## Seminar on Probability and Statistics

Organizer(s) Nakahiro Yoshida, Teppei Ogihara, Yuta Koike

Seminar information archive

### 2020/12/16

14:30-16:00   Room #Zoom (Graduate School of Math. Sci. Bldg.)
Register at least 3 days before at the reference URL. The URL for participation sent before the seminar.
Parthanil Roy (Indian Statistical Institute, Bangalore)
How to tell a tale of two tails? (ENGLISH)
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics https://sites.google.com/view/apsps/home

Branching random walk is a system of growing particles that starts with one particle. This particle branches into a random number of particles, and each new particle makes a random displacement independently of each other and of the branching mechanism. The same dynamics goes on and gives rise to a branching random walk. This model arises in statistical physics, and has connections to various probabilistic objects, mathematical biology, ecology, etc. In this overview talk, we shall discuss branching random walks and their long run behaviour. More precisely, we shall try to answer the following question: if we run a branching random walk for a very long time and take a snapshot of the particles, how would the system look like? We shall investigate how the tails of the progeny and displacement distributions change the answer to this question.
This talk is based on a series of joint papers with Ayan Bhattacharya, Rajat Subhra Hazra, Krishanu Maulik, Zbigniew Palmowski, Souvik Ray and Philippe Soulier.
[ Reference URL ]

### 2020/11/27

17:00-18:10   Room #on-line (Graduate School of Math. Sci. Bldg.)
Register at least 3 days before at the reference URL. The URL for participation sent before the seminar.
Yuliya Mishura (Taras Shevchenko National University of Kyiv)
Processes with small ball estimate: properties, examples, statistical inference (ENGLISH)
[ Abstract ]
The notion of a process with small ball estimate is introduced and studied. In particular, divergence of integral functional of such process is established and applied to statistical estimation. Several interesting examples are provided, and various modifications of the main group of properties are considered. The talk is based on the common research with Prof. Nakahiro Yoshida.
[ Reference URL ]

### 2020/11/18

14:30-16:00   Room #Zoom (Graduate School of Math. Sci. Bldg.)
Please register at least 3 days before at the reference URL. The URL for participation will be sent before the seminar.
Sanjay Chaudhuri (National University of Singapore)
Hamiltonian Monte Carlo In Bayesian Empirical Likelihood Computation (English)
[ Abstract ]
Asia-Pacific Seminar in Probability and Statistics https://sites.google.com/view/apsps/home

Abstract: We consider Bayesian empirical likelihood estimation and develop an efficient Hamiltonian Monte Carlo method for sampling from the posterior distribution of the parameters of interest. The proposed method uses hitherto unknown properties of the gradient of the underlying log-empirical likelihood function. It is seen that these properties hold under minimal assumptions on the parameter space, prior density and the functions used in the estimating equations determining the empirical likelihood. We overcome major challenges posed by complex, non-convex boundaries of the support routinely observed for empirical likelihood which prevents efficient implementation of traditional Markov chain Monte Carlo methods like random walk Metropolis-Hastings etc. with or without parallel tempering. Our method employs finite number of estimating equations and observations but produces valid semi-parametric inference for a large class of statistical models including mixed effects models, generalised linear models, hierarchical Bayes models etc. A simulation study confirms that our proposed method converges quickly and draws samples from the posterior support efficiently. We further illustrate its utility through an analysis of a discrete data-set in small area estimation.

Keywords: Constrained convex optimisation; Empirical likelihood; Generalised linear models; Hamiltonian Monte Carlo; Mixed effect models; Score equations; Small area estimation; Unbiased estimating equations.

This is a joint work with Debashis Mondal, Oregon State University and Yin Teng, E&Y, Singapore.
[ Reference URL ]

### 2020/10/19

10:30-11:30   Room #Zoom (Graduate School of Math. Sci. Bldg.)
Yuta Koike (University of Tokyo)
Development of high-dimensional CLTs arising from high-frequency data analysis (日本語)
[ Reference URL ]

### 2020/05/01

17:00-18:10   Room #φ (Graduate School of Math. Sci. Bldg.)
Xiao Fang (Chinese University of Hong Kong)
High order distributional approximations by Stein's method (ENGLISH)
[ Abstract ]
Stein's method is a powerful tool to proving distributional approximations with error bounds. In this talk, we present two recent developments of Stein's method for high order approximations. (1) Together with Li Luo and Qi-Man Shao, we consider skewness correction in normal approximation. We prove a refined Cram¥'er-type moderate deviation result for a class of statistics possessing a local structure. We discuss applications to k-runs, U-statistics and subgraph counts. (2) Together with Anton Braverman and Jim Dai, we derive and justify new diffusion approximations with state-dependent diffusion coefficients for stationary distributions of Markov chains. We discuss applications to the Erlang-C system, a hospital inpatient flow model and the auto-regressive model.
[ Reference URL ]

### 2020/04/16

17:00-18:10   Room #φ (Graduate School of Math. Sci. Bldg.)
Masatoshi Goda (University of Tokyo)
Hawkes process and Edgeworth expansion with application to Maximum Likelihood Estimator (JAPANESE)
[ Abstract ]
The Hawkes process is a point process with a self-exciting property. It has been used to model earthquakes, social media events, infections, etc. and is getting a lot of attention. However, as a real problem, there are often situations where we can not obtain data with sufficient observation time. In such cases, it is not appropriate to approximate the error distribution of an estimator by the normal distribution. We established the Edgeworth expansion for a functional of a geometric mixing process, and applied this scheme to a functional of the Hawkes process with an exponential kernel. Furthermore, we gave a more appropriate asymptotic distribution for the error of the Maximum Likelihood Estimator of the Hawks process, i.e. a higher-order asymptotic distribution than the normal distribution. Here, in addition to the details of these statements, we also present the simulation results.
[ Reference URL ]

### 2020/02/12

15:00-16:10   Room #126 (Graduate School of Math. Sci. Bldg.)
Lorenzo Mercuri (University of Milan)
Handling the underlying noise of Stochastic Differential Equations in YUIMA project

[ Abstract ]
Some advances in the implementation of advanced mathematical tools and numerical methods for an object of class yuima.law are presented and discussed. An object of yuima.law-class refers to the mathematical description of the underlying noise specified in the formal definition of a general Stochastic Differential Equation. Its aim is to create a link between YUIMA and other R packages available on CRAN for managing specific Lévy noises. Here we present as examples the simulation and the estimation of a CARMA(p,q) and Point Process regression models.

### 2019/12/27

15:00-16:10   Room #126 (Graduate School of Math. Sci. Bldg.)
Xiao Fang (Chinese University of Hong Kong)
High order distributional approximations by Stein's method
[ Abstract ]
Stein's method is a powerful tool to proving distributional approximations with error bounds. In this talk, we present two recent developments of Stein's method for high order approximations. (1) Together with Li Luo and Qi-Man Shao, we consider skewness correction in normal approximation. We prove a refined Cram\'er-type moderate deviation result for a class of statistics possessing a local structure. We discuss applications to $k$-runs, U-statistics and subgraph counts. (2) Together with Anton Braverman and Jim Dai, we derive and justify new diffusion approximations with state-dependent diffusion coefficients for stationary distributions of Markov chains. We discuss applications to the Erlang-C system, a hospital inpatient flow model and the auto-regressive model.

### 2019/12/27

16:30-17:40   Room #126 (Graduate School of Math. Sci. Bldg.)
Nikolai Leonenko (Cardiff University)
Heavy-Tailed Fractional Pearson Diffusions
[ Abstract ]
We define fractional Pearson diffusions [5,7,8] by non-Markovian time change in the corresponding Pearson diffusions [1,2,3,4]. They are governed by the time-fractional diffusion equations with polynomial coefficients depending on the parameters of the corresponding Pearson distribution. We present the spectral representation of transition densities of fractional Pearson diffusions, which depend heavily on the structure of the spectrum of the infinitesimal generator of the corresponding non-fractional Pearson diffusion. Also, we present the strong solutions of the Cauchy problems associated with heavy-tailed fractional Pearson diffusions and the correlation structure of these diffusions  .
Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs) [9,10,11]. The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model, Wright-Fisher model and Ehrenfest-Brillouin-type models. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments.

This is a joint work with M. Meerschaert (Michigan State University, USA), I. Papic (University of Osijek, Croatia), N.Suvak (University of Osijek, Croatia) and A. Sikorskii (Michigan State University and Arizona University, USA).

References:
 Avram, F., Leonenko, N.N and Suvak, N. (2013), On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusion, Markov Processes and Related Fields, Volume 19, N 2 , 249-298
 Avram, F., Leonenko, N.N and Suvak, N., (2013), Spectral representation of transition density of Fisher-Snedecor diffusion, Stochastics, 85 (2013), no. 2, 346—369
 Bourguin, S., Campese, S., Leonenko, N. and Taqqu,M.S. (2019) Four moments theorems on Markov chaos, Annals of Probability, 47, N3, 1417–1446
 Kulik, A.M. and Leonenko, N.N. (2013) Ergodicity and mixing bounds for the Fisher-Snendecor diffusion, Bernoulli, Vol. 19, No. 5B, 2294-2329
 Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Fractional Pearson diffusions, Journal of Mathematical Analysis and Applications, vol. 403, 532-546
 Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Correlation Structure of Fractional Pearson diffusion, Computers and Mathematics with Applications, 66, 737-745
 Leonenko,N.N., Meerschaert,M.M., Schilling,R.L. and Sikorskii, A. (2014) Correlation Structure of Time-Changed Lévy Processes, Communications in Applied and Industrial Mathematics, Vol. 6 , No. 1, p. e-483 (22 pp.)
 Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2017) Heavy-tailed fractional Pearson diffusions, Stochastic Processes and their Applications, 127, N11, 3512-3535
 Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2018) Correlated continuous time random walks and fractional Pearson diffusions, Bernoulli, Vol. 24, No. 4B, 3603-3627
 Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Ehrenfest-Brillouin-type correlated continuous time random walks and fractional Jacoby diffusion, Theory Probablity and Mathematical Statistics, Vol. 99,123-133.
 Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod topology, submitted

### 2019/06/18

11:00-12:10   Room #052 (Graduate School of Math. Sci. Bldg.)
Xiaohui Chen (University of Illinois at Urbana–Champaign)
Gaussian and bootstrap approximations of high-dimensional U-statistics with applications and extensions　※変更の可能性あり

[ Abstract ]
We shall first discuss the Gaussian approximation of high-dimensional and non-degenerate U-statistics of order two under the supremum norm. A two-step Gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed. Subject to mild moment conditions on the kernel, we establish the explicit rate of convergence that decays polynomially in sample size for a high-dimensional scaling limit, where the dimension can be much larger than the sample size. We also provide computable approximation methods for the quantiles of the maxima of centered U-statistics. Specifically, we provide a unified perspective for the empirical, the randomly reweighted, and the multiplier bootstraps as randomly reweighted quadratic forms, all asymptotically valid and inferentially first-order equivalent in high-dimensions.

The bootstrap methods are applied on statistical applications for high-dimensional non-Gaussian data including: (i) principled and data-dependent tuning parameter selection for regularized estimation of the covariance matrix and its related functionals; (ii) simultaneous inference for the covariance and rank correlation matrices. In particular, for the thresholded covariance matrix estimator with the bootstrap selected tuning parameter, we show that the Gaussian-like convergence rates can be achieved for heavy-tailed data, which are less conservative than those obtained by the Bonferroni technique that
ignores the dependency in the underlying data distribution. In addition, we also show that even for subgaussian distributions, error bounds of the bootstrapped thresholded covariance matrix estimator can be much tighter than those of the minimax estimator with a universal threshold.

Time permitting, we will discuss some extensions to the infinite-dimensional version (i.e., U-processes of increasing complexity) and to the randomized inference via the incomplete U-statistics whose computational cost can be made independent of the order.

### 2019/03/13

14:00-15:10   Room #052 (Graduate School of Math. Sci. Bldg.)
Yury A. Kutoyants (Laboratoire Manceau de Mathématiques, Le Mans University)
On parameter estimation of hidden Markov processes
[ Abstract ]
We present a survey of several results devoted to parameter estimation of partially observed models. The hidden processes are Ornstein-Uhlenbeck process and Telegraph process. We describe the asymptotic behavior of the MLE and BE of the unknown parameters of hidden processes and special attention is paid to a new class of estimators called Multi-step MLE-processes, which have the same asymptotic properties as the MLE but can be calculated much easier than MLE.

The corresponding articles are

1.Kutoyants Yu. A., " On the multi-step MLE-process for ergodic
diffusion", (arXiv 1504.01869) Stochastic Processes and their
Applications, 2017, 127, 2243-2261.

2.Khasminskii, R. Z. and Kutoyants, Yu. A. "On parameter estimation of
hidden telegraph process". (arXiv:1509.02704 ) Bernoulli, 2018, 24, 3,
2064-2090.

3.Kutoyants, Yu. A. "On parameter estimation of hidden
Ornstein-Uhlenbeck process", Journal of Multivariate Analysis. 2019,
169, 1, 248-263.

4.Kutoyants, Yu. A. "On parameter estimation of hidden ergodic
Ornstein-Uhlenbeck process", 2019, submitted (arXiv:1902.08500)

### 2019/02/06

16:30-18:00   Room #270 (Graduate School of Math. Sci. Bldg.)
Ioane Muni Toke (Centrale Supelec Paris)
Testing the causality of Hawkes processes with time reversal
[ Abstract ]
We show that univariate and symmetric multivariate Hawkes processes are only weakly causal: the true log-likelihoods of real and reversed event time vectors are almost equal, thus parameter estimation via maximum likelihood only weakly depends on the direction of the arrow of time. In ideal (synthetic) conditions, tests of goodness of parametric fit unambiguously reject backward event times, which implies that inferring kernels from time-symmetric quantities, such as the autocovariance of the event rate, only rarely produce statistically significant fits. Finally, we find that fitting financial data with many-parameter kernels may yield significant fits for both arrows of time for the same event time vector, sometimes favouring the backward time direction. This goes to show that a significant fit of Hawkes processes to real data with flexible kernels does not imply a definite arrow of time unless one tests it.

### 2018/12/05

13:00-15:00   Room #156 (Graduate School of Math. Sci. Bldg.)
Yuliia Mishura (The Taras Shevchenko National University of Kiev)
Lecture 2:Representation results for the Gaussian processes. Financial applications of fractional Brownian motion

[ Abstract ]
Arbitrage with fBm: why it appears. How to present any contingent claim via self-financing strategy on the financial market involving fBm. Absence of arbitrage for the mixed models. Fractional -Uhlenbeck and fractional Cox-Ingersoll-Ross processes as the models for stochastic volatility.

### 2018/12/05

15:00-17:00   Room #156 (Graduate School of Math. Sci. Bldg.)
Yuliia Mishura (The Taras Shevchenko National University of Kiev)
Lecture 3:Statistical parameter estimation for the diffusion processes and in the models involving fBm

[ Abstract ]
Drift parameter estimation in the standard diffusion model and its strong consistency. Hurst and drift parameter estimation in the models involving fBm and in the mixed models. Asymptotic properties. Estimation of the diffusion parameter.

### 2018/12/04

15:00-17:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Yuliia Mishura (The Taras Shevchenko National University of Kiev)
Lecture 1: Elements of fractional calculus
How to connect the fractional Brownian motion to the Wiener process. Stochastic integration w.r.t. fBm and stochastic differential equations involving fB
[ Abstract ]
Fractional integrals and fractional derivatives. Wiener and stochastic integration w.r.t. the fractional Brownian motion. Representations of fBm via a Wiener process and vice versa. Elements of the fractional stochastic calculus. Stochastic differential equations involving fBm: existence, uniqueness, properties of the solutions. Simplest models: fractional Ornstein-Uhlenbeck and fractional Cox-Ingersoll-Ross processes.

### 2018/11/09

11:00-12:00   Room #123 (Graduate School of Math. Sci. Bldg.)
Frédéric Abergel (CentraleSupélec)
Market impact and option hedging in the presence of liquidity costs

[ Abstract ]
The phenomenon of market (or: price) impact is well-known among practicioners, and it has long been recognized as a key feature of trading in electronic markets. In the first part of this talk, I will present some new, recent results on market impact, especially for limit orders. I will then propose a theory for option hedging in the presence of liquidity costs.(Based on joint works with E. Saïd, G. Loeper).

### 2018/10/30

15:30-16:40   Room #126 (Graduate School of Math. Sci. Bldg.)
Ciprian A. Tudor (Université de Lille 1, Université de Panthéon-Sorbonne Paris 1)
Asymptotic expansion for random vectors
[ Abstract ]
We develop the asymptotic expansion theory for vector-valued sequences $F_{N}$ of random variables. We find the second-order term in the expansion of the density of $F_{N}$, based on assumptions in terms of the convergence of the Stein-Malliavin matrix associated to the sequence $F_{N}$ . Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of $F_{N}$ and we discuss the main ideas on higher order asymptotic expansion. We illustrate our results by several examples.

### 2018/05/23

14:00-15:10   Room #052 (Graduate School of Math. Sci. Bldg.)
Lorenzo Mercuri (University of Milan)
"yuima.law": From mathematical representation of general Lévy processes to a numerical implementation
[ Abstract ]
We present a new class called yuima.law that refers to the mathematical description of a general Lévy process used in the formal definition of a general Stochastic Differential Equation. The final aim is to have an object, defined by the user, that contains all possible information about the Lévy process considered. This class creates a link between YUIMA and other R packages available on CRAN that manage specific Lévy processes.

An example of yuima.law is shown based the Mixed Tempered Stable(MixedTS) Lévy processes. A review of the univariate MixedTS is given and some new results on the asymptotic tail behaviour are derived. The multivariate version of the Mixed Tempered Stable, which is a generalisation of the Normal Variance Mean Mixtures, is discussed. Characteristics of this distribution, its capacity in fitting tails and in capturing dependence structure between components are investigated.

### 2018/05/23

15:30-16:40   Room #052 (Graduate School of Math. Sci. Bldg.)
Emanuele Guidotti (University of Milan)
Latest Development in yuimaGUI - Interactive Platform for Computational Statistics and Finance

[ Abstract ]
The yuimaGUI package provides a user-friendly interface for the yuima package, including additional tools related to Quantitative Finance. It greatly simplifies tasks such as estimation and simulation of stochastic processes, data retrieval, time series clustering, change point and lead-lag analysis. Today we are going to discuss the latest development in yuimaGUI, extending the Platform with multivariate modeling and simulation, Levy processes, Point processes, broader model selection tools and more general distributions thanks to the new yuima-Law object.

### 2018/05/08

15:00-16:10   Room #052 (Graduate School of Math. Sci. Bldg.)

### 2018/03/15

16:00-17:10   Room #052 (Graduate School of Math. Sci. Bldg.)
Stefano Iacus (University of Milan)
On Hypotheses testing for discretely observed SDE (Joint work with Alessandro De Gregorio, University of Rome)
[ Abstract ]
In this talk we consider parametric hypotheses testing for discretely observed ergodic diffusion processes. We present the different test statistics proposed in literature and recall their asymptotic properties. We also compare the empirical performance of different tests in the case of small sample sizes.

### 2018/03/02

15:00-16:10   Room #270 (Graduate School of Math. Sci. Bldg.)
Arnaud Gloter (Université d'Evry Val d'Essonne)
"Estimating functions for SDE driven by stable Lévy processes"
Joint work with Emmanuelle Clément (Ecole Centrale)
[ Abstract ]
In this talk we will discuss about parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an α-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of α ∈ (0, 2).

### 2018/02/04

12:00-18:00   Room #118 (Graduate School of Math. Sci. Bldg.)
Yukai Yang (Uppsala University) 12:30-15:00
Nonlinear Economic Time Series Models

[ Abstract ]
The lecture goes through several chapters in the book “Modelling Nonlinear Economic Time Series” by Teräsvirta, Tjøstheim and Granger in 2010. The lecture serves as an introduction for the students and researchers who are interested in this area. It introduces a number of examples of families of nonlinear time series parametric models in economic theory. It also talks about testing linearity against parametric alternatives with the presence of a characterization of the identification problem in many situations. Different ways of solving the identification problem are presented and their merits and disadvantages are discussed.
Yuliia Mishura (The Taras Shevchenko National University of Kiev ) 15:30-18:00
Analytic, representation and statistical aspects related to fractional Gaussian processes.

[ Abstract ]
We consider the properties of fractional Gaussian processes whose covariance function is situated between two self-similarities, or, in other words, these processes belong to the generalized quasi-helix, according to geometric terminology of Kahane. For such processes we consider the two-sided bounds for maximal functionals and the representation results. We consider stochastic differential equations involving fractional Brownian motion and present also several results on statistical estimations for them.

### 2018/02/02

13:30-14:40   Room #052 (Graduate School of Math. Sci. Bldg.)
Ioane Muni Toke (Centrale Supelec Paris)
Estimation of ratios of intensities in a Cox-type model of limit order books
[ Abstract ]
We introduce a Cox-type model for relative intensities of orders flows in a limit order book. The Cox-like intensities of the counting processes of events are assumed to share an unobserved and unspecified baseline intensity, which in finance can be identified to a global market activity affecting all events. The model is formulated in terms of relative responses of the intensities to covariates, and relative parameters can be estimated by quasi likelihood maximization. Consistency and asymptotic normality of the estimators are proven. Computationally intensive inferences are run on large samples of tick-by-tick data (35+ stocks and 220+ trading days, adding to more than one billion events). Penalization methods are also investigated. Results of the model are interpreted in terms of probability of occurrence of events. Excellent agreement with empirical data is found. Estimated model reproduces known empirical facts on imbalance, spread and queue sizes, and helps identifying trading signals of interests on a given stock.

Joint work with N.Yoshida.

### 2017/11/16

13：00-16：00   Room #123 (Graduate School of Math. Sci. Bldg.)