## Seminar on Probability and Statistics

Seminar information archive ～09/27｜Next seminar｜Future seminars 09/28～

Organizer(s) | Nakahiro Yoshida, Teppei Ogihara, Yuta Koike |
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**Seminar information archive**

### 2007/07/06

15:00-16:10 Room #122 (Graduate School of Math. Sci. Bldg.)

Estimating multidimensional densities through the Malliavin-Thalmaier formula

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/03.html

**Arturo KOHATSU-HIGA**(大阪大学大学院基礎工学研究科)Estimating multidimensional densities through the Malliavin-Thalmaier formula

[ Abstract ]

TBA

[ Reference URL ]TBA

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/03.html

### 2007/06/27

16:20-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Empirical likelihood method for time series analysis

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/02.html

**小方 浩明**(早稲田大学, 国際教養学部)Empirical likelihood method for time series analysis

[ Abstract ]

For a class of vector-valued non-Gaussian stationary processes with unkown parameters, we develop the empirical likelihood approach which was proposed in the i.i.d. setting. In the time series analysis it is known that Whittle likelihood is one of fundamental tools to get a good estimator of unknown parameters and that the score functions are asymptotically normal. Motivated by the Whittle likelihood, we take its score as an estimating function and obtain the asymptotic distribution of our test statistic. Since the fitted spectral model may be different from true spectral structure, the results enable us to construct confidence rigions for various important time series parameters without knowing true spectral structure. We also consider the approach to a minimum contrast estimation and Cressie-Read power-divergence statistic. Numerical studies are introduced and illuminate some interesting features of the empirical approach.

[ Reference URL ]For a class of vector-valued non-Gaussian stationary processes with unkown parameters, we develop the empirical likelihood approach which was proposed in the i.i.d. setting. In the time series analysis it is known that Whittle likelihood is one of fundamental tools to get a good estimator of unknown parameters and that the score functions are asymptotically normal. Motivated by the Whittle likelihood, we take its score as an estimating function and obtain the asymptotic distribution of our test statistic. Since the fitted spectral model may be different from true spectral structure, the results enable us to construct confidence rigions for various important time series parameters without knowing true spectral structure. We also consider the approach to a minimum contrast estimation and Cressie-Read power-divergence statistic. Numerical studies are introduced and illuminate some interesting features of the empirical approach.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/02.html

### 2007/06/06

16:20-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

非正則な位置尺度母数分布族における位置母数の逐次点推定について

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/01.html

**小池 健一**(筑波大学大学院数理物質科学研究科)非正則な位置尺度母数分布族における位置母数の逐次点推定について

[ Abstract ]

有界な台をもつ非正則な位置尺度母数分布族に対して,その位置母数の逐次点推定を考える.ここでは,平均二乗誤差に費用も加えてリスクを考える.レンジに基づく停止則を提案し,これが漸近有効であることを示す.また,良く知られているRobbinsの逐次推定方式との比較を行い,密度関数の台の端点で密度関数が急激に変化する場合には,提案する逐次推定方式が標本数やリスクの意味で優れていることを示す.この結果は,逐次区間推定に関するKoike (2007)のものと同様であることが分かる

[ Reference URL ]有界な台をもつ非正則な位置尺度母数分布族に対して,その位置母数の逐次点推定を考える.ここでは,平均二乗誤差に費用も加えてリスクを考える.レンジに基づく停止則を提案し,これが漸近有効であることを示す.また,良く知られているRobbinsの逐次推定方式との比較を行い,密度関数の台の端点で密度関数が急激に変化する場合には,提案する逐次推定方式が標本数やリスクの意味で優れていることを示す.この結果は,逐次区間推定に関するKoike (2007)のものと同様であることが分かる

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/01.html

### 2007/05/23

16:20-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

New Evidence of Asymmetric Dependence Structures in International Equity Markets

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/00.html

**沖本 竜義**(横浜国立大学経済学部・大学院国際社会科学研究科)New Evidence of Asymmetric Dependence Structures in International Equity Markets

[ Abstract ]

A number of recent studies found two asymmetries in dependence structures in international equity markets; specifically, dependence tends to be high in (1) highly volatile markets and (2) bear markets. In this paper, a further investigation on asymmetric dependence structures in international equity markets is performed under the use of the Markov switching model and copula theory. Combining these two theories enables us to model dependence structures with sufficient flexibility. Using this flexible framework we indeed found that there are two distinct regimes in the US-UK market. We also showed that, for the US-UK market, the bear regime is better described by an asymmetric copula with lower tail dependence with clear rejection of the Markov switching multivariate Normal model. In addition, we showed ignorance of this further asymmetry in bear markets is very costly for risk management. Lastly, we conducted similar analysis for other G7 countries, where we found other c ases where the use of a Markov switching multivariate Normal model would be inappropriate.

[ Reference URL ]A number of recent studies found two asymmetries in dependence structures in international equity markets; specifically, dependence tends to be high in (1) highly volatile markets and (2) bear markets. In this paper, a further investigation on asymmetric dependence structures in international equity markets is performed under the use of the Markov switching model and copula theory. Combining these two theories enables us to model dependence structures with sufficient flexibility. Using this flexible framework we indeed found that there are two distinct regimes in the US-UK market. We also showed that, for the US-UK market, the bear regime is better described by an asymmetric copula with lower tail dependence with clear rejection of the Markov switching multivariate Normal model. In addition, we showed ignorance of this further asymmetry in bear markets is very costly for risk management. Lastly, we conducted similar analysis for other G7 countries, where we found other c ases where the use of a Markov switching multivariate Normal model would be inappropriate.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/00.html

### 2007/01/31

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

A Sequential Unit Root Test

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/18.html

**西山 慶彦**(京都大学経済研究所)A Sequential Unit Root Test

[ Abstract ]

It is well known that conventional unit root tests such as Dickey=Fuller and its variants do not have good power properties when sample size is not large. Lai and Siegmund (1983, AS) proved that OLS estimator of the AR(1) coefficient is asymptotically normally distributed in a sequential framework even if the time series has a unit root unlike the OLS estimator under a standard sampling scheme. We pursue this direction to propose a unit root test under a sequential sampling. The proposed test uses not only the OLS estimator of the AR(1) coefficient, which is asymptotically normal, but also the stopping time to construct the critical region, anticipating a better power property. We obtain analytic expressions of the joint distribution of the two statistics as well as its marginals under the null. We also consider the distribution of the statistics under local alternatives. The properties of the stopping time, to the best of our knowledge, have not been studied in the unit root literature. We calculate its expectation and variance.

[ Reference URL ]It is well known that conventional unit root tests such as Dickey=Fuller and its variants do not have good power properties when sample size is not large. Lai and Siegmund (1983, AS) proved that OLS estimator of the AR(1) coefficient is asymptotically normally distributed in a sequential framework even if the time series has a unit root unlike the OLS estimator under a standard sampling scheme. We pursue this direction to propose a unit root test under a sequential sampling. The proposed test uses not only the OLS estimator of the AR(1) coefficient, which is asymptotically normal, but also the stopping time to construct the critical region, anticipating a better power property. We obtain analytic expressions of the joint distribution of the two statistics as well as its marginals under the null. We also consider the distribution of the statistics under local alternatives. The properties of the stopping time, to the best of our knowledge, have not been studied in the unit root literature. We calculate its expectation and variance.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/18.html

### 2007/01/17

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Second order optimality for estimators in time series regression models

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/17.html

**玉置 健一郎**(早稲田大学)Second order optimality for estimators in time series regression models

[ Abstract ]

We consider the second order asymptotic properties of an efficient frequency domain regression coefficient estimator $\\hat{\\beta}$ proposed by Hannan (1963). This estimator is a semiparametric estimator based on nonparametric spectral estimators. We derive the second order Edgeworth expansion of the distribution of $\\hat{\\beta}$. Then it is shown that the second order asymptotic properties are independent of the bandwidth choice for residual spectral estimator, which implies that $\\hat{\\beta}$ has the same rate of convergence as in regular parametric estimation. This is a sharp contrast with the general semiparametric estimation theory. We also examine the second order Gaussian efficiency of $\\hat{\\beta}$. Numerical studies are given to confirm the theoretical results.

[ Reference URL ]We consider the second order asymptotic properties of an efficient frequency domain regression coefficient estimator $\\hat{\\beta}$ proposed by Hannan (1963). This estimator is a semiparametric estimator based on nonparametric spectral estimators. We derive the second order Edgeworth expansion of the distribution of $\\hat{\\beta}$. Then it is shown that the second order asymptotic properties are independent of the bandwidth choice for residual spectral estimator, which implies that $\\hat{\\beta}$ has the same rate of convergence as in regular parametric estimation. This is a sharp contrast with the general semiparametric estimation theory. We also examine the second order Gaussian efficiency of $\\hat{\\beta}$. Numerical studies are given to confirm the theoretical results.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/17.html

### 2006/12/06

15:00-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Inference problems for the standard and geometric telegraph process

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/16.html

**Stefano IACUS**(Department of Economics Business and Statistics, University of Milan, Italy)Inference problems for the standard and geometric telegraph process

[ Abstract ]

The telegraph process {X(t), t>0}, has been introduced (see Goldstein, 1951) as an alternative model to the Brownian motion B(t). This process describes a motion of a particle on the real line which alternates its velocity, at Poissonian times, from +v to -v. The density of the distribution of the position of the particle at time t solves the hyperbolic differential equation called telegraph equation and hence the name of the process. Contrary to B(t) the process X(t) has finite variation and continuous and differentiable paths. At the same time it is mathematically challenging to handle.

In this talk we will discuss inference problems for the estimation of the intensity of the Poisson process, either homogeneous and non homogeneous, from continuous and discrete time observations of X(t). We further discuss estimation problems for the geometric telegraph process S(t) = S(0) * exp{m - 0.5 * s^2) * t + s X(t)} where m is a known constant and s>0 and the intensity of the underlying Poisson process are two parameter of interest to be estimated. The geometric telegraph process has been recently introduced in Mathematical Finance to describe the dynamics of assets as an alternative to the usual geometric Brownian motion.

For discrete time observations we consider the "high frequency" approach, which means that data are collected at n+1 equidistant time points Ti=i * Dn, i=0,1,..., n, n*Dn = T, T fixed and such that Dn shrinks to 0 as n increases.

The process X(t) in non Markovian, non stationary and not ergodic thus we use approximation arguments to derive estimators. Given the complexity of the equations involved only estimators on the standard telegraph process can be studied analytically. We will also present a Monte Carlo study on the performance of the estimators for small sample size, i.e. Dn not shrinking to 0.

[ Reference URL ]The telegraph process {X(t), t>0}, has been introduced (see Goldstein, 1951) as an alternative model to the Brownian motion B(t). This process describes a motion of a particle on the real line which alternates its velocity, at Poissonian times, from +v to -v. The density of the distribution of the position of the particle at time t solves the hyperbolic differential equation called telegraph equation and hence the name of the process. Contrary to B(t) the process X(t) has finite variation and continuous and differentiable paths. At the same time it is mathematically challenging to handle.

In this talk we will discuss inference problems for the estimation of the intensity of the Poisson process, either homogeneous and non homogeneous, from continuous and discrete time observations of X(t). We further discuss estimation problems for the geometric telegraph process S(t) = S(0) * exp{m - 0.5 * s^2) * t + s X(t)} where m is a known constant and s>0 and the intensity of the underlying Poisson process are two parameter of interest to be estimated. The geometric telegraph process has been recently introduced in Mathematical Finance to describe the dynamics of assets as an alternative to the usual geometric Brownian motion.

For discrete time observations we consider the "high frequency" approach, which means that data are collected at n+1 equidistant time points Ti=i * Dn, i=0,1,..., n, n*Dn = T, T fixed and such that Dn shrinks to 0 as n increases.

The process X(t) in non Markovian, non stationary and not ergodic thus we use approximation arguments to derive estimators. Given the complexity of the equations involved only estimators on the standard telegraph process can be studied analytically. We will also present a Monte Carlo study on the performance of the estimators for small sample size, i.e. Dn not shrinking to 0.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/16.html

### 2006/11/22

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

A Note on Haplotype Estimation

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/15.html

**鎌谷 研吾**(東京大学大学院数理科学研究科)A Note on Haplotype Estimation

[ Abstract ]

Haplotype information is important for many analyses but it is not always possible to obtain. This work is motivated to seek haplotype information from diploid population data. We present a new approach to know the haplotype information using classical methods. We do not intend to say that our method is better than the well-known EM based approache for practical purposes, but our way is attractive in some sense.

[ Reference URL ]Haplotype information is important for many analyses but it is not always possible to obtain. This work is motivated to seek haplotype information from diploid population data. We present a new approach to know the haplotype information using classical methods. We do not intend to say that our method is better than the well-known EM based approache for practical purposes, but our way is attractive in some sense.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/15.html

### 2006/11/17

15:00-16:10 Room #118 (Graduate School of Math. Sci. Bldg.)

Functional estimation of L'evy measure for jump-type processes

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/13.html

**清水 泰隆**(大阪大学大学院基礎工学研究科)Functional estimation of L'evy measure for jump-type processes

[ Abstract ]

Recently, stochastic processes with Poissonian jumps are frequently used in finance and insurance. In their applications, it often becomes important to estimate some functionals of integral types with respect to L'evy measures. In this talk, we propose a nonparametric estimator of their functionals based on both continuous and discrete observations. If time permits, we shall also mention the application to the mathematical insurance, in particular, the estimates of ruin probabilities for genelarized risk processes.

[ Reference URL ]Recently, stochastic processes with Poissonian jumps are frequently used in finance and insurance. In their applications, it often becomes important to estimate some functionals of integral types with respect to L'evy measures. In this talk, we propose a nonparametric estimator of their functionals based on both continuous and discrete observations. If time permits, we shall also mention the application to the mathematical insurance, in particular, the estimates of ruin probabilities for genelarized risk processes.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/13.html

### 2006/11/08

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

LAN Theorem for Non-Gaussian Locally Stationary Processes and Their Discriminant and Cluster Analyses

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/13.html

**蛭川 潤一**(早稲田大学)LAN Theorem for Non-Gaussian Locally Stationary Processes and Their Discriminant and Cluster Analyses

[ Abstract ]

This talk is concerned with asymptotic inference for non-Gaussian locally stationary processes. Lucien LeCam established the most important and sophisticated foundation of the general statistical asymptotic theory. He introduced the concept of local asymptotic normality (LAN) for the likelihood ratio of general statistical models. Once LAN is proved, the asymptotic optimality of estimators and tests is described in terms of the LAN property. The techniques of statistical inference for stationary time series have been well established. However, stationary time series model is not plausible to describe the real world. One of the difficult problem when we deal with nonstationary processes is how to set up an adequate model. Otherwise, the observation in the future will bring no information for the present structure. Recently, Dahalhaus has proposed an important class of nonstationary processes, called locally stationary processes. Locally stationary processes have the time varying densities whose spectral structures smoothly change in time. In this talk, we first show the LAN results for locally stationary processes under the assumption of the non-Gaussianity. Then, we apply the LAN theorem to estimation and testing theory, non-Gaussian robustness and adaptive estimation. Our LAN theorem elucidates various non-Gaussian asymptotics. Next, we develop asymptotic theory for discriminant and cluster analyses of non-Gaussian locally stationary processes. We discuss about non-Gaussian robustness of our classification statistic. Furthermore, we execute the clustering of stock returns in Tokyo Stock Exchanges. Consequently, we observe that the clustering results well extract features of relationships among companies.

[ Reference URL ]This talk is concerned with asymptotic inference for non-Gaussian locally stationary processes. Lucien LeCam established the most important and sophisticated foundation of the general statistical asymptotic theory. He introduced the concept of local asymptotic normality (LAN) for the likelihood ratio of general statistical models. Once LAN is proved, the asymptotic optimality of estimators and tests is described in terms of the LAN property. The techniques of statistical inference for stationary time series have been well established. However, stationary time series model is not plausible to describe the real world. One of the difficult problem when we deal with nonstationary processes is how to set up an adequate model. Otherwise, the observation in the future will bring no information for the present structure. Recently, Dahalhaus has proposed an important class of nonstationary processes, called locally stationary processes. Locally stationary processes have the time varying densities whose spectral structures smoothly change in time. In this talk, we first show the LAN results for locally stationary processes under the assumption of the non-Gaussianity. Then, we apply the LAN theorem to estimation and testing theory, non-Gaussian robustness and adaptive estimation. Our LAN theorem elucidates various non-Gaussian asymptotics. Next, we develop asymptotic theory for discriminant and cluster analyses of non-Gaussian locally stationary processes. We discuss about non-Gaussian robustness of our classification statistic. Furthermore, we execute the clustering of stock returns in Tokyo Stock Exchanges. Consequently, we observe that the clustering results well extract features of relationships among companies.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/13.html

### 2006/11/01

15:00-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Some problems related to the estimation of the invariant measure of an ergodic diffusion.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/12.html

**Ilia NEGRI**(Department of Management and Information Technology, University of Bergamo, Italy)Some problems related to the estimation of the invariant measure of an ergodic diffusion.

[ Abstract ]

We consider a one dimensional ergodic diffusion process solution of a stochastic differential equation. We suppose that the diffusion coefficient is known whereas the drift coefficient $S$ is unknown. Our interest is the invariant measure of the process denoted as $\\mu $. We denote by $f_S$ and $F_S$ the invariant density and the invariant distribution function of $\\mu$ respectively. We present the problems of finding efficient estimators when we observe the trajectory of the diffusion in continuos time over $[0,T]$ and we study asymptotic properties of the estimators when $T$ goes to infinity.

[ Reference URL ]We consider a one dimensional ergodic diffusion process solution of a stochastic differential equation. We suppose that the diffusion coefficient is known whereas the drift coefficient $S$ is unknown. Our interest is the invariant measure of the process denoted as $\\mu $. We denote by $f_S$ and $F_S$ the invariant density and the invariant distribution function of $\\mu$ respectively. We present the problems of finding efficient estimators when we observe the trajectory of the diffusion in continuos time over $[0,T]$ and we study asymptotic properties of the estimators when $T$ goes to infinity.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/12.html

### 2006/10/18

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Fourier analysis of irregularly spaced data on R^d

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/11.html

**松田 安昌**(東北大学大学院経済学研究科)Fourier analysis of irregularly spaced data on R^d

[ Abstract ]

1970年代にはじまった時系列解析の研究は、1変量時系列からはじまり、多変量時系列、空間系列、時空間系列へと対象が拡張されてきている。近年の空間系列、時空間系列の文献を調べてみれば、時系列と同じく等間隔に観測されたデータ(regularly spaced data) を前提としている場合が多い。しかし時系列とは異なり、空間系列ではランダムに散らばった地点で観測されたデータ (irregularly spaced data)を仮定する方が、応用範囲がひろい。そこで本発表では、irregularly spaced dataによる空間相関の推定問題を扱う。具体的にはデータを周波数領域に変換してスペクトルを推定する方法を提案し、推定量が一致性、漸近正規性を持つための条件を示す。本方法による東京地価データ分析例も紹介する。本研究は、矢島美寛教授(東京大学大学院経済学研究科)との共同研究です。

[ Reference URL ]1970年代にはじまった時系列解析の研究は、1変量時系列からはじまり、多変量時系列、空間系列、時空間系列へと対象が拡張されてきている。近年の空間系列、時空間系列の文献を調べてみれば、時系列と同じく等間隔に観測されたデータ(regularly spaced data) を前提としている場合が多い。しかし時系列とは異なり、空間系列ではランダムに散らばった地点で観測されたデータ (irregularly spaced data)を仮定する方が、応用範囲がひろい。そこで本発表では、irregularly spaced dataによる空間相関の推定問題を扱う。具体的にはデータを周波数領域に変換してスペクトルを推定する方法を提案し、推定量が一致性、漸近正規性を持つための条件を示す。本方法による東京地価データ分析例も紹介する。本研究は、矢島美寛教授(東京大学大学院経済学研究科)との共同研究です。

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/11.html

### 2006/10/11

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Optimal stopping problem associated with jump-diffusion processes

**石川保志**(愛媛大学)Optimal stopping problem associated with jump-diffusion processes

[ Abstract ]

We study an optimal stopping problem of some performance function with respect to a jump-diffusion process.

We study an optimal stopping problem of some performance function with respect to a jump-diffusion process.

### 2006/09/25

13:00-14:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Nonparametric testing time-homogeneity for L'evy processes

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/09.html

**西山 陽一**(統計数理研究所)Nonparametric testing time-homogeneity for L'evy processes

[ Abstract ]

First, a review about uniform central limit theorems for martingales is given. The main part of the talk is concerned with a change point problem for L'evy processes. The null hypothesis is that the L'evy process is time-homogeneous, and the alternative is that the L'evy measure changes at a certain time point of the observation period. We propose an empirical process type statistics, and derive its asymptotic behaviour under the null and the alternative hypotheses. The limiting distribution under the null hypothesis is a functional of the standard Brownian motion.

[ Reference URL ]First, a review about uniform central limit theorems for martingales is given. The main part of the talk is concerned with a change point problem for L'evy processes. The null hypothesis is that the L'evy process is time-homogeneous, and the alternative is that the L'evy measure changes at a certain time point of the observation period. We propose an empirical process type statistics, and derive its asymptotic behaviour under the null and the alternative hypotheses. The limiting distribution under the null hypothesis is a functional of the standard Brownian motion.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/09.html

### 2006/08/22

15:30-16:40 Room #128 (Graduate School of Math. Sci. Bldg.)

A unifying approach to inference in semimartingale and long-memory models

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/07.html

**Jeannette H.C. WOERNER**(University of Gottingen)A unifying approach to inference in semimartingale and long-memory models

[ Abstract ]

Over the recent years classical stochastic volatility models based on Brownian motion have been generalized in different ways, either replacing the Brownian motion by a pure jump Levy process, which leads to a pure jump model, or by a fractional Brownian motion, which makes it possible to model both long memory or turbulent behaviour. We consider robust and easily computable estimators for the inte- grated volatility, providing insight in the level of volatility, as needed for risk measurement and pricing of variance and volatility swaps. We discuss consistency and distributional results for the power and multi- power variation estimates based on high frequency data. Furthermore, we consider robustness against additive components and compare the results for the different classes of semimartingale and fractional Brow- nian motion models.

[ Reference URL ]Over the recent years classical stochastic volatility models based on Brownian motion have been generalized in different ways, either replacing the Brownian motion by a pure jump Levy process, which leads to a pure jump model, or by a fractional Brownian motion, which makes it possible to model both long memory or turbulent behaviour. We consider robust and easily computable estimators for the inte- grated volatility, providing insight in the level of volatility, as needed for risk measurement and pricing of variance and volatility swaps. We discuss consistency and distributional results for the power and multi- power variation estimates based on high frequency data. Furthermore, we consider robustness against additive components and compare the results for the different classes of semimartingale and fractional Brow- nian motion models.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/07.html

### 2006/08/22

16:50-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A computation of Theta in a jump diffusion model by integration by parts

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/08.html

**Delphine DAVID**(Departement de Mathematiques, Universite de La Rochelle)A computation of Theta in a jump diffusion model by integration by parts

[ Abstract ]

Using Malliavin weights in a jump-diffusion model we obtain an expression for Theta (the sensitivity of an option price with respect to the time remaining until exercise), with application to non-smooth payoff functions. Optimal weights are computed by minimization of variance and numerical simulations are presented for digital and European options. Some results are also presented for Asian options. Our representation formula for Theta differs in general from the one obtained from the Black-Scholes PDE in terms of Delta and Gamma.

[ Reference URL ]Using Malliavin weights in a jump-diffusion model we obtain an expression for Theta (the sensitivity of an option price with respect to the time remaining until exercise), with application to non-smooth payoff functions. Optimal weights are computed by minimization of variance and numerical simulations are presented for digital and European options. Some results are also presented for Asian options. Our representation formula for Theta differs in general from the one obtained from the Black-Scholes PDE in terms of Delta and Gamma.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/08.html

### 2006/07/19

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Edgeworth Expansion for Likelihood Analysis on Ergodic Diffusions with applications to Bootstrap

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/06.html

**深澤 正彰**(東京大学大学院数理科学研究科)Edgeworth Expansion for Likelihood Analysis on Ergodic Diffusions with applications to Bootstrap

[ Abstract ]

We shall consider the maximal lilelihood estimator for the drift coefficient of a given one-dimensional diffusion. An Edgeworth expansion formula will be presented and verify a second-order correct confidence interval we shall newly propose. We are also going to mention the likelihood ratio statistic, which enjoys second-order correctness. There are Bootstrap methods closely related to the subject and introduced recently by the author. Some generalized results on those methods will be also introduced in this talk.

[ Reference URL ]We shall consider the maximal lilelihood estimator for the drift coefficient of a given one-dimensional diffusion. An Edgeworth expansion formula will be presented and verify a second-order correct confidence interval we shall newly propose. We are also going to mention the likelihood ratio statistic, which enjoys second-order correctness. There are Bootstrap methods closely related to the subject and introduced recently by the author. Some generalized results on those methods will be also introduced in this talk.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/06.html

### 2006/06/21

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Malliavin calculus applied to mathematical finance and a new formulation of the intgration-by-parts

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/05.html

**石川 保志**(愛媛大学理学部)Malliavin calculus applied to mathematical finance and a new formulation of the intgration-by-parts

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/05.html

### 2006/05/31

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Bhattacharyya inequality for quantum state estimation II

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/04.html

**津田 美幸**(統計数理研究所)Bhattacharyya inequality for quantum state estimation II

[ Abstract ]

前回導出した三種類(S型, R型, L型)の量子Bhattacharyya不等式を量子ガウス状態の複素振幅θの多項式g(θ)の推定問題に応用する. 量子ガウス状態は, レーザ光の量子状態の典型的なモデルであり, 量子光学や量子情報で重要な研究対象である. 未知の複素振幅θを推定する方法としては, θが実軸にある場合はホモダイン測定, 一般の複素数の場合はヘテロダイン測定が知られており, それぞれS型とR型の量子Cramer-Rao不等式の下限を達成するUMVUEである. さらにここでは, θが実数の場合に (1), (2)を示し, θが複素数の場合に (3), (4)を示す.

(1) g(θ)=θ^2に対するUMVUEは存在してS型Bhattacharyya下限を達成する. その推定量は, 物理系にスクイジングと呼ばれる操作を施した後の個数測定によって与えられる.

(2) g(θ)=θ^3に対するUMVUEは, 生成消滅作用素の多項式の形では存在しない.

(3) g(θ)が正則, 或いは反正則, ならば, ヘテロダイン測定によってUMVUEが与えられ, それぞれR型, L型のBhattacharyya下限を達成する.

(4) g(θ)が実数値ならば, ある測定によりUMVUEが与えられ, R型, L型両方の下限を達成する.

量子ガウス状態は古典の正規分布に似ている. しかし, 古典では, 平均の多項式は Hermite多項式により常にUMVUEを構成できるが, 量子では上記のように事情が異なる.

[ Reference URL ]前回導出した三種類(S型, R型, L型)の量子Bhattacharyya不等式を量子ガウス状態の複素振幅θの多項式g(θ)の推定問題に応用する. 量子ガウス状態は, レーザ光の量子状態の典型的なモデルであり, 量子光学や量子情報で重要な研究対象である. 未知の複素振幅θを推定する方法としては, θが実軸にある場合はホモダイン測定, 一般の複素数の場合はヘテロダイン測定が知られており, それぞれS型とR型の量子Cramer-Rao不等式の下限を達成するUMVUEである. さらにここでは, θが実数の場合に (1), (2)を示し, θが複素数の場合に (3), (4)を示す.

(1) g(θ)=θ^2に対するUMVUEは存在してS型Bhattacharyya下限を達成する. その推定量は, 物理系にスクイジングと呼ばれる操作を施した後の個数測定によって与えられる.

(2) g(θ)=θ^3に対するUMVUEは, 生成消滅作用素の多項式の形では存在しない.

(3) g(θ)が正則, 或いは反正則, ならば, ヘテロダイン測定によってUMVUEが与えられ, それぞれR型, L型のBhattacharyya下限を達成する.

(4) g(θ)が実数値ならば, ある測定によりUMVUEが与えられ, R型, L型両方の下限を達成する.

量子ガウス状態は古典の正規分布に似ている. しかし, 古典では, 平均の多項式は Hermite多項式により常にUMVUEを構成できるが, 量子では上記のように事情が異なる.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/04.html

### 2006/05/24

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Bhattacharyya inequality for quantum state estimation I

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/03.html

**津田 美幸**(統計数理研究所)Bhattacharyya inequality for quantum state estimation I

[ Abstract ]

量子状態推定のBhattacharyya不等式の導出とその応用例を前後二回に分けて紹介する. 今回は, 一般的な形で問題設定を行い, 量子Cramer-Rao不等式と量子 Bhattacharyya不等式について述べる.

量子状態推定は量子力学系の未知の状態に関する統計的推定問題である. 古典的な統計モデルの推定問題との違いは, データを観測するための測定を, 量子力学的に可狽ネ範囲で, 選択する点にある. 実数または複素数でパラメトライズされたモデルに対しては, 不偏推定量の分散の最小化が基本的な問題である. ただしここでは, 複素パラメータz=x+iyの分散とは, (x,y)の二次元の共分散行列のトレースをさす. この問題に対しては Yuen and Lax (1973) 等により, パラメータの一階微分に基づいた量子Cramer-Rao不等式が導出されており, 量子ガウス状態の複素振幅θのUMVUEが知られている. 二階以上の微分に基づくBhattacharyya型の不等式は, Brody and Hughston (1998) により, ある特殊なモデルにおいて導入され, 漸近論へ応用された. ここではより一般的なモデルに適用可能な形で量子Bhattacharyya不等式を三種類定式化する.

[ Reference URL ]量子状態推定のBhattacharyya不等式の導出とその応用例を前後二回に分けて紹介する. 今回は, 一般的な形で問題設定を行い, 量子Cramer-Rao不等式と量子 Bhattacharyya不等式について述べる.

量子状態推定は量子力学系の未知の状態に関する統計的推定問題である. 古典的な統計モデルの推定問題との違いは, データを観測するための測定を, 量子力学的に可狽ネ範囲で, 選択する点にある. 実数または複素数でパラメトライズされたモデルに対しては, 不偏推定量の分散の最小化が基本的な問題である. ただしここでは, 複素パラメータz=x+iyの分散とは, (x,y)の二次元の共分散行列のトレースをさす. この問題に対しては Yuen and Lax (1973) 等により, パラメータの一階微分に基づいた量子Cramer-Rao不等式が導出されており, 量子ガウス状態の複素振幅θのUMVUEが知られている. 二階以上の微分に基づくBhattacharyya型の不等式は, Brody and Hughston (1998) により, ある特殊なモデルにおいて導入され, 漸近論へ応用された. ここではより一般的なモデルに適用可能な形で量子Bhattacharyya不等式を三種類定式化する.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/03.html

### 2006/05/17

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Second-order efficiency in the semiparametric problem of estimating the shift of a signal

**Arnak DALALYAN**(Universite Paris 6, France)Second-order efficiency in the semiparametric problem of estimating the shift of a signal

### 2006/05/10

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Asymptotic statistical equivalence for diffusion processes II

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/01.html

**Arnak DALALYAN**(Universite Paris 6, France)Asymptotic statistical equivalence for diffusion processes II

[ Abstract ]

We consider the experiment of a continuously observed scalar diffusion process with unknown drift function. In the stationary case, we prove that this experment is locally asymptotically equivalent to a simple Gaussian white noise experiment. We also derive the rate of convergence of the Le Cam's distance and describe the Markov kernel attaining this rate of convergence. These results are obtained in collaboration with Markus Reiss.

[ Reference URL ]We consider the experiment of a continuously observed scalar diffusion process with unknown drift function. In the stationary case, we prove that this experment is locally asymptotically equivalent to a simple Gaussian white noise experiment. We also derive the rate of convergence of the Le Cam's distance and describe the Markov kernel attaining this rate of convergence. These results are obtained in collaboration with Markus Reiss.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/01.html

### 2006/04/26

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Asymptotic statistical equivalence for diffusion processes I (JAPANESE)

**Arnak DALALYAN**(Universite Paris 6, France)Asymptotic statistical equivalence for diffusion processes I (JAPANESE)

[ Abstract ]

This is the first talk of a series of three talks devoted to the asymptotic statistical equivalence for diffusion processes. We will introduce the notion of Le Cam's distance between statistical experiments and will present its properties with some easy examples. Then we will show that the experiment of a discretely observed diffusion process with unknown drift is asymptoically equivalent to the experiment of continuously observed diffusion process provided that the step of discretisation is small enough (this result is due to Milstein and Nussbaum).

This is the first talk of a series of three talks devoted to the asymptotic statistical equivalence for diffusion processes. We will introduce the notion of Le Cam's distance between statistical experiments and will present its properties with some easy examples. Then we will show that the experiment of a discretely observed diffusion process with unknown drift is asymptoically equivalent to the experiment of continuously observed diffusion process provided that the step of discretisation is small enough (this result is due to Milstein and Nussbaum).