Gaussian models. When a statistical model has long-range dependence and/or heavy-tails, the limit distributions of fundamental statistics (e.g., sample mean) are not expressed in a closed form, and the rate of convergence contains the Hurst-index of long-range dependence and the tail-index of the underlying innovation density. Such properties make the situation complicated, and it is unfeasible to use the classical maximum likelihood method or the method of moments directly. To overcome the hurdles, we make use of some statistical methodologies involving the empirical likelihood, self-weighting and self-normalization methods. The empirical likelihood method proposed by Owen (1988, Biometrika) is a modern important statistical framework without knowledge of the underlying distribution. In particular, we integrate the concepts of the empirical likelihood and the least absolute deviations-based self-weighting method proposed by Ling (2005, J. Roy. Stat. Soc.), and construct the robust empirical likelihood statistic which is not affected by the nuisance parameters of the model and has the standard chi-square limit distribution. On the other hand, we also overcome the difficulties brought by the long-range dependence by using the self-normalized sabsampling method proposed by Bai et al. (2016, Ann. Stat.). Finally, a unified, feasible and robust framework for various time series models under the non-standard situation is established.

SUMMARY: We show that an inversion formula for probability measures on the real line holds in a sense of the theory of hyperfunctions. As an application of our inversion formula, we give a representation of probability density functions utilizing characteristic functions.

SUMMARY: For a non-generic, yet dense subset of \(C^1\) expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a single periodic orbit. A key ingredient is a new \(C^1\) perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

SUMMARY: We consider the random iteration of two expanding diffeomorphisms on the real-line without a common fixed point. We derive the spectral gap property of an associated transition operator acting on spaces of Hölder continuous functions. We introduce generalised Takagi functions on the real-line and we investigate their regularity properties.

SUMMARY: We give an upper bound for the discrepancy of irrational rotations \(\{n \alpha \}\) in terms of the continued fraction expansion of \(\alpha \) and the related Ostrowski expansion. Our result improves earlier bounds in the literature and substantially simplifies their proofs.

SUMMARY: We consider the reconstruction problem of a random function from the system of its stochastic Fourier coefficients (SFC in abbr.). We employ arbitrary orthonormal basis and Ogawa integral to construct SFCs. We first discuss the representation of Ogawa integral of a random function from its SFCs and \(H^1\) basis, and then solve the reconstruction problem.

SUMMARY: We consider the question whether a random function (or a stochastic derivative as an extension) is identified from the stochastic Fourier coefficient (SFC). We give an answer for the stochastic derivatives driven by finite variation processes. Especially, any finite variation process is identified from the SFC of Ogawa type. Also, we reconstruct, independently of values of the Brownian motion, nonnegative absolutely continuous noncausal Wiener functionals from the SFC of Skorokhod type.

SUMMARY: We show the global-in-time well-posedness of the complex Ginzburg–Landau (CGL) equation with a space-time Gaussian white noise on the 3-dimensional torus. The local well-posedness was obtained by Hoshino, Inahama and Naganuma, as an application of the theory of paracontrolled calculus. For the global well-posedness, we use a similar argument to Mourrat and Weber’s work about the global well-posedness of the dynamical \(\Phi _3^4\) model. By improving their method, we show a priori \(L^{2p}\) estimate of the solution for \(p>\frac 32\).

SUMMARY: We discuss the multi-component coupled Kardar–Parisi–Zhang (KPZ) equation and its two types of approximations. By applying the paracontrolled calculus introduced by Gubinelli, Imkeller and Perkowski, we show that these approximations have a common limit under well adjusted choices of renormalization factors. Moreover, if the coupling constants satisfy the so-called “trilinear” condition, then the Wiener measure becomes stationary for the limit, so that this limit exists globally in time when the initial value is sampled under the stationary measure.

SUMMARY: Let \(\{X_t\}_{t \geq 0}\) be the symmetric \(\alpha \)-stable process with generator \(\mathcal {L} = -(-\Delta )^{\alpha /2}\) for \(0 < \alpha \leq 2\) and \(\mu \) be a positive Radon measure in a certain class. We define the Schrödinger operator \(\mathcal {L}^\mu = \mathcal {L} + \mu \) and consider the fundamental solution of the equation \(\partial u/\partial t = \mathcal {L}^{\mu } u\). If \(\mu \) is critical, the behavior of the fundamental solution is different from that of the transition density function of \(\{X_t\}_{t \geq 0}\). In this talk, we give large time asymptotics for fundamental solutions of critical Schrödinger operators.

SUMMARY: In this talk, we give an order statistic with random vectors to construct multivariate compound Poisson processes. The method works well for some processes, especially generated by zeta distributions. For example, Aoyama and Nakamura introduced generalized Euler products attached to a subclass of multidimensional infinitely divisible distributions. We will construct the compound Poisson processes corresponding their infinitely divisible distributions generated by the Euler products.

SUMMARY: In the recent years, finite frame theory has come to draw a lot of attention since there exist many applications, e.g, numerical analysis, algebraic design theory, directional statistics, compressed sensing and so on. In this talk we show that determinantal point processes on the sphere give almost tight finite frames. We give two expectations of frame potentials of spherical ensembles and harmonic ensembles, which are the typical types of DPPs on the sphere. We also discuss random matrices induced by determinantal point processes on the sphere.

SUMMARY: We consider vertex-reinforced random walks on complete bipartite graphs, and study their limiting behavior.

SUMMARY: In this study, we consider a new decision process model with a converging branch system which is one of the nonserial transition systems. We give the formulation of the model and introduce a recursive method to solve it by using dynamic programming.

SUMMARY: The existence of pairwise additive cyclic balanced incomplete block (BIB) designs with \(k=2\) and \(\lambda =1\) has been discussed in the literature. In this talk, for an odd prime \(p\ge 5\), \(2\)-pairwise additive cyclic BIB designs with \((v,k,\lambda ) = (3p,2,1), (2p,2,2)\) are mainly constructed through methods of block replacements. Finally, the existence of \(2\)-pairwise additive cyclic BIB designs with \(k=2\) and \(\lambda \ge 1\) is shown entirely.

SUMMARY: We consider the third-order linear model for \(3^{m}\) factorials. In previous talks (MSJ Autumn Meeting 2016; MSJ Spring Meeting 2017), we have established a necessary and sufficient condition for a simple array (SA) to be a balanced third-order (\(3^{m}\)-BTO) design of resolution \(\mathrm {R}^{\ast }(\{10,01\}),\) where the number of assemblies (\(=N\)) is less than the number of non-negligible factorial effects (\(=\nu (m)\)) and \(m\ge 6.\)

Let \(T\) be a \(3^{m}\)-BTO design of resolution \(\mathrm {R}^{\ast }(\{10,01\})\) derived from an SA with \(N\) assemblies, and further let \(\sigma ^{2}S_{T}\) be the trace of the variance-covariance matrix of the estimators concerning with all the main effects based on \(T\). If \(S_{T}\le S_{T^{\ast }}\) for any \(T^{\ast },\) then \(T\) is said to be A\(^{\ast }\)-optimal, where \(T^{\ast }\) is a \(3^{m}\)-BTO design of resolution \(\mathrm {R}^{\ast }(\{10,01\})\) derived from an SA with \(N\) assemblies. In this talk, we give A\(^{\ast }\)-optimal \(3^{m}\)-BTO designs of resolution \(\mathrm {R}^{\ast }(\{10,01\})\) derived from SA’s for \(m=6,7,8,\) where \(N<\nu (m).\)

SUMMARY: Locating arrays are introduced for identifying interaction faults and their locations in component-based systems. This class of problems are closely related with covering arrays and group testing, but the constructions are less known. Under the assumption that the system contains (at most) \(d\) faults, each involving (at most) \(t\) interacting factors, the notion of a \((\bar {d}, t)\)-locating array is proposed. In this talk, I will focus on \((\bar {1},t)\)-locating arrays. Moreover, by taking the similar consideration to error-correcting codes, the notion of locating arrays with error-correcting ability will be introduced.

SUMMARY: This talk is concerned with regression based on the empirical risk minimization. The estimator is composed as a convex combination of the word (learner) in dictionary. The word is selected in each step of the proposed stagewise algorithm, which minimizes a certain divergence measure. A non-asymptotic error bound of the estimator is developed, and it is seen that the error bound becomes sharp as the number of iteration of the algorithm increases.

SUMMARY: Let \(f(x)\) and \(g(x)\) denote probability density functions and \(g(x_0)\neq 0\). In this paper we discuss the density ratio \(f(x_0)/g(x_0)\). A naive estimator is constituted from separate estimators of \(f(x_0)\) and \(g(x_0)\), which we call an indirect estimaotr. The other estimator is proposed by Cwik and Mielniczuk (1989), which we call a direct estimator. Here we propose a new direct estimator, and derive asymptotic mean squared error. We also prove central limit theorem of the new estimator. We also compare mean squared errors of the proposed estimator and others by simulation

SUMMARY: We consider estimation of the probability density for nonnegative data. In that case, the standard kernel density estimator is, in general, inconsistent near the boundary, due to the so-called boundary bias. Many authors have suggested various remedies, e.g., renormalization, reflection, and generalized jackknifing (see Jones (1993) for a review). On the other hand, over the last decade, there has been growing interest in the use of asymmetric kernel (AK), whose support matches the support of the density to be estimated. We propose a new AK density estimator using a q-MIG kernel. Here, “MIG” is a mixture of symmetrical-based inverse Gaussian (IG) and its reciprocal (RIG), and “q-MIG” is its generalization via Yang’s (2006) dual transformation, including a subfamily of log-symmetrical densities as a special case.

SUMMARY: For square contingency table with the same row and column ordinal classifications, this paper shows that the diamond model holds if and only if the weighted covariance for the difference between the row and column classifications and the sum of them equals zero and the uniform association diamond model holds.

SUMMARY: For the analysis of square contingency tables with ordered categories, the present paper proposes a model that indicates the structure of asymmetry for cell probabilities. The model is the closest to the symmetry model in terms of the \(f\)-divergence under certain conditions, and includes the asymmetry models, which have been proposed by many statisticians, in the special cases. Also, it is shown the theorem that the symmetry model can be separated into some models by using the proposed model. It may be useful to see the reason for the poor fit of the symmetry model.

SUMMARY: We consider the null distribution of the Hotelling’s \(T^2\)-type statistic for testing equality of two mean vectors when the two data matrices are of the same monotone missing pattern. As with the one-sample problem, a simplified \(T^2\) statistic and an asymptotic expansion of its null distribution using decomposition of the test statistic are derived. Decomposition of the test statistic proposed in the study allowed to calculate the asymptotic expansion more easily. Further, we present the transformed test statistics based on the Bartlett adjustment. Finally, by a Monte Carlo simulation, we investigate the accuracy and asymptotic behavior of the approximation for chi-squared distribution.

SUMMARY: In this talk, we propose an \(L^2\)-norm-based statistic and its asymptotic distribution for simultaneous test of the mean vector and covariance matrix for multi-sample problem. An asymptotic distribution of test statistic are derived under a high-dimensional framework to deal with high-dimensional problems. This result is used for asymptotic size adjustment and derivation of asymptotic power. Finally, we study the finite sample and dimension performance of this test via Monte Carlo simulations.

SUMMARY: The Rao structure about the dispersion matrix in the general linear model is a well-known necessary and sufficient condition which guarantees that the ordinary least square estimator becomes the best linear unbiased estimator. In this presentation, we discuss a general ridge estimator to derive an extension of the Rao structure, that is, a necessary and sufficient condition under which two general ridge estimators coincide with each other.

SUMMARY: In this talk, we consider the equality test of covariance matrices for high-dimensional data. Aoshima and Yata (2017, SS) proposed two eigenvalue models for high-dimensional data. One is called the strongly spiked eigenvalue (SSE) model and the other one is called the non-SSE (NSSE) model. Li and Chen (2012) proposed a test statistic under the NSSE model. We verify that the statistic is asymptotically distributed as a chi-squared distribution under the SSE model. With the help of the asymptotic distribution, we proposed an equality test under the SSE model.

SUMMARY: In this talk, we consider the asymptotic normality for inference on high-dimensional mean vectors under two disjoint models: the strongly spiked eigenvalue (SSE) model and the non-SSE (NSSE) model. We first consider a distance-based statistics. We verify that it is asymptotically distributed as a normal distribution under the NSSE model. We also show that the asymptotic normality does not hold under the SSE model. We propose a new statistics by the estimation of eigenstructures for the SSE model. We verify that the proposed statistics is asymptotically distributed as a normal distribution under the SSE model. With the help of the asymptotic normality, we consider inferences on multi-sample and mean vectors under the SSE model.

SUMMARY: In this talk, we consider asymptotic properties of the support vector machine (SVM) in high-dimension, low-sample-size (HDLSS) settings. We show that the SVM holds a consistency property in which misclassification rates tend to zero as the dimension goes to infinity under certain severe conditions. We show that the SVM is very biased in HDLSS settings and its performance is affected by the bias directly. In order to overcome such difficulties, we propose a bias-corrected SVM (BC-SVM). We show that the BC-SVM gives preferable performances in HDLSS settings for typical kernel functions. Finally, we check the performance of the BC-SVM by numerical simulations.

SUMMARY: When we deal with actual problems by model building, it is often commonly assumed that the response variable and covariates satisfy linear relationship. One of the usual assumptions is that the disturbances follow identically independent distribution. Nevertheless the correlation of them may occur when the data are collected sequentially in time, especially in the field of economics and geophysics. In this talk, we assume the errors are strongly dependent. Then the asymptotic theory for modified LASSO estimators is discussed.

SUMMARY: The Dantzig selector, which was proposed by Candés and Tao in 2007, is an estimation procedure for regression models in a high-dimensional and sparse setting. In this presentation, linear models of diffusion processes with unknown drift matrices and diagonal diffusion matrices are discussed. We will consider the estimation problems for drift and diffusion matrices based on the discrete time observation in high-dimensional and sparse settings for drift matrices. To estimate drift matrices, we will apply the Dantzig selector and prove the \(l_q\) consistency of the estimator for every \(q \in [1,\infty ]\) under some appropriate conditions.

SUMMARY: For independent observations, analysis of variance (ANOVA) has been enoughly tailored. Recently there has been much demand for ANOVA of high dimensional and dependent observations in many fields. However ANOVA for high dimensional and dependent observations has been immature. In this paper, we study ANOVA for high dimensional and dependent observations. Specifically, we show asymptotics of classical tests proposed for independent observations and give a sufficient condition for them to be asymptotically normal. Some numerical examples for simulated and real financial data are given as applications of these results. The extension in this paper is not straightforward and contains a lot of novel aspects for the analysis of variance for high dimensional time series.

SUMMARY: We study a system space of autoregressive process of degree 1. System spaces of time series in information geometry have a Fisher metric as a Riemannian metric, and admit an \(\alpha \)-connection which is defined by the power spectrum. This space is a two dimensional \(\alpha \)-flat statistical manifold. Moreover, for \(\alpha =-1,0,1\) we discuss \(\alpha \)-geodesics and almost complex structures which are parallel with respect to the \(\alpha \)-connection.

SUMMARY: We consider a hypothesis testing problem on the stationarity in locally stationary processes. In the existing literature, several test statistics have been proposed in the framework of local periodogram generated from the local stationary processes. However, we need sufficient moments of the stochastic processes under that framework. Thus, we propose a new test statistic constructed from the local quantile periodogram, where the measure of stationarity is also redefined in the quantiles. In addition, we extend our test statistic to the empirical likelihood ratio statistic to test the hypothesis. The theoretical results and numerical results under the alternative hypotheses will be given in the talk.