## 統計数学セミナー

担当者 吉田朋広、荻原哲平、小池祐太 http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/ 確率統計学およびその関連領域に関する研究発表, 研究紹介を行う.

### 2023年03月08日(水)

14:00-   数理科学研究科棟(駒場) 号室
Evgeny Spodarev 氏 ( Ulm University, Germany)
Non-ergodic statistics for hamonizable stable processes (English)
[ 講演概要 ]
We consider stationary real harmonizable symmetric α-stable processes X={X(t):t∈ℝ} with a finite control measure. Assuming the control measure is symmetric and absolutely continuous with respect to the Lebesgue measure on the real line, we refer to its density function as the spectral density of X. Standard methods for statistical inference on stable processes cannot be applied as harmonizable stable processes are non-ergodic.
A stationary real harmonizable symmetric α-stable process X admits a LePage series representation and is conditionally Gaussian which allows us to derive the non-ergodic limit of sample functions on X. In particular, we give an explicit expression for the non-ergodic limits of the empirical characteristic function of X and the lag process {X(t+h)−X(t):t∈ℝ} with h>0, respectively.
The process admits an equivalent representation as a series of sinusoidal waves with random frequencies whose probability density function is in fact the (normalized) spectral density of X. Using the strongly consistent frequency estimation via periodograms we present a strongly consistent estimator of the spectral density which is based only on one sampled path of X. The periodogram computation is fast and efficient, and our method is not affected by the non-ergodicity of X. Most notably no prior knowledge on parameters of the process such as its index of stability α is needed.

References:
[1] L.V. Hoang, E. Spodarev, "Inversion of alpha-sine and alpha-cosine transforms on R", Inverse Problems 37 (2021), 085008
[2] L.V. Hoang, E. Spodarev, "Non-ergodic statistics and spectral density estimation for stationary real harmonizable symmetric α-stable processes", Preprint arXiv:2209.04315, submitted, 2022.
[ 参考URL ]

### 2023年01月10日(火)

10:50-11:30   数理科学研究科棟(駒場) 号室

Parameter Estimation with Increased Precision for Elliptic and Hypo-elliptic Diffusions

[ 講演概要 ]
Parametric inference for multi-dimensional diffusion processes has been studied over the past decades. Established approaches for likelihood-based estimation invoke a numerical time-discretisation scheme for the approximation of the (typically intractable) transition dynamics of the Stochastic Differential Equation (SDE) over finite time periods. Especially in the setting of some class of hypo-elliptic models, recent research (Ditlevsen and Samson 2019, Gloter and Yoshida 2021) has highlighted the critical effect of the choice of numerical scheme on the behaviour of derived parameter estimates. In our work, first, we develop two weak second order ‘sampling schemes' (to cover both the hypo-elliptic and elliptic classes) and generate accompanying ‘transition density schemes' of the SDE (i.e., approximations of the SDE transition density). Then, we produce a collection of analytic results, providing a complete theoretical framework that solidifies the proposed schemes and showcases advantages from their incorporation within SDE calibration methods, in both high and low frequency observations regime. We also present numerical results from carrying out classical or Bayesian inference. This is a joint work with Alexandros Beskos and Matthew Graham, and the preprint is available at https://arxiv.org/abs/2211.16384.
[ 参考URL ]

### 2022年12月05日(月)

14:40-15:50   数理科学研究科棟(駒場) 号室

Michael Choi 氏 (National University of Singapore and Yale-NUS College)
A binary branching model with Moran-type interactions (English)
[ 講演概要 ]
Branching processes naturally arise as pertinent models in a variety of applications such as population size dynamics, neutron transport and cell proliferation kinetics. A key result for understanding the behaviour of such systems is the Perron Frobenius decomposition, which allows one to characterise the large time average behaviour of the branching process via its leading eigenvalue and corresponding left and right eigenfunctions. However, obtaining estimates of these quantities can be challenging, for example when the branching process is spatially dependent with inhomogeneous rates. In this talk, I will introduce a new interacting particle model that combines the natural branching behaviour of the underlying process with a selection and resampling mechanism, which allows one to maintain some control over the system and more efficiently estimate the eigenelements. I will then present the main result, which provides an explicit relation between the particle system and the branching process via a many-to-one formula and also quantifies the L^2 distance between the occupation measures of the two systems. Finally, I will discuss some examples in order to illustrate the scope and possible extensions of the model, and to provide some comparisons with the Fleming Viot interacting particle system. This is based on work with Alex Cox (University of Bath) and Denis Villemonais (Université de Lorraine).
[ 参考URL ]

### 2022年10月21日(金)

①14:30-15:40- ②16:20-17:30   数理科学研究科棟(駒場) 126号室
ハイブリッド開催
Estate Khmaladze 氏 (Victoria University of Wellington)
On the theory of distribution free testing of statistical hypothesis
①Empirical processes for discrete and continuous observations: structure, difficulties and resolution.
②Further testing problems: parametric regression and Markov chains. (ENGLISH)
[ 講演概要 ]
The concept of distribution free testing is familiar to all. Everybody, who heard about rank statistics, knows that the distribution of ranks is independent from the distribution of underlying random variables, provided this later is a continuous distribution on the real line. Everybody, who ever used classical goodness of fit tests like Kolmogorov - Smirnov test or Cram\'er-von Mises test, knows that the distribution of statistics of these tests is independent from the distribution of the underlying random variables, again, provided this distribution is a continuous distribution on the real line.

Development in subsequent decades revealed many cracks in existing theory and difficulties in extending the concept of distribution free testing to majority of interesting models. It gradually became clear that the new starting point is needed to expand the theory to these models.

In our lectures we first describe the current situation in empirical and related processes. Then we describe how the new approaches have been developed and what progress has been made.

Then we hope to show how the new approach can be naturally extended to the domain of stochastic processes, and how the important probabilistic models of the processes can be tested in distribution free way. In discrete time, results for Markov chains have been published in 2021. Extension to continuous time will be explored during the current visit to University of Tokyo.
[ 参考URL ]

### 2022年10月19日(水)

10:30-11:40   数理科学研究科棟(駒場) 号室

fractional Brownian motion(fBm)に関係する汎関数のオーダー評価とfBmで駆動される確率微分方程式の二次変分の漸近展開について

[ 講演概要 ]

[ 参考URL ]

### 2022年07月21日(木)

13:30-14:40   数理科学研究科棟(駒場) -号室
オンライン開催

クロネッカー積表現に基づくテンソルデータに対する共通成分分析
[ 講演概要 ]

[ 参考URL ]
https://forms.gle/JrtVRcQNgn9pug3F7

### 2022年02月16日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Teppei Ogihara 氏 (University of Tokyo)
Efficient estimation for ergodic jump-diffusion processes
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

We study the estimation problem of the parametric model for ergodic jump-diffusion processes. Shimizu and Yoshida (Stat. Inference Stoch. Process. 2006) proposed a quasi-maximum-likelihood estimator based on a thresholding likelihood function that detects the existence of jumps.
In this talk, we consider the efficiency of estimators by using local asymptotic normality (LAN). To show the LAN property, we need to specify the asymptotic behavior of log-likelihood ratios, which is complicated for the jump-diffusion model because the transition probability for no jump is quite different from that for the presence of jumps. We develop techniques to show the LAN property based on transition density approximation. By applying these techniques to the thresholding likelihood function, we obtain the LAN property for the jump-diffusion model. Moreover, we have the asymptotic efficiency of
the quasi-maximum-likelihood estimator in Shimizu and Yoshida (2006) and a Bayes-type estimator proposed in Ogihara and Yoshida (Stat.Inference Stoch. Process. 2011). This is a joint work with Yuma Uehara (Kansai University).
[ 参考URL ]

### 2022年01月20日(木)

15:00-16:10   数理科学研究科棟(駒場) 号室

On weak factor models
[ 講演概要 ]

[ 参考URL ]

### 2022年01月19日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Martin Hazelton 氏 (Otago University)
Dynamic fibre samplers for linear inverse problems
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

Statistical inverse problems occur when we wish to learn about some random process that is observed only indirectly. Inference in such situations typically involves sampling possible values for the latent variables of interest conditional on the indirect observations. For count data, the latent variables are constrained to lie on a fibre (solution set for the linear system) comprising the integer lattice within a convex polytope.

Sampling the latent counts can be conducted using MCMC methods,through a random walk on this fibre. A major challenge is finding a set of basic moves that ensures connectedness of the walk over the fibre. In principle this can be done by computing a Markov basis of potential moves, but the resulting sampler can be hugely inefficient even when such a basis is computable. In this talk I will describe some current work on developing a dynamic Markov basis that generates moves on the fly. This approach can guarantee irreducibility of the sampler while gaining efficiency by increasing the probability of selecting serviceable sampling directions.
[ 参考URL ]

### 2021年12月15日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Estate Khmaladze 氏 (Victoria University of Wellington)
Theory of Distribution-free Testing
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

The aim of the talk is to introduce transformations of empirical-type processes by a group of unitary operators. Recall that if v_{nP} is empirical process on real line, based on a sample from P, it can be mapped into empirical process v_{nQ} by appropriate change of time

v_{nP}(h(x))=v_{nQ}(x)

where h(x) is continuous and increasing. This is the basis for distribution-free theory of goodness of fit testing. If w(\phi) is a function-parametric “empirical-type” process (i.e. has functions \phi from a space L as a time) and if K* is a unitary operator on L, then transformed process Kw we define as
Kw(\phi) = w(K*\phi)

These two formulas have good similarity, but one transformation in on the real line, while the other transformation in on functional space.This later one turns out to be of very broad use, and allows to base distribution-free theory upon it. Examples, we have specific results for, are parametric empirical
processes in R^d, regression empirical processes, those in GLM, parametric models for point processes and for Markov processes in discrete time. Hopefully, further examples will follow.
[ 参考URL ]

### 2021年11月17日(水)

15:30-17:00   数理科学研究科棟(駒場) 号室
Jean Bertoin 氏 (Institut of Mathematics, University of Zurich (UZH))
On the local times of noise reinforced Bessel processes
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

Bessel processes form a one-parameter family of self-similar diffusion on $[0,\infty)$ with the same Hurst exponent 1/2 as Brownian motion. Loosely speaking, in this setting, linear noise reinforcement with reinforcement parameter $p$ consists of repeating (if $p>0$) or counterbalancing (if $p<0$)infinitesimal increments of the process, uniformly at random and at a fixed rate as time passes. In this talk, we will investigate the effect of noise reinforcement on the local time at level $0$, that is, informally, the time that the process spends at $0$. A connection with increasing self-similar Markov processes will play a key role.
[ 参考URL ]

### 2021年10月13日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Li Cheng 氏 (National University of Singapore (NUS))
Bayesian Fixed-domain Asymptotics for Covariance Parameters in Gaussian Random Field Models
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

Gaussian random field models are commonly used for modeling spatial processes. In this work we focus on the Gaussian process with isotropic Matern covariance functions. Under fixed-domain asymptotics,it is well known that when the dimension of data is less than or equal to three, the microergodic parameter can be consistently estimated with asymptotic normality while the range (or length-scale) parameter cannot. Motivated by this frequentist result, we prove that under a Bayesian fixed-domain framework, the posterior distribution of the microergodic parameter converges in total variation norm to a normal distribution with shrinking variance, while the posterior of the range parameter does not necessarily converge. Built on this new theory, we further show that the Bayesian kriging predictor satisfies the posterior asymptotic efficiency in linear prediction. We illustrate these asymptotic results in numerical examples.
[ 参考URL ]

### 2021年09月15日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Anup Biswas 氏 (Indian Institute of Science Education and Research (IISER), Pune)
Ergodic risk-sensitive control: history, new results and open problems
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

Risk-sensitive control became popular because of the robustness it provides to the optimal control. Its connection to the theory of large deviation also made it a natural candidate of mathematical interest. In this talk, we shall give an overview of the history of risk-sensitive control problems and some of its applications. We shall then (informally) discuss the ways of tackling this problem and the main questions of interest. At the end, we shall see some important open problems.
[ 参考URL ]

### 2021年08月18日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Gery Geenens 氏 (The University of New South Wales (UNSW Sydney))
Dependence, Sklar's copulas and discreteness
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

Copulas have now become ubiquitous statistical tools for describing, analysing and modelling dependence between random variables. Yet the classical copula approach, building on Sklar’s theorem, cannot be legitimised if the variables of interest are not continuous. Indeed in the presence of discreteness, copula models are (i) unidentifiable, and (ii) not margin-free, and this by construction. In spite of the serious inconsistencies that this creates, downplaying statements are widespread in the literature, where copula methods are devised and used in discrete settings. In this work we call to reconsidering this current practice. To reconcile copulas with discreteness, we argued that they should be apprehended from a more fundamental perspective. Inspired by century-old ideas of Yule, we propose a novel construction which allows all the pleasant properties of copulas for modelling dependence (in particular:‘margin-freeness’) to smoothly carry over to the discrete setting.
[ 参考URL ]

### 2021年07月14日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Anirvan Chakraborty 氏 ( IISER Kolkata, India)
Statistics for Functional Data
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

With the advancement in technology, statisticians often have to analyze data which are curves or functions observed over a domain. Data of this type is usually called functional data and is very common these days in various fields of science. Statistical modelling of this type of data is usually done by viewing the data as a random sample from a probability distribution on some infinite dimensional function space. This formulation, however, implies that one has to delve into the mathematical rigour and complexity of dealing with infinite dimensional objects and probability distributions in function spaces. As such, standard multivariate statistical methods are far from useful in analyzing such data. We will discuss some statistical techniques for analyzing functional data as well as outline some of the unique challenges faced and also discuss some interesting open problems in this frontline research area.
[ 参考URL ]

### 2021年07月03日(土)

10:55-17:10   数理科学研究科棟(駒場) 号室

- 氏 (-)
-
[ 講演概要 ]

7月3日（土）
10:55 – 12:00 YUIMAパッケージの基本（Zoomサーバ不具合のため時間変更）
13:00 – 14:10 qmle, 漸近正規性，信頼区間，統計推測
14:30 – 15:40 qmle, 漸近正規性，信頼区間，統計推測
16:00 – 17:10 高頻度データ解析入門

7月4日（日）

14:30 – 15:40 レヴィ過程の基本と応用
16:00 – 17:10 レヴィ過程の基本と応用
17:20 – フリーディスカッション

YUIMAパッケージを通じて，確率微分方程式の直感的理解とシミュレーション，およびモデリングについてのスキルを習得できます．PCを用いた実習も行います．大学初年次程度の微分積分の知識が必要です．また，R言語の知識があるとよりスムーズです． 幅広い分野の学生・研究者・社会人の参加を歓迎します．

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・参加無料
[ 参考URL ]
http://www.sigmath.es.osaka-u.ac.jp/statmodel/?page_id=2028

### 2021年06月16日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Hiroki Masuda 氏 (Kyushu University)
Levy-Ornstein-Uhlenbeck Regression
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

We will present some of recent developments in parametric inference for a linear regression model driven by a non-Gaussian stable Levy process, when the process is observed at high frequency over a fixed time period. The model depends on a covariate process and the finite-dimensional parameter: the stability index (activity index) and the scale in the noise term, and the (auto)regression coefficients in the trend term, all being unknown. The maximum-likelihood estimator is shown to be asymptotically mixed-normally distributed with maximum concentration property. In order to bypass possible multiple-root problem and heavy numerical optimization, we also consider some easily computable initial estimator with which the one-step improvement does work. The asymptotic properties hold true in a unified manner regardless of whether the model is stationary and/or ergodic, almost without taking care of character of the
covariate process. Also discussed will be model-selection issues and some possible model extensions.
[ 参考URL ]

### 2021年05月19日(水)

14:30-16:00   オンライン開催

Federico Camia 氏 (NYU Abu Dhabi)
Limit Theorems and Random Fractal Curves in Statistical Mechanics (ENGLISH)
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

Statistical mechanics deals with systems that have a large number of components whose behavior can often be considered random. For this reason, probability theory plays an essential role in the mathematical analysis of the subject. After a gentle introduction to statistical mechanics, I will give a nontechnical overview of recent, exciting developments that combine in a beautiful and unexpected way discrete probability, stochastic processes and complex analysis. For concreteness, I will discuss two specific models, percolation and the Ising model, which have a long history, have played an important part in the development of statistical mechanics, and occupy a central place in modern probability theory.
[ 参考URL ]

### 2021年05月19日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Federico Camia 氏 (NYU Abu Dhabi)
Limit Theorems and Random Fractal Curves in Statistical Mechanics
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

Statistical mechanics deals with systems that have a large number of components whose behavior can often be considered random. For this reason, probability theory plays an essential role in the mathematical analysis of the subject. After a gentle introduction to statistical mechanics, I will give a nontechnical overview of recent, exciting developments that combine in a beautiful and unexpected way discrete probability, stochastic processes and complex analysis. For concreteness, I will discuss two specific models, percolation and the Ising model, which have a long history, have played an important part in the development of statistical mechanics, and occupy a central place in modern probability theory.
[ 参考URL ]

### 2021年04月21日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Zoomでの開催となります。３日前までに講演参考URLから参加申込みをしてください。
Han Liang Gan 氏 (University of Waikato)
Stationary distribution approximations for two-island and seed bank models (ENGLISH)

[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

Two-island Wright-Fisher models are used to model genetic frequencies and variability for subdivided populations. One of the key components of the model is the level of migration between the two
islands. We show that as the population size increases, the appropriate approximation and limit for the stationary distribution of a two-island Wright-Fisher Markov chain depends on the level of migration. In a seed bank model, individuals in one of the islands stay dormant rather than reproduce. We give analogous results for the seed bank model, compare and contrast the differences and examine the effect the seed bank has on genetic variability. Our results are derived from a new development of Stein's method for the two-island diffusion model and existing results for Stein's method for the Dirichlet distribution.

This talk is based on joint work with Adrian Röllin, Nathan Ross and Maite Wilke Berenguer.
[ 参考URL ]

### 2021年04月21日(水)

14:30-16:00   数理科学研究科棟(駒場) 号室
Han Liang Gan 氏 (University of Waikato)
Stationary distribution approximations for two-island and seed bank models
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics (APSPS)

Two-island Wright-Fisher models are used to model genetic frequencies and variability for subdivided populations. One of the key components of the model is the level of migration between the two islands. We show that as the population size increases, the appropriate approximation and limit for the stationary distribution of a two-island Wright-Fisher Markov chain depends on the level of migration. In a seed bank model, individuals in one of the islands stay dormant rather than reproduce. We give analogous results for the seed bank model, compare and contrast the differences and examine the effect the seed bank has on genetic variability. Our results are derived from a new development of Stein's method for the two-island diffusion model and existing results for
Stein's method for the Dirichlet distribution.

This talk is based on joint work with Adrian Röllin, Nathan Ross and Maite Wilke Berenguer.
[ 参考URL ]

### 2021年03月29日(月)

14:00-15:10   オンライン開催

ガウス近似を用いたM推定量の統計的推論 (JAPANESE)
[ 講演概要 ]
M推定量とは、経験基準関数の最大化として定義される推定量で、最尤推定量や経験誤差最小化推定量を含む広い推定量の広いクラスである。M推定量の分布を近似することは、多くの統計的推論の基盤をなす重要な研究トピックであ理、これまで各論および一般論を問わず多くの研究が行われてきた。本研究では、既存の極限分布を用いたアプローチとは対照的に、非漸近的なガウス過程による近似法を採用し、M推定量の分布近似理論を構成した。加えて、実用的なガウス係数ブートストラップ近似法を提案した。これらのアプローチは、近年発展著しい経験過程の最大値の分布近似理論を拡張することで得られている。本研究は、最小絶対偏差推定量のような正則的な推定量だけでなく、non-Donsker級やcubic-root推定量のような、漸近分布の導出や数値計算が困難な非正則な場合を扱うことができる。
[ 参考URL ]

### 2021年03月24日(水)

14:30-16:00   オンライン開催

Rachel Fewster 氏 (University of Auckland)
Stochastic modelling in ecology: why is it interesting? (ENGLISH)
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics https://sites.google.com/view/apsps/home

The ecological sciences offer rich pickings for stochastic modellers. There is currently an abundance of new technologies for　monitoring wildlife and biodiversity, for which no practicable data-analysis methods exist. Often, modelling approaches that are motivated by a specific problem with relatively narrow focus can turn out to have　surprisingly broad application elsewhere. As the generality of the problem structure becomes clear, this can also motivate new statistical theory.

I will describe some ecological modelling scenarios that have led to interesting developments from methodological and theoretical perspectives.　As time allows, these will include: saddlepoint approximations for dealing　with data corrupted by non-invertible linear transformations; information theory for assuring that it is a good idea to unite data from multiple sources; and methods for dealing with so-called 'enigmatic' data from remote sensors, involving a blend of ideas from point processes, queuing theory, and trigonometry. All scenarios will be generously illustrated with　pictures of charismatic wildlife.

This talk covers joint work with numerous collaborators, especially Joey Wei Zhang, Mark Bravington, Peter Jupp, Jesse Goodman, Martin Hazelton,　Godrick Oketch, Ben Stevenson, David Borchers, Paul van Dam-Bates, and　Stephen Marsland.
[ 参考URL ]

### 2021年02月17日(水)

14:30-15:30   数理科学研究科棟(駒場) Zoom号室

Nakahiro Yoshida 氏 (University of Tokyo)
Quasi-likelihood analysis for stochastic differential equations: volatility estimation and global jump filters (ENGLISH)
[ 講演概要 ]
Asia-Pacific Seminar in Probability and Statistics https://sites.google.com/view/apsps/home

The quasi likelihood analysis (QLA) is a framework of statistical inference for stochastic processes, featuring the quasi-likelihood random field and the polynomial type large deviation inequality. The QLA enables us to systematically derive limit theorems and tail probability estimates for the associated QLA estimators (quasi-maximum likelihood estimator and quasi-Bayesian estimator) for various dependent models. The first half of the talk will be devoted to an introduction to the QLA for stochastic differential equations. The second half presents recent developments in a filtering problem to estimate volatility from the data contaminated with jumps. A QLA for volatility for a stochastic differential equation with jumps is constructed, based on a "global jump filter" that uses all the increments of the process to decide whether an increment has jumps.

Key words: stochastic differential equation, high frequency data, Le Cam-Hajek theory, Ibragimov-Has'minskii-Kutoyants program, polynomial type large deviation inequality, quasi-maximum likelihood estimator, quasi-Bayesian estimator, L^p-estimates of the error, non-ergodic statistics, asymptotic (mixed) normality.
[ 参考URL ]