Kyushu University [Hiroki Masuda (Professor) Faculty of Mathematics, Department of Mathematical Sciences]

I have been studying statistical inference for stochastic processes, especially for non-Gaussian Levy driven models with jumps.
My research goal is to establish a standard machinery which can flexibly capture non-Gaussianity of dependent data and provide us with high-accuracy quantitative prediction tools. Of primary interests are: asymptotic distributional theory of various statistics consisting of complex dependent data set; higher order approximation of statistical functionals; and development of how to compute conditional expectations which do not admit an explicit expression. Derivation of the mixing property and its rate of stochastic processes, which is crucial when considering ergodic models, is also in my scope. Furthermore, I have been addressing implementation of the obtained statistical tools in the "YUIMA package" in R (YUIMA project), with a view toward applications to life and physical sciences.

Research

Research Interests

Statistical inference for stochastic processes, its implementation and applications
keyword : Asymptotic statistics, Stochastic process, Dependent data, Large-scale high-frequency data analysis, software development
2000.04Statistical inference for stochastic processes, distributional theory for statistical functionals.

Current and Past Project

Workshops are organized in Hiroshima University and Moscow State University. We issue statements and exchange information about asymptotic theory for stochastic processes and its applications to insurance, high-dimensional data analysis and related topics.

Academic Activities

Reports

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1.	Hiroki Masuda, Approximate quadratic estimating function for discretely observed Lévy driven SDEs with application to a noise normality test, RIMS Kokyuroku, RIMS Kokyuroku 1752 (2011), 113--131., 2011.07.

Papers

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1.	Hiroki Masuda, Lorenzo Mercuri, Yuma Uehara, Noise inference for ergodic Lévy driven SDE, Electronic Journal of Statististics, 10.1214/22-EJS2006, 16, 1, 2432-2474, 2022.04, We study inference for the driving Lévy noise of an ergodic stochastic differential equation (SDE) model, when the process is observed at high-frequency and long time and when the drift and scale coefficients contain finite-dimensional unknown parameters. By making use of the Gaussian quasi-likelihood function for the coefficients, we derive a stochastic expansion for functionals of the unit-time residuals, which clarifies some quantitative effect of plugging in the estimators of the coefficients, thereby enabling us to take several inference procedures for the driving-noise characteristics into account. We also present new classes and methods available in YUIMA for the simulation and the estimation of a Lévy SDE model. We highlight the flexibility of these new advances in YUIMA using simulated and real data..
2.	Hiroki Masuda, Yuma Uehara, Estimating diffusion with compound Poisson jumps based on self-normalized residuals, Statistical Planning and Inference, https://doi.org/10.1016/j.jspi.2021.02.008, 215, 158-183, 2021.12, We consider parametric estimation of the continuous part of a class of ergodic diffusions with jumps based on high-frequency samples. Various papers previously proposed threshold based methods, which enable us to distinguish whether observed increments have jumps or not at each small-time interval, hence to estimate the unknown parameters separately. However, a data-adapted and quantitative choice of the threshold parameter is known to be a subtle and sensitive problem. In this paper, we present a simple alternative based on the Jarque–Bera normality test for the Euler residuals. Different from the threshold based method, the proposed method does not require any sensitive fine tuning, hence is of practical value. It is shown that under suitable conditions the proposed estimator is asymptotically equivalent to an estimator constructed by the unobserved fluctuation of the continuous part of the solution process, hence is asymptotically efficient. Some numerical experiments are conducted to observe finite-sample performance of the proposed method..
3.	Ajay Jasra, Kengo Kamatani, Hiroki Masuda, Bayesian inference for stable Lévy–driven stochastic differential equations with high-frequency data, Scandinavian Journal of Statistics, 10.1111/sjos.12362, 2019.06, In this paper, we consider parametric Bayesian inference for stochastic differential equations driven by a pure-jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available; hence, we use a quasi-likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein–von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo algorithm for Bayesian inference, and assisted with theoretical results, we show how to scale Metropolis–Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio from going to zero in the large data limit. Our algorithm is presented on numerical examples that help verify our theoretical findings..
4.	Hiroki Masuda, Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable Lévy process, Stochastic Processes and their Applications, 10.1016/j.spa.2018.04.004, 129, 3, 1013-1059, 2019.03, We address estimation of parametric coefficients of a pure-jump Lévy driven univariate stochastic differential equation (SDE) model, which is observed at high frequency over a fixed time period. It is known from the previous study (Masuda, 2013) that adopting the conventional Gaussian quasi-maximum likelihood estimator then leads to an inconsistent estimator. In this paper, under the assumption that the driving Lévy process is locally stable, we extend the Gaussian framework into a non-Gaussian counterpart, by introducing a novel quasi-likelihood function formally based on the small-time stable approximation of the unknown transition density. The resulting estimator turns out to be asymptotically mixed normally distributed without ergodicity and finite moments for a wide range of the driving pure-jump Lévy processes, showing much better theoretical performance compared with the Gaussian quasi-maximum likelihood estimator. Extensive simulations are carried out to show good estimation accuracy. The case of large-time asymptotics under ergodicity is briefly mentioned as well, where we can deduce an analogous asymptotic normality result..
5.	Alexandre Brouste, Hiroki Masuda, Efficient estimation of stable Lévy process with symmetric jumps, Statistical Inference for Stochastic Processes, 10.1007/s11203-018-9181-0, 1-19, 2018.07, Efficient estimation of a non-Gaussian stable Lévy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a non-singular Fisher information matrix through the use of a non-diagonal norming matrix. The asymptotic normality and efficiency of a sequence of roots of the associated likelihood equation are shown as well. Moreover, we show that a simple preliminary method of moments can be used as an initial estimator of a scoring procedure, thereby conveniently enabling us to bypass numerically demanding likelihood optimization. Our simulation results show that the one-step estimator can exhibit quite similar finite-sample performance as the maximum likelihood estimator..
6.	Shoichi Eguchi, Hiroki Masuda, Schwarz type model comparison for LAQ models, Bernoulli, 10.3150/17-BEJ928, 24, 3, 2278-2327, 2018.08, For model-comparison purpose, we study asymptotic behavior of the marginal quasi-log likelihood associated with a family of locally asymptotically quadratic (LAQ) statistical experiments. Our result entails a far-reaching extension of applicable scope of the classical approximate Bayesian model comparison due to Schwarz, with frequentist-view theoretical foundation. In particular, the proposed statistics can deal with both ergodic and non-ergodic stochastic process models, where the corresponding M-estimator may of multi-scaling type and the asymptotic quasi-information matrix may be random. We also deduce the consistency of the multistage optimal-model selection where we select an optimal sub-model structure step by step, so that computational cost can be much reduced. Focusing on some diffusion type models, we illustrate the proposed method by the Gaussian quasi-likelihood for diffusion-type models in details, together with several numerical experiments..
7.	Dmytro Ivanenko, Alexey M. Kulik, Hiroki Masuda, Uniform LAN property of locally stable Lévy process observed at high frequency, ALEA - Latin American Journal of Probability and Mathematical Statistics, 12, 835-862, 2015.10, Suppose we have a high-frequency sample from the {¥lp} of the form $X_t^¥theta=¥beta t+¥gamma Z_t+U_t$, where $Z$ is a possibly asymmetric locally $¥al$-stable {¥lp}, and $U$ is a nuisance {¥lp} less active than $Z$. We prove the LAN property about the explicit parameter $¥theta=(¥beta,¥gam)$ under very mild conditions without specific form of the {¥lm} of $Z$, thereby generalizing the LAN result of ¥cite{AJ07}. In particular, it is clarified that a non-diagonal norming may be necessary in the truly asymmetric case. Due to the special nature of the local $¥al$-stable property, the asymptotic Fisher information matrix takes a clean-cut form..
8.	Hiroki Masuda, Convergence of Gaussian quasi-likelihood random fields for ergodic Levy driven SDE observed at high frequency, Annals of Statistics, 10.1214/13-AOS1121, 41, 3, 1593-1641, 2013.06.
9.	Hiroki Masuda, Asymptotics for functionals of self-normalized residuals of discretely observed stochastic processes, Stochastic Processes and their Applications, 10.1016/j.spa.2013.03.013, 123, 7, 2752-2778, 2013.07, The purpose of this paper is to derive the stochastic expansion of self-normalized-residual functionals stemming from a class of diffusion type processes observed at high frequency, where total observing period may or may not tend to infinity. The result enables us to construct some explicit statistics for goodness of fit tests, consistent against “presence of a jump component” and “diffusion-coefficient misspecification”; then, the acceptance of the null hypothesis may serve as a collateral evidence for using the correctly specified diffusion type model. Especially, our asymptotic result clarifies how to remove the bias caused by plugging in a diffusion-coefficient estimator..
10.	Hiroki Masuda, Reiichiro Kawai, Local asymptotic normality for normal inverse Gaussian Levy processes with high-frequency sampling, ESAIM: Probability and Statistics, 10.1051/ps/2011101, 17, 13-32, 2013.01, We prove the local asymptotic normality for the full parameters of the normal inverse Gaussian Lévy process X, when we observe high-frequency data XΔn,X2Δn,...,XnΔn with sampling mesh Δn → 0 and the terminal sampling time nΔn → ∞. The rate of convergence turns out to be (√nΔn, √nΔn, √n, √n) for the dominating parameter (α,β,δ,μ), where α stands for the heaviness of the tails, β the degree of skewness, δ the scale, and μ the location. The essential feature in our study is that the suitably normalized increments of X in small time is approximately Cauchy-distributed, which specifically comes out in the form of the asymptotic Fisher information matrix..
11.	Hiroki Masuda, Approximate self-weighted LAD estimation of discretely observed ergodic ornstein-uhlenbeck processes, Electronic Journal of Statistics, 10.1214/10-EJS565, 4, 525-565, 2010.01, We consider drift estimation of a discretely observed OrnsteinUhlenbeck process driven by a possibly heavy-tailed symmetric Lévy process with positive activity index β. Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being √ nh^1−1/β _n, where n denotes sample size and h_n > 0 the sampling mesh satisfying that h_n → 0 and nh_n → ∞. This implies that the rate of convergence is determined by the most active part of the driving Lévy process; the presence of a driving Wiener part leads to √ nh_n, which is familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving Lévy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving Lévy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions..
12.	Hiroki Masuda, Joint estimation of discretely observed stable L\'evy processes with symmetric L\'evy density, The Journal of The Japan Statistical Society, Vol.39, no.1, pp.49-75, 2009.06.
13.	Hiroki Masuda, Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps, Stochastic Processes and their Applications, 10.1016/j.spa.2006.04.010, 117, 1, 35-56, 2007.01, Let X be a multidimensional diffusion with jumps. We provide sets of conditions under which: X fulfils the ergodic theorem for any initial distribution; and X is exponentially β-mixing. Utilizing the Foster-Lyapunov drift criteria developed by Meyn and Tweedie, we extend several existing results concerning diffusions. We also obtain the boundedness of moments of g (X_t) for a suitable unbounded function g. Our results can cover a wide variety of diffusions with jumps by selecting suitable test functions, and serve as fundamental tools for statistical analyses concerning the processes..
14.	Hiroki Masuda, Simple estimators for parametric Markovian trend of ergodic processes based on sampled data, Journal of the Japan Statistical Society, 35, no.2, 147-170, 2005.01.
15.	Hiroki Masuda, Nakahiro Yoshida, Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model, Stochastic Processes and their Applications, 10.1016/j.spa.2005.02.007, 115, 7, 1167-1186, 115, 1167-1185., 2005.01.

Presentations

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1.	Hiroki Masuda, Estimation and selection of ergodic Lévy driven SDE: an overview of some recent developments, MFO-RIMS Tandem Workshop ''Nonlocality in Analysis, Probability and Statistics'', 2022.03, We consider estimation and relative model comparison for the parametric coefficients of a semiparametric ergodic non-linear Levy-driven model observed at high-frequency. Our asymptotics is based on the fully explicit two-stage Gaussian quasi-likelihood function (GQLF) of the Euler-approximation type. For selections of the scale and drift coefficients, we propose explicit Gaussian quasi-AIC (GQAIC) and Gaussian quasi-BIC (GQBIC) statistics through the stepwise inference procedure. In particular, we show that the mixed-rates structure of the joint GQLF, which does not emerge in the case of diffusions, gives rise to the non-standard forms of the regularization terms in the selection of the scale coefficient, quantitatively clarifying the relation between estimation precision and sampling frequency. Our study clearly shows that the reach and the limit of the GQLF, the explict character and wide applicability of which must remain valuable despite of its possible sub-opitimality. Some numerical experiments are given to illustrate our theoretical findings..
2.	Hiroki Masuda, On mixed-rates structure in Gaussian quasi-likelihood inference for Lévy SDE, Workshop on Statistical modeling for stochastic processes and related fields, 2021.09, [URL].
3.	Hiroki Masuda, Estimation and selection for ergodic Lévy driven models, PVSeminar, 2021.09, [URL].
4.	Hiroki Masuda, Lévy-Ornstein-Uhlenbeck regression, Asia-Pacific Seminar in Probability and Statistics (APSPS), 2021.06, [URL].
5.	Shoichi Eguchi, Hiroki Masuda, Gaussian quasi-information criterion for ergodic Lévy driven model, Virtual 63rd ISI World Statistics Congress, 2021.07.
6.	Hiroki Masuda, Alexei Kulik, LAD estimation of locally stable SDE, Computational and Methodological Statistics, 2020.12, Our goal is to prove the asymptotic (mixed) normality of the least absolute deviation (LAD) type estimator of locally stable SDE observed at high frequency, where the target drift coefficient may be nonlinear in both state variable and parameter. The proof essentially relies on the recently developed general representation result about small-time stable approximation for general locally stable processes. The result is a far-reaching extension of the previous study Masuda (Electronic Journal of Statistics, 2010)..
7.	Hiroki Masuda, Noise estimation for ergodic Levy driven SDE and YUIMA package, Séminaire AgroParisTech, 2020.01, Levy driven stochastic differential equation (SDE) is a flexible building block for modeling non-Gaussian high-frequency data observed in many application fields such as biology and ecology. It is, however, a common knowledge that a closed form of the likelihood function is rarely available except for quite special cases, making estimation of characteristics of the driving Levy noise difficult. In this talk, we begin with an overview of the related previous studies, and then propose a multistep estimation procedure based on the Euler residuals constructed from the Gaussian quasi-maximum likelihood estimator (GQMLE). Specifically, we first estimate the parametric coefficient by the GQMLE, next approximate “unit" time increments of the driving noise by partially summing up the Euler residuals; this strategy would be useful when the underlying Levy process has a tractable unit-time density while having very complicated Levy measure. We will present large-sample properties of the proposed estimator, followed by how to implement it to the YUIMA package in R (https://yuimaproject.com). This talk is based on the ongoing joint work with Yuma Uehara (The Institute of Statistical Mathematics) and Lorenzo Mercuri (University of Milan)..
8.	Hiroki Masuda, Mercuri, Lorenzo, Yuma Uehara, Noise estimation for ergodic Lévy driven SDE in YUIMA package, CMStatistics 2019, 2019.12, To describe non-Gaussian activity in high frequency data obtained from financial, biological, and technological phenomenon, Levy driven stochastic differential equations serve as good candidates. Since the closed form of its genuine likelihood is generally not obtained, the estimation of its driving noise is often done by empirical moment fittings with respect to its Levy measure. However, the measure sometimes takes complex form, and thus intractable. For such a problem, we consider the approximation of unit time increments of the driving noise based on the Euler residual. By making use of this approximation, we can conduct parametric estimation methods of the driving noise with bias correction. We will present its theoretical properties and show some numerical experiments..
9.	Hiroki Masuda, Nonlinear locally stable regression, Dynstoch meeting 2019, 2019.06, We consider statistical inference for a class of non-ergodic locally stable regression models with parametric trend and scale coefficients, when the process is observed at high frequency and the local stable (activity) index is unknown. A detailed asymptotics of the associated (conditional) stable quasi-likelihood estimator is given. In particular, we show that the asymptotic property of the estimator is affected in an essential way by a sort of nonlinearity of the scale coefficient, resulting in a significant generalization of the previous finding in Brouste and Masuda (2018, SISP). We will also mention how to implement the proposed statistics into the YUIMA package (2014)..
10.	Hiroki Masuda, Locally stable regression with unknown activity index, CMStatistics 2018, 2018.12, Typically, transition of large-scale dependent data, such as those sampled at ultra high-frequency, are highly non-Gaussian. One of natural ways of modeling such data would be to use continuous-time stochastic processes driven by a non-Gaussian pure-jump noise. The related existing literature is, however, still far from being well-developed. In this talk, we present tailor-made quasi-likelihood inference results that can efficiently handle such locally and highly non-Gaussian statistical models with the activity index of the driving noise process being unknown. The model setup includes not only Markovian stochastic differential equations but also a class of semimartingale regression models. Of primary interest are cases where estimation target includes not only the rapidly varying scale structure but also the slowly varying trend one..
11.	Alexandre Brouste, Hiroki Masuda, Efficient estimation of stable Lévy process, ASC2018, Asymptotic Statistics and Computations, 2018.02.
12.	Hiroki Masuda, Local limit theorem in non-Gaussian quasi-likelihood inference, Asymptotic Statistics of Stochastic Processes and Applications XI, 2017.07, We consider parameter estimation of the finite-dimensional parameter in the stochastic differential equation (SDE) model driven by a highly non-Gaussian noise. We will present handy sufficient conditions for the L1-local limit theorem with convergence rate, which is the key assumption for the asymptotic mixed normality. The sufficient conditions are given only in terms of the driving Levy measure and/or the characteristic exponent of the driving noise. Specific examples satisfying them include stable, exponentially tempered $¥beta$-stable, and generalized hyperbolic Levy processes..
13.	Hiroki Masuda, Stable quasi-likelihood regression, EcoSta 2017, 2017.06.
14.	Hiroki Masuda, Shoichi Eguchi, Yuma Uehara, Lévy SDE inference in Yuima package, Dynstoch meeting 2017, 2017.04.
15.	Hiroki Masuda, Locally stable regression without ergodicity and finite moments, Hokkaido International Symposium "Recent Developments of Statistical Theory in Statistical Science", 2016.10.
16.	Hiroki Masuda, On Asymptotics of multivariate non-Gaussian quasi-likelihood, World Congress in Probability and Statistics, 2016.07, We consider (semi-)parametric inference for a class of stochastic differential equation (SDE) driven by a locally stable Levy process, focusing on multivariate setting and some computational aspects. The process is supposed to be observed at high frequency over a fixed time domain. This setting naturally gives rise to a theoretically fascinating quasi-likelihood which brings about a novel unified estimation strategy for targeting a broad spectrum of driving Levy processes. The limit experiment is mixed normal with a clean-cut random information structure, based on which it is straightforward to make several conventional asymptotic statistical decisions. The infill-asymptotics adopted here makes the popular Gaussian quasi-likelihood useless, while instead enabling us not only to incorporate any exogenous and/or observable endogenous data into the trend and/or scale coefficients without essential difficulty, but also to sidestep most crucial assumptions on the long-term stability such as ergodicity and moment boundedness. The proposed quasi-likelihood estimator is asymptotically efficient in some special cases..
17.	Hiroki Masuda, On Asymptotics of multivariate non-Gaussian quasi-likelihood, The 4th Institute of Mathematical Statistics Asia Pacific Rim Meeting, 2016.06.
18.	Hiroki Masuda, Lévy in quasi-likelihood estimation of SDE, Statistics for Stochastic Processes and Analysis of High Frequency Data V, 2016.03, We try to give a clear whole picture about the local stable approximation in estimating a L\'{e}vy driven SDE under infill asymptotics without ergodicity. Our finding here is that the completely analogous strategy as in the local Gauss approximation in estimating a diffusion does a good job, when the activity degree is equal to or greater than 1 (the Cauchy-like case). The proposed estimator is indeed asymptotically efficient in some instances..
19.	Hiroki Masuda, Computational aspects of estimating Lévy driven models, The 9th IASC-ARS conference, 2015.12, We consider estimation problem concerning stochastic differential equations driven by a Levy process with jumps. The model is supposed to be observed at high-frequency, allowing us to incorporate a small-time approximation of the underlying likelihood. An overview of some existing theories based on the Gaussian and non-Gaussian quasi-likelihoods is presented, together with their computational aspects. Also to be demonstrated is how to implement the theory in the YUIMA package: an R framework for simulation and inference of stochastic differential equations..
20.	Hiroki Masuda, On variants of stable quasi-likelihood for Levy driven SDE, Statistique Asymptotique des Processus Stochastiques X, 2015.03.
21.	Hiroki Masuda, On sampling problem for pure-jump SDE , 3rd APRM, Taipei, 2014.07.
22.	Hiroki Masuda, LAD-based estimation of locally stable Ornstein-Uhlenbeck processes, Waseda International Symposium on "Stable Process, Semimartingale, Finance & Pension Mathematics", 2014.03, [URL], The LAD type estimator for discretely observed Levy driven OU process is much more efficient than the LSE type one. We prove that the proposed estimator under a random norming is asymptotically standard-normally distributed, making construction of confidence intervals easy..
23.	Hiroki Masuda, Stable quasi-likelihood: Methodology and computational aspects, ERCIM 2013 London, 2013.12, [URL], We consider the semi-parametric model described by the parametric locally stable pure-jump stochastic differential equation. We wish to estimate the parametric coefficients based on a high-frequency sample over a fixed interval. In this talk, we introduce a novel, tailor-made estimator based on the stable approximation of the one-step transition distribution. Under suitable regularity conditions, it is shown that the proposed estimator is asymptotically mixed-normal. The result reveals that, in case of the stable-like driving Levy process, the proposed estimator is much more efficient than the conventional Gaussian quasi-maximum likelihood estimator, which requires the large-time asymptotics and leads to a slower rates of convergence. Nevertheless, evaluation of the proposed estimator is computationally more involved compared with the Gaussian case. Also discussed in some detail is the computational aspects of the proposed methodology. .
24.	Osaka University, [URL].
25.	Hiroki Masuda, On statistical inference for Levy-driven models, The 59th World Statistics Congress (WSC), 2013.08, [URL], 保険数理分野では局所安定型確率微分方程式によるモデリングが有用である．モデルを適合させる対象期間を固定しつつ統計的分布論の理論基盤を確保できるという点において，ノイズの非正規性が如実に現れる当該分野での推測問題に新たな視点・展開を与えた．.
26.	Hiroki Masuda, Estimation of stable-like stochastic differential equations, 29th European Meeting of Statisticians, 2013.07, [URL], We consider the stochastic differential equation of pure-jumps type with parametric coefficients. We wish to estimate the unknown parameters based on a discrete-time but high-frequency sample. A naive way would be to use the Gaussian quasi likelihood. However, although the Gaussian quasi likelihood is known to be well-suited for the case of diffusions, it leads to asymptotically suboptimal estimator in the pure-jump case; in particular, the Gaussian quasi-maximum likelihood estimation inevitably needs a large-time asymptotics. In this talk, we will introduce another kind of quasi-maximum likelihood estimator based on the local-stable approximation of the one-step transition distribution; the proposed estimation procedure is a pure-jump counterpart to the Gaussian quasi-maximum likelihood estimation. Under some regularity conditions, we will show the asymptotic mixed normality of the proposed estimator, revealing that the proposed estimator is asymptotically much more efficient than the Gaussian quasi-maximum likelihood estimator..
27.	Hiroki Masuda, On optimal estimation of stable Ornstein-Uhlenbeck processes, Dynstoch meeting 2013, 2013.04, [URL], Ornstein-Uhlenbeck (OU) processes driven by a Levy process form a particular tractable class of Markovian stochastic differential equations with jumps. Among them, the non-Gaussian stable driven ones, the study of which dates back to Doob's work in 1942, are known to have a pretty inherent character. Especially, a special property of stable integrals allows us to exactly generate the discrete-time sample from the process, and more importantly, to study in a transparent way the likelihood ratio associated with discrete-time sampling. We are concerned with optimal estimation of the stable OU processes observed at high-frequency. We clarify that, due to the infinite-variance character of the model, the likelihood ratio exhibits entirely different asymptotic behaviors according to whether or not the terminal sampling time tends to infinity. When the terminal time is a fixed time, we present the LAMN (Local Asymptotic Mixed Normality) structure of the statistical model, entailing the notion of asymptotic efficiency of a regular estimator. Also presented is how to construct some simple rate-efficient estimators having asymptotic mixed normality, together with numerical experiments..
28.	Hiroki Masuda, Non-Gaussian quasi-likelihoods for estimating jump SDE, 8th World Congress in Probability and Statistics, 2012.07, We consider a stochastic differential equation driven by a stable-like Levy process, which is observed at high frequency. In this talk, we will introduce a quasi-maximum likelihood estimator based on the local-stable approximation of the transition laws. This is a pure-jump counterpart to the local-Gauss contrast function, well-suited for the case of diffusions. Under some regularity conditions, we will present asymptotic distribution results, which is entirely different from the Gaussian quasi-likelihood case and much more efficient. In particular, the rate of convergence of the estimator obtained is much better and they are jointly asymptotically normal and mixed-normal according as the terminal sampling tends to infinity or not. .
29.	Hiroki Masuda, Non-Gaussian quasi likelihood in estimating jump SDE, 2nd Asian Pacific Rim Meeting, 2012.07, 非正規安定レヴィ過程で微小時間近似できる確率微分方程式モデルの推定問題を考察した．当該モデルでは従来の正規型擬似最尤推定は効率が悪いことが知られており，新たな推定手法が要求される．筆者は，データ増分の非正規安定近似を介した新しい擬似尤度推定法を考案し，その漸近挙動を導出した．特に，ドリフト推定量の有界時間区間上での漸近混合正規性，および推定量の収束率の改善など，正規型では決して得られない（好ましい）現象が明らかとなった．.
30.	Hiroki Masuda, Local-stable contrast function, Dynstoch meeting 2012, 2012.06, We consider a stochastic differential equation driven by a stable-like Levy process, which is observed at high frequency. In this talk, we will introduce a quasi-maximum likelihood estimator based on the local-stable approximation of the transition laws. This is a pure-jump counterpart to the local-Gauss contrast function, well-suited for the case of diffusions. Under some regularity conditions, we will present asymptotic distribution results, which is entirely different from the Gaussian quasi-likelihood case and much more efficient. In particular, the rate of convergence of the estimator obtained is much better and they are jointly asymptotically normal and mixed-normal according as the terminal sampling tends to infinity or not. .

Membership in Academic Society

ISI (International Statistical Institute)

Awards

Studies in statistical inference for stochastic processes with jumps and their implementation
Simple estimators for parametric Markovian trend of ergodic processes based on sampled data

Educational

Educational Activities

Lectures: Mainly, both elementary and advanced mathematical statistics, and statistical tools for data analyses, including various multivariate analyses.
I was in charge of the exercise classes: (i) metric and topological spaces; (ii) statistics; (iii) basic set theory and logics; and (iv) Measure theory.

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