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Seminar on Probability and Statistics
Seminar information archive ～05/21｜Next seminar｜Future seminars 05/22～
| Organizer(s) | Nakahiro Yoshida, Hiroki Masuda, Teppei Ogihara, Yuta Koike |
|---|
Seminar information archive
2015/11/18
17:00-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)
Ioane Muni Toke (University of New Caledonia)
Order flow intensities for limit order book modelling
Ioane Muni Toke (University of New Caledonia)
Order flow intensities for limit order book modelling
[ Abstract ]
Limit order books are at the core of electronic financial markets. Mathematical models of limit order books use point processes to model the arrival of limit, market and cancellation orders in the order book, but it is not clear what a "good" parametric model for the intensities of these point processes should be.
In the first part of the talk, we show that despite their simplicity basic Poisson processes can be used to accurately model a few features of the order book that more advanced models reproduce with volume-dependent intensities.
In the second part of the talk we present ongoing investigations in a more advanced statistical modelling of these order flow intensities using in particular normal mixture distributions and exponential models.
Limit order books are at the core of electronic financial markets. Mathematical models of limit order books use point processes to model the arrival of limit, market and cancellation orders in the order book, but it is not clear what a "good" parametric model for the intensities of these point processes should be.
In the first part of the talk, we show that despite their simplicity basic Poisson processes can be used to accurately model a few features of the order book that more advanced models reproduce with volume-dependent intensities.
In the second part of the talk we present ongoing investigations in a more advanced statistical modelling of these order flow intensities using in particular normal mixture distributions and exponential models.
2015/10/19
13:00-16:40 Room #052 (Graduate School of Math. Sci. Bldg.)
2015/09/17
15:00-16:10 Room #052 (Graduate School of Math. Sci. Bldg.)
Stefano Iacus (University of Milan)
The use of S4 classes and methods in the Yuima R package
Stefano Iacus (University of Milan)
The use of S4 classes and methods in the Yuima R package
[ Abstract ]
In this talk we present the basic concept of S4 classes and methods approach for object oriented programming in R. As a working example, we introduce the structure of the Yuima package for simulation and inference of stochastic differential equations. We will describe the basic classes and objects as well as some recent extensions which allows for Carma and Co-Garch processes handling in Yuima.
In this talk we present the basic concept of S4 classes and methods approach for object oriented programming in R. As a working example, we introduce the structure of the Yuima package for simulation and inference of stochastic differential equations. We will describe the basic classes and objects as well as some recent extensions which allows for Carma and Co-Garch processes handling in Yuima.
2015/08/07
14:40-15:50 Room #052 (Graduate School of Math. Sci. Bldg.)
UBUKATA, Masato (Kushiro Public University of Economics)
Effectiveness of time-varying minimum value at risk and expected shortfall hedging
UBUKATA, Masato (Kushiro Public University of Economics)
Effectiveness of time-varying minimum value at risk and expected shortfall hedging
[ Abstract ]
This paper assesses the incremental value of time-varying minimum value at risk (VaR) and expected shortfall (ES) hedging strategies over unconditional hedging strategy. The conditional futures hedge ratios are calculated through estimation of multivariate volatility models under a skewed and leptokurtic distribution and Monte Carlo simulation for conditional skewness and kurtosis of hedged portfolio returns. We examine DCC-GJR models with or without encompassing realized covariance measure (RCM) from high-frequency data under a multivariate skewed Student's t-distribution. In the out-of-sample analysis with a daily rebalancing approach, the empirical results show that the conditional minimum VaR and ES hedging strategies outperform the unconditional hedging strategy. We find that the use of RCM improves the futures hedging performance for a short hedge, although the degree of improvement is small relative to that when switching from unconditional to conditional.
This paper assesses the incremental value of time-varying minimum value at risk (VaR) and expected shortfall (ES) hedging strategies over unconditional hedging strategy. The conditional futures hedge ratios are calculated through estimation of multivariate volatility models under a skewed and leptokurtic distribution and Monte Carlo simulation for conditional skewness and kurtosis of hedged portfolio returns. We examine DCC-GJR models with or without encompassing realized covariance measure (RCM) from high-frequency data under a multivariate skewed Student's t-distribution. In the out-of-sample analysis with a daily rebalancing approach, the empirical results show that the conditional minimum VaR and ES hedging strategies outperform the unconditional hedging strategy. We find that the use of RCM improves the futures hedging performance for a short hedge, although the degree of improvement is small relative to that when switching from unconditional to conditional.
2015/08/07
13:20-14:30 Room #052 (Graduate School of Math. Sci. Bldg.)
Yoann Potiron (University of Chicago)
ESTIMATION OF INTEGRATED QUADRATIC COVARIATION BETWEEN TWO ASSETS WITH ENDOGENOUS SAMPLING TIMES
Yoann Potiron (University of Chicago)
ESTIMATION OF INTEGRATED QUADRATIC COVARIATION BETWEEN TWO ASSETS WITH ENDOGENOUS SAMPLING TIMES
[ Abstract ]
When estimating integrated covariation between two assets based on high-frequency data,simple assumptions are usually imposed on the relationship between the price processes and the observation times. In this paper, we introduce an endogenous 2-dimensional model and show that it is more general than the existing endogenous models of the literature. In addition, we establish a central limit theorem for the Hayashi-Yoshida estimator in this general endogenous model in the case where prices follow pure-diffusion processes.
When estimating integrated covariation between two assets based on high-frequency data,simple assumptions are usually imposed on the relationship between the price processes and the observation times. In this paper, we introduce an endogenous 2-dimensional model and show that it is more general than the existing endogenous models of the literature. In addition, we establish a central limit theorem for the Hayashi-Yoshida estimator in this general endogenous model in the case where prices follow pure-diffusion processes.
2015/06/05
16:20-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)
2015/04/10
14:50-16:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Yacine Ait-Sahalia (Princeton University)
Principal Component Analysis of High Frequency Data (joint with Dacheng Xiu)
Yacine Ait-Sahalia (Princeton University)
Principal Component Analysis of High Frequency Data (joint with Dacheng Xiu)
[ Abstract ]
We develop a methodology to conduct principal component analysis of high frequency financial data. The procedure involves estimation of realized eigenvalues, realized eigenvectors, and realized principal components and we provide the asymptotic distribution of these estimators. Empirically, we study the components of the constituents of Dow Jones Industrial Average Index, in a high frequency version, with jumps, of the Fama-French analysis. Our findings show that, excluding jump variation, three Brownian factors explain between 50 and 60% of continuous variation of the stock returns. Their explanatory power varies over time. During crises, the first principal component becomes increasingly dominant, explaining up to 70% of the variation on its own, a clear sign of systemic risk.
We develop a methodology to conduct principal component analysis of high frequency financial data. The procedure involves estimation of realized eigenvalues, realized eigenvectors, and realized principal components and we provide the asymptotic distribution of these estimators. Empirically, we study the components of the constituents of Dow Jones Industrial Average Index, in a high frequency version, with jumps, of the Fama-French analysis. Our findings show that, excluding jump variation, three Brownian factors explain between 50 and 60% of continuous variation of the stock returns. Their explanatory power varies over time. During crises, the first principal component becomes increasingly dominant, explaining up to 70% of the variation on its own, a clear sign of systemic risk.
2015/02/19
16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)
Dobrislav Dobrev (Board of Governors of the Federal Reserve System, Division of International Finance)
TBA
Dobrislav Dobrev (Board of Governors of the Federal Reserve System, Division of International Finance)
TBA
[ Abstract ]
TBA
TBA
2015/02/10
16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)
Ioane Muni Toke (Ecole Centrale Paris and University of New Caledonia)
Zero-intelligence modelling of limit order books
Ioane Muni Toke (Ecole Centrale Paris and University of New Caledonia)
Zero-intelligence modelling of limit order books
[ Abstract ]
Limit order books (LOB) are at the core of electronic financial markets.
A LOB centralizes all orders of all market participants on a given
exchange, matching buy and sell orders of all types.
In a first part, we observe that a LOB is a queueing system and that
this analogy is fruitful to derive stationary properties of these
structures. Using a basic Poisson model, we compute analytical formulas
for the average shape of the LOB. Our model allows for non-unit size of
limit orders, leading to new predictions on the granularity of financial
markets that turn out to be empirically valid.
In a second part, we study the LOB during the call auction, a market
design often used during the opening and closing phases of the trading
day. We show that in a basic Poisson model of the call auction, the
distributions for the traded volume and the range of clearing prices are
analytically computable. In the case of a liquid market, we derive weak
limits of these distributions and test them empirically.
Limit order books (LOB) are at the core of electronic financial markets.
A LOB centralizes all orders of all market participants on a given
exchange, matching buy and sell orders of all types.
In a first part, we observe that a LOB is a queueing system and that
this analogy is fruitful to derive stationary properties of these
structures. Using a basic Poisson model, we compute analytical formulas
for the average shape of the LOB. Our model allows for non-unit size of
limit orders, leading to new predictions on the granularity of financial
markets that turn out to be empirically valid.
In a second part, we study the LOB during the call auction, a market
design often used during the opening and closing phases of the trading
day. We show that in a basic Poisson model of the call auction, the
distributions for the traded volume and the range of clearing prices are
analytically computable. In the case of a liquid market, we derive weak
limits of these distributions and test them empirically.
2015/01/16
14:00-15:30 Room #052 (Graduate School of Math. Sci. Bldg.)
Ajay Jasra (National University of Singapore)
A stable particle filter in high-dimensions
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/06.html
Ajay Jasra (National University of Singapore)
A stable particle filter in high-dimensions
[ Abstract ]
We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in $\mathbb{R}^d$ with $d$ large. For low dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed as particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in $d$ for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space-time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provide consistent Monte Carlo estimates for any fixed $d$, as do standard particle filters. Moreover, under a simple i.i.d. model structure, we show that in order to achieve some stability properties this new filter has cost $\mathcal{O}(nNd^2)$, where $n$ is the time parameter and $N$ is the number of Monte Carlo samples, that are fixed and independent of $d$. Similar results hold, under a more general structure than the i.i.d. one. Here we show that, under additional assumptions and with the same cost, the asymptotic variance of the relative estimate of the normalizing constant grows at most linearly in time and independently of the dimension. Our theoretical results are supported by numerical simulations. The results suggest that it is possible to tackle some high dimensional filtering problems using the space-time particle filter that standard particle filters cannot.
This is joint work with: Alex Beskos (UCL), Dan Crisan (Imperial), Kengo Kamatani (Osaka) and Yan Zhou (NUS).
[ Reference URL ]We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in $\mathbb{R}^d$ with $d$ large. For low dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed as particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in $d$ for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space-time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provide consistent Monte Carlo estimates for any fixed $d$, as do standard particle filters. Moreover, under a simple i.i.d. model structure, we show that in order to achieve some stability properties this new filter has cost $\mathcal{O}(nNd^2)$, where $n$ is the time parameter and $N$ is the number of Monte Carlo samples, that are fixed and independent of $d$. Similar results hold, under a more general structure than the i.i.d. one. Here we show that, under additional assumptions and with the same cost, the asymptotic variance of the relative estimate of the normalizing constant grows at most linearly in time and independently of the dimension. Our theoretical results are supported by numerical simulations. The results suggest that it is possible to tackle some high dimensional filtering problems using the space-time particle filter that standard particle filters cannot.
This is joint work with: Alex Beskos (UCL), Dan Crisan (Imperial), Kengo Kamatani (Osaka) and Yan Zhou (NUS).
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/06.html
2014/11/26
16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)
KATAYAMA, Shota (Tokyo Institute of Technology)
Sparse and robust linear regression: Iterative algorithm and its statistical convergence
KATAYAMA, Shota (Tokyo Institute of Technology)
Sparse and robust linear regression: Iterative algorithm and its statistical convergence
2014/11/11
16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)
Terada, Yoshikazu (CiNet / Center for Information and Neural Networks)
Local Ordinal Embedding
Terada, Yoshikazu (CiNet / Center for Information and Neural Networks)
Local Ordinal Embedding
2014/11/04
16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)
YANAGIHARA, Hirokazu (Graduate School of Science, Hiroshima University)
Conditions for consistency of a log-likelihood-based information criterion in normal multivariate linear regression models under the violation of normality assumption
YANAGIHARA, Hirokazu (Graduate School of Science, Hiroshima University)
Conditions for consistency of a log-likelihood-based information criterion in normal multivariate linear regression models under the violation of normality assumption
2014/05/20
13:00-14:10 Room #052 (Graduate School of Math. Sci. Bldg.)
OGIHARA, Teppei (Center for the Study of Finance and Insurance, Osaka University)
Maximum likelihood type estimation of diffusion processes with non synchronous observations contaminated by market microstructure noise (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/02.html
OGIHARA, Teppei (Center for the Study of Finance and Insurance, Osaka University)
Maximum likelihood type estimation of diffusion processes with non synchronous observations contaminated by market microstructure noise (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/02.html
2014/05/13
13:00-14:10 Room #052 (Graduate School of Math. Sci. Bldg.)
Selma Chaker (Bank of Canada)
On High Frequency Estimation of the Frictionless Price: The Use of Observed Liquidity Variables (ENGLISH)
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/01.html
Selma Chaker (Bank of Canada)
On High Frequency Estimation of the Frictionless Price: The Use of Observed Liquidity Variables (ENGLISH)
[ Abstract ]
Observed high-frequency prices are always contaminated with liquidity costs or market microstructure noise. Inspired by the market microstructure literature, I explicitly model this noise and remove it from observed prices to obtain an estimate of the frictionless price. I then formally test whether the prices adjusted for the estimated liquidity costs are either totally or partially free from noise. If the liquidity costs are only partially removed, the residual noise is smaller and closer to an exogenous white noise than the original noise is. To illustrate my approach, I use the adjusted prices to improve volatility estimation in the presence of noise. If the noise is totally absorbed, I show that the sum of squared returns - which would be inconsistent for return variance when based on observed returns - becomes consistent when based on adjusted returns.
[ Reference URL ]Observed high-frequency prices are always contaminated with liquidity costs or market microstructure noise. Inspired by the market microstructure literature, I explicitly model this noise and remove it from observed prices to obtain an estimate of the frictionless price. I then formally test whether the prices adjusted for the estimated liquidity costs are either totally or partially free from noise. If the liquidity costs are only partially removed, the residual noise is smaller and closer to an exogenous white noise than the original noise is. To illustrate my approach, I use the adjusted prices to improve volatility estimation in the presence of noise. If the noise is totally absorbed, I show that the sum of squared returns - which would be inconsistent for return variance when based on observed returns - becomes consistent when based on adjusted returns.
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/01.html
2014/04/08
13:00-14:10 Room #052 (Graduate School of Math. Sci. Bldg.)
Alexandre Brouste (Universite du Maine, France)
Parametric estimation in fractional Ornstein-Uhlenbeck process (ENGLISH)
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/00.html
Alexandre Brouste (Universite du Maine, France)
Parametric estimation in fractional Ornstein-Uhlenbeck process (ENGLISH)
[ Abstract ]
Several statistical models that imply the fractional Ornstein-Uhlenbeck (fOU) process will be presented: direct observations of the process or partial observations in an additive independent noise, continuous observations or discrete observations. In this different settings, we exhibit large sample (or high-frequency) asymptotic properties of the estimators (maximum likelihood estimator, quadratic variation based estimator, moment estimator, …) for all parameters of interest of the fOU. We also illustrate our results with the R package yuima.
[ Reference URL ]Several statistical models that imply the fractional Ornstein-Uhlenbeck (fOU) process will be presented: direct observations of the process or partial observations in an additive independent noise, continuous observations or discrete observations. In this different settings, we exhibit large sample (or high-frequency) asymptotic properties of the estimators (maximum likelihood estimator, quadratic variation based estimator, moment estimator, …) for all parameters of interest of the fOU. We also illustrate our results with the R package yuima.
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/00.html
2013/12/04
13:30-14:40 Room #052 (Graduate School of Math. Sci. Bldg.)
HINO, Hideitsu (University of Tsukuba)
A quantile-based likelihood estimator for information theoretic clustering (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/09.html
HINO, Hideitsu (University of Tsukuba)
A quantile-based likelihood estimator for information theoretic clustering (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/09.html
2013/11/27
13:30-14:40 Room #052 (Graduate School of Math. Sci. Bldg.)
TAKEUCHI, Atsushi (Osaka City University)
Density of solutions to stochastic functional differential equations (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/08.html
TAKEUCHI, Atsushi (Osaka City University)
Density of solutions to stochastic functional differential equations (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/08.html
2013/11/20
13:30-14:40 Room #052 (Graduate School of Math. Sci. Bldg.)
NOMURA, Ryosuke (Graduate school of Mathematical Sciences, Univ. of Tokyo)
TD法における価値関数への収束アルゴリズム (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/07.html
NOMURA, Ryosuke (Graduate school of Mathematical Sciences, Univ. of Tokyo)
TD法における価値関数への収束アルゴリズム (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/07.html
2013/11/11
14:50-16:00 Room #052 (Graduate School of Math. Sci. Bldg.)
NINOMIYA, Yoshiyuki (Kyusyu University)
LASSO に対する AIC タイプの情報量規準 (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/06.html
NINOMIYA, Yoshiyuki (Kyusyu University)
LASSO に対する AIC タイプの情報量規準 (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/06.html
2013/10/25
14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)
MURATA, Noboru (Waseda University)
Sparse coding and structured dictionary learning (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/05.html
MURATA, Noboru (Waseda University)
Sparse coding and structured dictionary learning (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/05.html
2013/10/23
13:00-15:30 Room #006 (Graduate School of Math. Sci. Bldg.)
Mark Podolskij (Universität Heidelberg)
Limit theorems for ambit processes (ENGLISH)
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/04.html
Mark Podolskij (Universität Heidelberg)
Limit theorems for ambit processes (ENGLISH)
[ Abstract ]
We present some recent limit theorems for high frequency observations of ambit processes. Ambit processes constitute a flexible class of models, which are usually used to describe turbulent motion in physics. Mathematically speaking, they have a continuous moving average structure with additional random component called intermittency. In the first part of the lecture we will demonstrate the asymptotic theory for ambit processes driven by Brownian motion. The second part will deal with Levy driven ambit processes. We will see that these two cases deliver completely different limiting results.
本講演は数物フロンティア・リーディング大学院のレクチャーとして行います.
[ Reference URL ]We present some recent limit theorems for high frequency observations of ambit processes. Ambit processes constitute a flexible class of models, which are usually used to describe turbulent motion in physics. Mathematically speaking, they have a continuous moving average structure with additional random component called intermittency. In the first part of the lecture we will demonstrate the asymptotic theory for ambit processes driven by Brownian motion. The second part will deal with Levy driven ambit processes. We will see that these two cases deliver completely different limiting results.
本講演は数物フロンティア・リーディング大学院のレクチャーとして行います.
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/04.html
2013/10/16
13:30-14:40 Room #052 (Graduate School of Math. Sci. Bldg.)
NAKATANI, Tomoaki (Hokkaido University)
統計解析環境Rにおける多変量GARCHモデルの推定とパッケージ化 (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/03.html
NAKATANI, Tomoaki (Hokkaido University)
統計解析環境Rにおける多変量GARCHモデルの推定とパッケージ化 (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/03.html
2013/07/04
14:50-16:00 Room #052 (Graduate School of Math. Sci. Bldg.)
SUZUKI, Taiji (Tokyo Institute of Technology)
低ランク行列推定におけるベイズ推定法の性質 (JAPANESE)
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/02.html
SUZUKI, Taiji (Tokyo Institute of Technology)
低ランク行列推定におけるベイズ推定法の性質 (JAPANESE)
[ Abstract ]
真のパラメータが低ランク行列の構造を持つような低ランク行列推定問題を考える. 低ランク行列推定問題の例としては,低ランク行列の一部が見えている時にその残りを 推定する行列補完の問題などがある.応用としてはユーザへの推薦システムなどがある. これまでの理論解析は主にスパース正則化を用いた経験誤差最小化を対象としてきたが, 本発表ではベイズ法を考え,その統計的性質を調べる.ベイズ法においては, 正則化付き経験誤差最小化による方法とは異なるやや緩い仮定のもと, ほぼ最適な収束レートが導けることを示す.また,テンソル型データ (多次元アレイデータ)へも同様の議論が拡張可能であることも述べる.
[ Reference URL ]真のパラメータが低ランク行列の構造を持つような低ランク行列推定問題を考える. 低ランク行列推定問題の例としては,低ランク行列の一部が見えている時にその残りを 推定する行列補完の問題などがある.応用としてはユーザへの推薦システムなどがある. これまでの理論解析は主にスパース正則化を用いた経験誤差最小化を対象としてきたが, 本発表ではベイズ法を考え,その統計的性質を調べる.ベイズ法においては, 正則化付き経験誤差最小化による方法とは異なるやや緩い仮定のもと, ほぼ最適な収束レートが導けることを示す.また,テンソル型データ (多次元アレイデータ)へも同様の議論が拡張可能であることも述べる.
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/02.html
2013/06/18
13:00-14:10 Room #052 (Graduate School of Math. Sci. Bldg.)
MASUDA, Hiroki (Institute of Mathematics for Industry, Kyushu University)
Locally stable distribution approximation of high-frequency data (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/01.html
MASUDA, Hiroki (Institute of Mathematics for Industry, Kyushu University)
Locally stable distribution approximation of high-frequency data (JAPANESE)
[ Reference URL ]
http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/01.html
