2018年度秋季総合分科会(於:岡山大学)

SUMMARY: We consider large deviation principles of continuous or discontinuous additive functionals generated by symmetric Markov processes. As a useful approach in proving the large deviation, the Gärtner–Ellis theorem is well-known. The Gärtner–Ellis theorem implies that the large deviation holds if there exists the logarithmic moment generating function of additive functional and it is differentiable. In the first half of this presentation, we introduce some examples which large deviations are obtained by proving the differentiability of the logarithmic moment generating function directly. But in many cases, it is known that logarithmic moment generating functions are not differentiable. In the second half, we consider the large deviation in a quite general setting. We can not expect to directly apply the Gärtner–Ellis theorem to the large deviation in such a situation. Especially, we show that the large deviation holds for joint additive functionals with continuous and discontinuous parts generated by Borel right processes on Lusin spaces.

SUMMARY: In the field of singular SPDEs, two prominent theories are recently established: the theory of regularity structures by Hairer and the paracontrolled calculus by Gubinelli, Imkeller, and Perkowski. They are written by different mathematical tools, so that we can use either of them according to the situation. However, the GIP theory applies to less number of equations, because the GIP theory is less algebraic. In this talk, we discuss how to fill this gap. Our goal is to show the equivalence of the Hairer’s theory and the “higher order” version of the GIP theory introduced by Bailleul and Bernicot, on the \(d\)-dimensional torus.

SUMMARY: This talk is on statistical inferences on some combinatorial stochastic processes. Especially, it discusses the intersection of three subjects: partitions, hypergeometric systems, and Ferguson’s Dirichlet processes. It is shown that these three subjects related to a common structure called exchangeability. Then, based on the algebraic nature, direct samplers with use of homogeneity of polynomials and dualities in Markov processes, and estimation with information geometry of polytopes will be presented.

SUMMARY: We consider prediction and interpolation problem of stationary processes and their application to parameter estimation problem. In a naive approach to defining linear prediction, the independence between the predictor and the prediction error is assumed, which turns out to be useful only for the Gaussian stationary process. Instead of independence, the concept is extended to the least prediction error, which is evaluated in an adequate normed space. This refinement of the definition makes the prediction problem available for a much richer class of stationary processes, such as harmonizable stable process. The interpolation problem for the class is defined along the same line. On the other hand, the estimation problem is to determine the parameters in the model, a parametric spectral density, from observations. The Whittle likelihood is introduced to estimate parameters of the Gaussian stationary process as an approximate Gaussian likelihood. In addition, the Whittle likelihood could also be interpreted as a method which minimizes the prediction error explained above. In this connection, we regard the prediction error and the interpolation error as a contrast between the Fourier transform of observations and the parametric spectral density. The parameters of the stationary process are estimated by the minimum contrast estimation. To precisely understand the estimation procedure, we first investigate the fundamental properties of the contrast functions. The new functions are not contained in the class of either location or scale disparities. Afterward, we discuss the asymptotic behaviors of the minimum contrast estimator applied to the different types of stationary processes. The estimator is shown to be asymptotic consistent. The asymptotic distribution of the estimator depends on the assumptions on the stochastic process. In particular, the estimator is robust against the fourth order cumulant when the process is Gaussian. Although it is shown that the Whittle estimator is asymptotically efficient in the sense that the family of parametric spectral densities is truly specified, the new class contains robust members to the randomly missing observations from the stationary process. We discuss this phenomenon in much more details.

SUMMARY: We formulate parameters of interest, nuisance parameter, and sufficient statistics as \(\sigma \)-algebra. We determine them in the case of \(\mathcal {A}\)-hypergeometric distribution.

SUMMARY: Khinchin proved that the arithmetic mean of continued fraction digits of Lebesgue almost every irrational number in (0,1) diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers or central limit theorems hold. Nevertheless, we prove the existence of a large deviations rate function which estimates exponential probabilities with which the arithmetic mean of digits stays away from infinity. This leads us to a contradiction to the widely-shared view that the Large Deviation Principle is a refinement of laws of large numbers.

SUMMARY: We study the word counting problem. In Regnier and Szpankowski (1998), generating functions of the number of pattern occurrences are shown. In Bassino et.al (2010), simplified form of generating functions of the number of pattern occurrences are shown in combination with inclusion-exclusion principles. In this paper, we show a new inclusion-exclusion formula in multivariate generating function form on partially ordered sets, and show a simpler expression of generating functions of the number of pattern occurrences in finite samples, which is also easy to implement in computer program. The advantage of our method is that it is easy to compute other statistics based on the statistics of the number of pattern occurrence, e.g., statistics of the number of occurrence of subset patterns.

SUMMARY: We consider the bond percolation problem on the square lattice. Our main object is the lowest open crossing \(r_n\) of a box \([0,n]^2\). On the event an open crossing of \([0,n]^2\) exists, we define the width \(\xi _n\) of the lowest open crossing by the maximum height of points on \(r_n\). We show that as \(n \to \infty \), \(\xi _n/(\log n)\) converges to a constant in probability when the system is in the supercritical regime. This improves the previous result obtained by Y. C. Zhang (1999).

SUMMARY: In this talk, we show a modified logarithmic Sobolev inequality for canonical Lévy processes. We provide a direct, intrinsic proof of a modified logarithmic Sobolev inequality. Moreover, we derive several previously known inequalities as corollaries of main theorem. As an application of main theorem, we also get a concentration inequality for canonical Lévy processes.

SUMMARY: We consider the distributions of the sum \(\sum _{i=1}^n \left ( \{i \alpha \} - \frac {1}{2} \right )\) of irrational rotations, which were studied by Hardy–Littlewood, Ostrowski, Hecke, Khintchine, and Beck. We give an exact formula for the first-order sums, by using Ostrowski expansion, rational rotation approximation and cancellation techniques to derive our results. We also give some mathematical explanations for the effects on the distributions of the sum, caused by large partial quotient in the continued fraction expansion of irrational number \(\alpha \).

SUMMARY: We consider the distributions of the second-order partial sum \(\sum _{i=1}^n \left \{ \left ( \{i \alpha \} - \frac {1}{2} \right )^2 - \frac {1}{12} \right \}\), which was originally studied in Behnke. We give estimates for the second-order sums by using rational rotation approximation, which explain their cubic-function-like behaviors. We also give some mathematical explanations for the effects on the distributions of the sum, caused by large partial quotient in the continued fraction expansion of irrational number \(\alpha \).

SUMMARY: We consider the question whether and how a random function is identified from the stochastic Fourier coefficients (SFCs). We give an affirmative answer when random functions are Ogawa-integrable with respect to regular CONS and the SFCs are given by Ogawa integral.

SUMMARY: In this talk, I discuss the chordal Komatu–Loewner equation on standard slit domains in a manner applicable not just to a simple curve but also a family of continuously growing hulls. Especially a conformally invariant characterization of the Komatu–Loewner evolution is obtained. As an application, we can extend the locality of the stochastic Komatu–Loewner evolution with coefficients \((\sqrt {6}, -b_{\mathrm {BMD}})\) in a full generality.

SUMMARY: We discussed Convergence of diffusion processes in a tube which is direct product diffusion processes \(\mathbb Y\) of one dimensional diffusion processes \(X^{(1)}\) and skew product diffusions \(\Xi \), or the time changed process \(\mathbb X\) which is based on a positive continuous additive functional \(\Phi (t)\). The skew product \(\Xi \) are given by one dimensional diffusion processes \(\rm R\) and a spherical Brownian motion \(\Theta \) by means of positive continuous additive functional \({\bf f}(t)\). We show Concrete expressions of the Dirichlet forms corresponding to time changed processes, which may be of non-local type caused by degeneracy of the underlying measures.

SUMMARY: Consider the maximum of independent and identically distributed random variables. The classical result says that the renormalized sample maxima converges to the extreme value distributions, under certain conditions on the distribution function. In this talk, we shall study the uniform rate of the convergence on the Kolmogorov distance in the framework of the Stein equations.

SUMMARY: We investigate the first hitting times of the radial Ornstein–Uhlenbeck processes. In this talk we will give an explicit form of the distribution function of the hitting time by means of the confluent hypergeometric function and its zeros with respect to the first parameter.

SUMMARY: In this talk, I will show some asymptotic behaviour of the random walk in cooling random environment (RWCRE). First we will give an introduction of the RWCRE, which is an environment resampled along some increasing time sequence according to some prescribed probability measure. Next, we show some ergodic theorem for the cooling environment which yields the strong law of large numbers and the quenched large deviation principle. The point is to understand the connection between the RWRE and the RWCRE. In the end of the talk, I will present the remaining open problems. This talk is based on a joint work with L. Avena, C. da Costa and F. den Hollander.

SUMMARY: We talk about free infinite divisibility for the class of Generalized Power distributions with Free Poisson term (GPFP) by methods of complex analysis. More specifically, we show that the class of GPFP satisfies the univalent inverse Cauchy transform property and the free regular property under some conditions. The class of GPFP contains important distributions in classical and free probability which are Marchenko-Pastur distributions, free Generalized Inverse Gaussian distributions, beta distributions, shifted semicircle laws and free positive stable laws with index 1/2. Thus, we lead some results of free infinite divisibility for these important distributions.

SUMMARY: In this study we consider a multiclass single-server queue with customer abandonment and establish a sufficient condition for the stability of the queue under the first-come, first-served service discipline. Our condition is a generalization of the corresponding condition of such a queue in the single-class case. To obtain the condition, we extend the methodology of fluid-limit stability for the stability of multiclass queueing networks to our multiclass queue with abandonment.

SUMMARY: We consider Lipschitz-type backward stochastic differential equations (BSDEs) driven by cylindrical martingales on the space of continuous functions. We show the existence and uniqueness of the solution of such infinite-dimensional BSDEs and prove that the sequence of solutions of corresponding finite-dimensional BSDEs approximates the original solution.

SUMMARY: We investigate some risk functions to evaluate risk over a finite or infinite horizon by Markov decision processes (MDPs).

SUMMARY: Event-related functional Magnetic Resonance Imaging (efMRI) is an imaging technique that enables one to estimate the shape of the hemodynamic response function (HRF), describing changes in the blood oxygen level dependent (BOLD) to neural activity in response to mental stimuli. Although efficient designs are useful for statistical inference on HRF, there are only a few publications on systematic constructions of such designs. Lin et al. (2017) thus introduced a new class of integer sequences to generate highly efficient designs, which produces a certain statistical concept called circulant almost orthogonal array (CAOA), and laid the foundation on a theory of systematic constructions of CAOA with strength \(2\) and bandwidth \(1\). In this talk, we establish a method for constructing CAOA with strength \(3\) and bandwidth \(1\), and thereby provide a number of infinite families of such CAOA.

SUMMARY: fMRI designs have been researched to be applied to investigate brain activity. In 2017, Lin, Phoa and Kao defined circulant almost orthogonal arrays (CAOAs) as a kind of fMRI designs. Recently, Satake, Yoshida and Sawa gave a systematic construction of CAOAs with strength 3 by using the idea of Hadamard 3-design in combinatorial design theory. Here we can only get families of CAOAs with constant constraints.

In this talk, by combining our construction and a combinatorial observation, we show the existence of families of CAOAs with non-constant constraints.

SUMMARY: The concept of a splitting-balanced block design, denoted by \((v,u\times k,\lambda )\)-SBD, has been defined with some applications for authentication codes in Ogata et al. (2004). On the other hand, the result on graph decompositions given in Fu and Mishima (2002) implies that there exists a \(1\)-rotationally resolvable \((v,2\times 2,2)\)-SBD if and only if \(v\equiv 0\) (mod 4). In this talk, a new direct construction of a cyclically near-resolvable \((v,2\times 2,2)\)-SBD is provided. Finally, it is shown that there exists a cyclically near-resolvable \((v,2\times 2,2)\)-SBD if and only if \(v\equiv 1\) (mod \(4\)).

SUMMARY: Dropout is used in deep learning. It is a method of learning by invalidating nodes with randomly for each layer in the multi-layer neural network. This is useful for over-learning which excessively adapts to training data. It becomes difficult for the model to generalize to new data which were not in the training set. A random sample of nodes cause more irregular frequency of dropout edges. We propose a combinatorial design of dropout nodes from each partite which balances frequency of edges. In this talk, we introduce the design called dropout design and give some its constructions.

SUMMARY: The proportional hazards model proposed by D. R. Cox in a high-dimensional and sparse setting is discussed. The regression parameter is estimated by the Dantzig selector, which will be proved to have the variable selection consistency under some appropriate regularity conditions. This fact enables us to reduce the dimension of the parameter and to construct asymptotically normal estimators for the regression parameter and the cumulative baseline hazard function. In this talk, the simple model which satisfies the regularity conditions and some numerical results of the variable selection consistency of the Dantzig selector for the proportional hazards model are provided.

SUMMARY: In the Bayesian analysis, it is well-known that ordinary Bayes estimator is not robust against outliers. Ghosh and Basu (2016) and Nakagawa and Hashimoto (2017) proposed robust Bayesian inferences against outliers using the density power divergence and \(\gamma \)-divergence, respectively. On the other hand, the selection of priors is also an important problem in the robust Bayesian inference. In this talk, we propose the two type objective priors for the robust Bayesian inference.

SUMMARY: We consider the maximum likelihood estimators (MLEs) of the mean parameter vector and the covariance matrix in the growth curve model when the data set has a monotone missing pattern. Throughout this talk, we assume that the data are missing completely at random (MCAR). As a result, we present the MLE of the mean parameter vector when the covariance matrix is known, and the MLE of the covariance matrix when the mean parameter vector is known. That is, we give the determining equation to obtain the MLEs of the mean parameter vector and the covariance matrix.

SUMMARY: The Poisson–Dirichlet distribution \(\mathsf {PD}(\theta )\) is a statistical model of random distributions with the scalar parameter \(\theta \), where the parameter corresponds to the diversity of realized distributions. The following two sampling schemes are considered: (i) drawing \(s\) samples of \(n\) elements from corresponding \(s\) populations which follow \(\mathsf {PD}(\theta )\) independently, and (ii) drawing a single sample of \(ns\) elements from a population which follows \(\mathsf {PD}(\theta )\). In this presentation, we demonstrate that the magnitude relation of the two Fisher information, which sample partitions converted from samples in (i) and (ii) possess, can change depending on the parameters \(n\), \(s\), and \(\theta \).

SUMMARY: We show the null distribution function of the Bartlett–Nanda–Pillai test (BNP test) can be shown as a ratio of expectations of some functions of several independent beta random distributions. Using the Laplace’s approximation method we obtain the limiting distribution of BNP test and its computable error bound when the sample size and the dimension tend to infinity.

SUMMARY: Asymmetric kernel density estimation is recently well-studied in the literature. Actually, applying several bias reduction methods for the classical (location-scale type) standard kernel density estimator, some asymmetric kernel density estimators can be bias-corrected in an additive or multiplicative way. In this talk, we revisit additive bias reduction method to reduce the bias up to the higher-order.

SUMMARY: In this talk we discuss the kernel-type estimators of mean residual life function \(m_X(t)=E(X-t|X>t)\). New estimators that can eliminate the boundary bias effect are proposed. Let \(X_1,X_2,...,X_n\) be independently and identically distributed nonnegative random variables with an absolutely continuous survival function \(S_X\) and a density \(f_X\). We study asymptotic properties of a kernel estimatior of \(m_X(t)\).

SUMMARY: Zero Crossing (ZC) statistic is the number of zero crossings observed in a time series. In this talk, a strictly stationary ellipsoidal \(\phi \)-mixing processes with mean zero, finite variance is discussed. We consider the estimation problems of the autocorrelation of a time series by using zero crossing. First, we will elucidate the joint asymptotic distribution of the ZC estimator. Next, we show that the ZC estimator has a good robustness when the spectral density of the process is contaminated by a sharp peak.

SUMMARY: Binary time series is after having been converted a some time series into 0 and 1. In this presentation, we propose a classification method based on binary time series. We will show that the misclassification probability tends to zero when the number observation tends to infinity. Next, we evaluate the asymptotic misclassification probability when the two categories are contiguous. Finally, we show that our classification method based on binary time series has a good robustness when the spectral density of the process is contaminated by a sharp peak.

SUMMARY: The empirical Bayesian shrinkage (EBS) estimator is expressed in terms of a shrinkage function \(\phi (\cdot )\), and includes the sample mean and the James–Stein estimator as special cases. We evaluate the mean squared error (MSE) of EBS estimator for the mean of a Gaussian vector stationary process. Then a sufficient condition for the proposed EBS estimator to improve the sample mean is given in terms of \(\phi (\cdot )\) and the spectral density matrix of the process. We also seek \(\phi (\cdot )\) which gives the largest improvement for the difference of MSE’s between EBS and the sample mean. The results have a potential to improve the a lot of estimators in various time series data.

SUMMARY: In this talk, we consider asymptotic properties of support vector machine (SVM) for high-dimensional imbalanced data.

We show that 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 performance of SVM is affected by the imbalance and the tuning parameter. In order to overcome such difficulties, we propose a robust SVM (RSVM). We show that RSVM gives preferable performances for high-dimensional data. Finally, we check the performance of RSVM by numerical simulations.

SUMMARY: In this talk, we consider a correlation test for high-dimensional data. Aoshima and Yata (2018) proposed two eigenvalue models for high-dimensional data and constructed two-sample test procedures. One is called strongly spiked eigenvalue (SSE) model and the other one is called non-SSE (NSSE) model. Yata and Aoshima (2013) proposed a correlation test for high-dimensional data under the NSSE model. We focus on the SSE model that is often seen when we analyze the microarray data set. We give a new test procedure by using the extended cross-data-matrix method given by Yata and Aoshima (2013). We also check the performances of our test procedure by simulation study.

SUMMARY: In this talk, we consider estimation of mean vectors in high-dimensional settings. First, we show that the sample mean vector is not a consistent estimator of the true mean vector in high-dimension, low-sample-size (HDLSS) settings. With the help of a threshold method, we propose a new estimation method for the mean vector. We show that it holds the consistency property even in HDLSS settings. We apply the new method to multi-sample problems. Finally, we demonstrate the new method by using actual microarray data sets.

SUMMARY: This paper deals with a variable selection procedure in a multivariate linear regression model with normality assumption, which is called a normal multivariate linear regression model, by minimizing a generalized \(C_p\) (\(GC_p\)) criterion. The \(GC_p\) criterion used in this paper is defined by adding a penalty term to the multivariate standardized residual of sum of squares. A purpose of this paper is to clarify a sufficient condition of the penalty term in the \(GC_p\) criterion to satisfy an asymptotically loss efficiency property from the large sample and high-dimensional asymptotic framework, such that \(n\to \infty \) under the condition \(p/n\to c_0 \in [0,1)\). Then, we can propose an asymptotically loss efficient \(GC_p\) criterion under any noncentrality parameter matrix.

SUMMARY: This talk considers a nonparametric test for directional data. Most of classical density functions on the unit spheres share the common important feature called rotational symmetry. However, recently we found real data which do not satisfy this condition, and Ley and Verdebout (J. Multivariate Anal. 2017, 159:67–81) proposed a family of skew-rotationally-symmetric distributions on spheres. On the other hand, it is often severe to assume certain parametric family for real data. To overcome such hurdle, this talk employees the measure of skewness proposed by Mardia (Biometrika 1970, 57(3):519–530), and constructs the generalized empirical likelihood statistic for the null hypothesis of rotational symmetry. The proposed statistic is shown to converge to standard chi-squared distribution, and some simulation experiments illustrate finite sample performance of the proposed method.

SUMMARY: Circular data are a set of observations which can be expressed as angles \([-\pi ,\pi )\). Bivariate circular data, comprised of pairs of circular observations \([-\pi ,\pi )^2\), arise in numerous contexts. In this talk we propose a general Fourier series based approach to obtaining the bivariate circular analogues of copulas recently coined ‘circulas’. As examples of the general construction we consider some classes of circulas arising from different patterns of non-zero Fourier coefficients. The shape and sparsity of such arrangements are found to play a key role in determining the properties of the resultant models. All the special cases of the circulas we consider have simple closed-form expressions for their densities and display different dependence structures from the existing circulas.

SUMMARY: Discrete multivariate probability models are examined in this presentation. Consider the hypothesis \(H_{2}\) where some parameters from the first sample and those from the second sample are proportional, and the hypothesis \(H_{1}\) where \(H_{2}\) is not satisfied. In two sample problems under the hypothesis \(H_{2}\), the null hypothesis where the samples are from the same population is tested against the hypothesis where the samples are from the different population. It is found that the total information is equal to the sum of the within information and the between information in some case,but not equal in several cases. To investigate this phenomenon, we calculate the Kullback information based on \(H_{2}\) against \(H_{1}\). Some interesting results are presented.