統計数学セミナー

過去の記録 ~07/24次回の予定今後の予定 07/25~

担当者 吉田朋広、増田弘毅、荻原哲平、小池祐太
セミナーURL http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/
目的 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.

過去の記録

2021年03月29日(月)

14:00-15:10   オンライン開催
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今泉允聡 氏 (東京大学)
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 ]
https://docs.google.com/forms/d/e/1FAIpQLSfjQhmmZjWUllB6pQeEMGDRcLCe_0JPgVbEA05rHtcDYAZzqg/viewform

2021年03月24日(水)

14:30-16:00   オンライン開催
参考URLのGoogle Formより3日前までに参加登録してください。 ご登録後、会議参加に必要なURLを送付いたします。
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 ]
https://docs.google.com/forms/d/e/1FAIpQLSf05P9fCZ5Wkasc7clW1XBpkeONPSjPKuCkNYb3oIqnOAu5Mg/viewform

2021年02月17日(水)

14:30-15:30   数理科学研究科棟(駒場) Zoom号室
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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 ]
https://docs.google.com/forms/d/e/1FAIpQLSeLrq_Ifc4WvJC6uvwIpMyrAVM9v-0J3FOaZbsplbU9d21ALw/viewform

2021年01月13日(水)

14:30-15:30   数理科学研究科棟(駒場) Zoom号室
Pierre Lafaye de Micheaux 氏 (UNSW)
Depth of Curve Data and Applications (ENGLISH)
[ 講演概要 ]
[ 参考URL ]
https://sites.google.com/view/apsps/previous-speakers

2020年12月16日(水)

14:30-16:00   数理科学研究科棟(駒場) Zoom号室
参考URLのGoogle Formより3日前までに参加登録してください。 ご登録後、会議参加に必要なURLを送付いたします。
Parthanil Roy 氏 (Indian Statistical Institute, Bangalore)
How to tell a tale of two tails? (ENGLISH)
[ 講演概要 ]
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.
[ 参考URL ]
https://docs.google.com/forms/d/e/1FAIpQLSf6XCBIUMnI9OJjNi6KP7QEixLnZVMsw8BVeNqiPFxlUC8rQQ/viewform

2020年11月27日(金)

17:00-18:10   数理科学研究科棟(駒場) on-line号室
参考URLのGoogle Formより3日前までに参加登録してください。 ご登録後、会議参加に必要なURLを送付いたします。
Yuliya Mishura 氏 (Taras Shevchenko National University of Kyiv)
Processes with small ball estimate: properties, examples, statistical inference (ENGLISH)
[ 講演概要 ]
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.
[ 参考URL ]
https://docs.google.com/forms/d/e/1FAIpQLSeYCDQS9c9i0gy-Y0YY-q5TPJlwGmWhCYnkKRE7udvMOoy0mw/viewform

2020年11月18日(水)

14:30-16:00   数理科学研究科棟(駒場) Zoom号室
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Sanjay Chaudhuri 氏 (National University of Singapore)
Hamiltonian Monte Carlo In Bayesian Empirical Likelihood Computation (English)
[ 講演概要 ]
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.

[ 参考URL ]
https://docs.google.com/forms/d/e/1FAIpQLSfbk6GTAzQuj0__YUtUMiAgbPWabT-M1vmbgldohiwPxPltuw/viewform

2020年10月19日(月)

10:30-11:30   数理科学研究科棟(駒場) Zoom号室
下記URLより事前参加登録をお願いします.
小池祐太 氏 (東京大学大学院数理科学研究科)
高頻度データ解析に端を発する高次元中心極限定理の展開について (日本語)
[ 講演概要 ]
本発表では、金融高頻度データ解析におけるリード・ラグ関係の有意性検定の問題に端を発して講演者が取り組んだ、高次元中心極限定理に関する最近の研究の進展について報告する。まず、元々の研究の背景であるリード・ラグ関係の有意性検定の問題について説明し、その問題を解くために講演者が証明した、多重Wiener-伊藤積分およびhomogeneous sumに対する高次元中心極限定理を述べる。次に、後者の結果の証明におけるアイディアを発展させることによって得られた、独立な確率ベクトルの和に対する高次元中心極限定理の精緻化に関する結果を紹介する。
[ 参考URL ]
https://docs.google.com/forms/d/e/1FAIpQLSfDhlzlC6haR8dsDn9_mCxi1s9RtXZxTi_U7Nb_Xl6q7Gw1dA/viewform

2020年05月01日(金)

17:00-18:10   数理科学研究科棟(駒場) φ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。
Xiao Fang 氏 (Chinese University of Hong Kong)
High order distributional approximations by Stein's method (ENGLISH)
[ 講演概要 ]
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.
[ 参考URL ]
https://docs.google.com/forms/d/e/1FAIpQLSeSVwYsjhyQQXzjt3ZpvRh9ZEO5qZXxxLxYDYOu301Mc89RCA/viewform

2020年04月16日(木)

17:00-18:10   数理科学研究科棟(駒場) φ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。
郷田昌稔 氏 (東京大学)
Hawkes process and Edgeworth expansion with application to Maximum Likelihood Estimator (JAPANESE)
[ 講演概要 ]
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.
[ 参考URL ]
https://docs.google.com/forms/d/18MDagC71CtB7SJmW2s0tPIBW9YDUNWH5XH1uby2W6Xc/edit

2020年02月12日(水)

15:00-16:10   数理科学研究科棟(駒場) 126号室
Lorenzo Mercuri 氏 (University of Milan)
Handling the underlying noise of Stochastic Differential Equations in YUIMA project

[ 講演概要 ]
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年12月27日(金)

15:00-16:10   数理科学研究科棟(駒場) 126号室
Xiao Fang 氏 (Chinese University of Hong Kong)
High order distributional approximations by Stein's method
[ 講演概要 ]
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年12月27日(金)

16:30-17:40   数理科学研究科棟(駒場) 126号室
Nikolai Leonenko 氏 (Cardiff University)
Heavy-Tailed Fractional Pearson Diffusions
[ 講演概要 ]
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 [6] .
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:
[1] 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
[2] 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
[3] Bourguin, S., Campese, S., Leonenko, N. and Taqqu,M.S. (2019) Four moments theorems on Markov chaos, Annals of Probability, 47, N3, 1417–1446
[4] Kulik, A.M. and Leonenko, N.N. (2013) Ergodicity and mixing bounds for the Fisher-Snendecor diffusion, Bernoulli, Vol. 19, No. 5B, 2294-2329
[5] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Fractional Pearson diffusions, Journal of Mathematical Analysis and Applications, vol. 403, 532-546
[6] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Correlation Structure of Fractional Pearson diffusion, Computers and Mathematics with Applications, 66, 737-745
[7] 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.)
[8] 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
[9] 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
[10] 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.
[11] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod topology, submitted

2019年06月18日(火)

11:00-12:10   数理科学研究科棟(駒場) 052号室
Xiaohui Chen 氏 (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

[ 講演概要 ]
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年03月13日(水)

14:00-15:10   数理科学研究科棟(駒場) 052号室
Yury A. Kutoyants 氏 (Laboratoire Manceau de Mathématiques, Le Mans University)
On parameter estimation of hidden Markov processes
[ 講演概要 ]
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年02月06日(水)

16:30-18:00   数理科学研究科棟(駒場) 270号室
Ioane Muni Toke 氏 (Centrale Supelec Paris)
Testing the causality of Hawkes processes with time reversal
[ 講演概要 ]
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年12月05日(水)

13:00-15:00   数理科学研究科棟(駒場) 156号室
Yuliia Mishura 氏 (The Taras Shevchenko National University of Kiev)
Lecture 2:Representation results for the Gaussian processes. Financial applications of fractional Brownian motion

[ 講演概要 ]
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年12月05日(水)

15:00-17:00   数理科学研究科棟(駒場) 156号室
Yuliia Mishura 氏 (The Taras Shevchenko National University of Kiev)
Lecture 3:Statistical parameter estimation for the diffusion processes and in the models involving fBm

[ 講演概要 ]
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年12月04日(火)

15:00-17:00   数理科学研究科棟(駒場) 126号室
Yuliia Mishura 氏 (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
[ 講演概要 ]
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年11月09日(金)

11:00-12:00   数理科学研究科棟(駒場) 123号室
Frédéric Abergel 氏 (CentraleSupélec)
Market impact and option hedging in the presence of liquidity costs

[ 講演概要 ]
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年10月30日(火)

15:30-16:40   数理科学研究科棟(駒場) 126号室
Ciprian A. Tudor 氏 (Université de Lille 1, Université de Panthéon-Sorbonne Paris 1)
Asymptotic expansion for random vectors
[ 講演概要 ]
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年05月23日(水)

14:00-15:10   数理科学研究科棟(駒場) 052号室
Lorenzo Mercuri 氏 (University of Milan)
"yuima.law": From mathematical representation of general Lévy processes to a numerical implementation
[ 講演概要 ]
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年05月23日(水)

15:30-16:40   数理科学研究科棟(駒場) 052号室
Emanuele Guidotti 氏 (University of Milan)
Latest Development in yuimaGUI - Interactive Platform for Computational Statistics and Finance

[ 講演概要 ]
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年05月08日(火)

15:00-16:10   数理科学研究科棟(駒場) 052号室
千葉 航平 氏 (東京大学)
LAN property for stochastic differential equations driven by fractional Brownian motion of Hurst parameter 1/4 < H < 1/2
[ 講演概要 ]
We consider the problem of estimating the drift parameter of solution to the stochastic differential equation driven by a fractional Brownian motion with Hurst parameter less than 1/2 under complete observation. We derive a formula for the likelihood ratio and prove local asymptotic normality when 1/4 < H < 1/2. Our result shows that the convergence rate is $T^{-1/2}$ for the parameters satisfying a certain equation and $T^{-(1-H)}$ for the others.
In this talk, we outline the proof of local asymptotic normality and explain how the different rates of convergence occur and where we use the assumption H > 1/4. We also mention some remaining problems and future directions. This talk is based on arXiv:1804.04108.

2018年03月15日(木)

16:00-17:10   数理科学研究科棟(駒場) 052号室
Stefano Iacus 氏 (University of Milan)
On Hypotheses testing for discretely observed SDE (Joint work with Alessandro De Gregorio, University of Rome)
[ 講演概要 ]
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.

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