Kyushu University Academic Staff Educational and Research Activities Database
List of Papers
Hiroki Masuda Last modified date:2021.11.01

Professor / Department of Mathematical Sciences / Faculty of Mathematics


Papers
1. Hiroki Masuda, Preface, Special feature: statistics for high-frequency data, Japanese Journal of Statistics and Data Science, 10.1007/s42081-021-00117-z, 4, 479-481, 2021.04, [URL].
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, 2021.04, 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. Shoichi Eguchi, Hiroki Masuda, Data driven time scale in Gaussian quasi-likelihood inference, Statistical Inference for Stochastic Processes, 10.1007/s11203-019-09197-x, 383-430, 22:(3), 2019.10, We study parametric estimation of ergodic diffusions observed at high frequency. Different from the previous studies, we suppose that sampling stepsize is unknown, thereby making the conventional Gaussian quasi-likelihood not directly applicable. In this situation, we construct estimators of both model parameters and sampling stepsize in a fully explicit way, and prove that they are jointly asymptotically normally distributed. High order uniform integrability of the obtained estimator is also derived. Further, we propose the Schwarz (BIC) type statistics for model selection and show its model-selection consistency. We conducted some numerical experiments and found that the observed finite-sample performance well supports our theoretical findings..
5. 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..
6. Yuta Umezu, Yusuke Shimizu, Hiroki Masuda, Yoshiyuki Ninomiya, AIC for the non-concave penalized likelihood method, Annals of the Institute of Statistical Mathematics, 10.1007/s10463-018-0649-x, 1-28, 2019.04, Non-concave penalized maximum likelihood methods are widely used because they are more efficient than the Lasso. They include a tuning parameter which controls a penalty level, and several information criteria have been developed for selecting it. While these criteria assure the model selection consistency, they have a problem in that there are no appropriate rules for choosing one from the class of information criteria satisfying such a preferred asymptotic property. In this paper, we derive an information criterion based on the original definition of the AIC by considering minimization of the prediction error rather than model selection consistency. Concretely speaking, we derive a function of the score statistic that is asymptotically equivalent to the non-concave penalized maximum likelihood estimator and then provide an estimator of the Kullback–Leibler divergence between the true distribution and the estimated distribution based on the function, whose bias converges in mean to zero..
7. 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..
8. Kei Hirose, Hiroki Masuda, Robust relative error estimation, Entropy, 10.3390/e20090632, 20, 9, 2018.08, Relative error estimation has been recently used in regression analysis. A crucial issue of the existing relative error estimation procedures is that they are sensitive to outliers. To address this issue, we employ the γ-likelihood function, which is constructed through γ-cross entropy with keeping the original statistical model in use. The estimating equation has a redescending property, a desirable property in robust statistics, for a broad class of noise distributions. To find a minimizer of the negative γ-likelihood function, a majorize-minimization (MM) algorithm is constructed. The proposed algorithm is guaranteed to decrease the negative γ-likelihood function at each iteration. We also derive asymptotic normality of the corresponding estimator together with a simple consistent estimator of the asymptotic covariance matrix, so that we can readily construct approximate confidence sets. Monte Carlo simulation is conducted to investigate the effectiveness of the proposed procedure. Real data analysis illustrates the usefulness of our proposed procedure..
9. 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..
10. Hiroki Masuda, Yusuke Shimizu, Moment convergence in regularized estimation under multiple and mixed-rates asymptotics, Mathematical Methods of Statistics, 10.3103/S1066530717020016, 26, 2, 81-110, 2017.04, In M-estimation under standard asymptotics, the weak convergence combined with the polynomial type large deviation estimate of the associated statistical random field Yoshida (2011) provides us with not only the asymptotic distribution of the associated M-estimator but also the convergence of its moments, the latter playing an important role in theoretical statistics. In this paper, we study the above program for statistical random fields of multiple and also possibly mixedrates type in the sense of Radchenko (2008) where the associated statistical random fields may be nondifferentiable and may fail to be locally asymptotically quadratic. Consequently, a very strong mode of convergence of a wide range of regularized M-estimators is ensured.Our results are applied to regularized estimation of an ergodic diffusion observed at high frequency..
11. Hiroki Masuda, Yuma Uehara, Two-step estimation of ergodic Lévy driven SDE, Statistical Inference for Stochastic Processes, 10.1007/s11203-016-9133-5, 20, 1, 105-137, 2017.04, We consider high frequency samples from ergodic Lévy driven stochastic differential equation with drift coefficient a(x, α) and scale coefficient c(x, γ) involving unknown parameters α and γ. We suppose that the Lévy measure ν0, has all order moments but is not fully specified. We will prove the joint asymptotic normality of some estimators of α, γ and a class of functional parameter ∫ φ(z) ν0(dz) , which are constructed in a two-step manner: first, we use the Gaussian quasi-likelihood for estimation of (α, γ) ; and then, for estimating ∫ φ(z) ν0(dz) we make use of the method of moments based on the Euler-type residual with the the previously obtained quasi-likelihood estimator..
12. 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..
13. Denis Belomestny, Fabienne Comte, Valentine Genon-Catalot, Hiroki Masuda, Markus Reiß, Lévy Matters IV
Estimation for Discretely Observed Lévy Processes, Lecture Notes in Mathematics, 10.1007/978-3-319-12373-8, 2128, 2015.01.
14. Alexandre Brouste, Masaaki Fukasawa, Hideitsu Hino, Stefano M. Iacus, Kengo Kamatani, Yuta Koike, Hiroki Masuda, Ryosuke Nomura, Teppei Ogihara, Yasutaka Shimuzu, Masayuki Uchida, Nakahiro Yoshida, The YUIMA project
A computational framework for simulation and inference of stochastic differential equations, Journal of Statistical Software, 10.18637/jss.v057.i04, 57, 4, 1-51, 2014.01, The YUIMA Project is an open source and collaborative effort aimed at developing the R package yuima for simulation and inference of stochastic differential equations. In the yuima package stochastic differential equations can be of very abstract type, multidimensional, driven by Wiener process or fractional Brownian motion with general Hurst parameter, with or without jumps specified as Ĺevy noise. The yuima package is intended to offer the basic infrastructure on which complex models and inference procedures can be built on. This paper explains the design of the yuima package and provides some examples of applications..
15. 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.
16. Hiroki Masuda, Nakahiro Yoshida, Edgeworth expansion for the integrated Levy driven Ornstein-Uhlenbeck process, Electronic Communications in Probability, 10.1214/ECP.v18-2726, 18, 94, 2013.12.
17. 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..
18. 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..
19. Raiichiro Kawai, Hiroki Masuda, Exact simulation of finite variation tempered stable Ornstein-Uhlenbeck processes, Monte Carlo Methods and Applications, 10.1515/MCMA.2011.012, 17, 3-4, 279-300, 2012.02, Exact yet simple simulation algorithms are developed for a wide class of Ornstein–Uhlenbeck processes with tempered stable stationary distribution of finite variation with the help of their exact transition probability between consecutive time points. Random elements involved can be divided into independent tempered stable and compound Poisson distributions, each of which can be simulated in the exact sense through acceptance-rejection sampling, respectively, with stable and gamma proposal distributions. We discuss various alternative simulation methods within our algorithms on the basis of acceptance rate in acceptance-rejection sampling for both high- and low-frequency sampling. Numerical results illustrate their advantage relative to the existing approximative simulation method based on infinite shot noise series representation..
20. Hiroki Masuda, Takayuki Morimoto, An optimal weight for realized variance based on intermittent high-frequency data, Japanese Economic Review, 10.1111/j.1468-5876.2011.00552.x, 63, 4, 497-527, 2012.12, Japanese stock markets have two types of breaks, overnight and lunch, during which no trading occurs, causing an inevitable increased variance in estimating daily volatility via a naive realized variance (RV). In order to perform a more stabilized estimation, we modify Hansen and Lunde's weighting technique. As an empirical study, we estimate optimal weights by using a particular approach for Japanese stock data listed on the Tokyo Stock Exchange, and then compare the forecast performance of weighted and non-weighted RV through an autoregressive fractionally integrated moving average model. The empirical result indicates that the appropriate use of the optimally weighted RV can lead to remarkably smaller estimation variance compared with the naive RV, in many series. Therefore a more accurate forecasting of daily volatility data is obtained. Finally, we perform a Monte Carlo simulation to support the empirical result..
21. Reiichiro Kawai, Hiroki Masuda, Infinite variation tempered stable Ornstein-Uhlenbeck processes with discrete observations, Communications in Statistics: Simulation and Computation, 10.1080/03610918.2011.582561, 41, 1, 125-139, 2012.01, We investigate transition law between consecutive observations of Ornstein-Uhlenbeck processes of infinite variation with tempered stable stationary distribution. Thanks to the Markov autoregressive structure, the transition law can be written in the exact sense as a convolution of three random components; a compound Poisson distribution and two independent tempered stable distributions, one with stability index in (0, 1) and the other with index in (1, 2). We discuss simulation techniques for those three random elements. With the exact transition law and proposed simulation techniques, sample paths simulation proves significantly more efficient, relative to the known approximative technique based on infinite shot noise series representation of tempered stable Lévy processes..
22. Hiroki Masuda, Ilia Negri, Yoichi Nishiyama, Goodness-of-fit test for ergodic diffusions by discrete-time observations
An innovation martingale approach, Journal of Nonparametric Statistics, 10.1080/10485252.2010.510186, 23, 2, 237-254, 2011.06, We consider a nonparametric goodness-of-fit test problem for the drift coefficient of one-dimensional ergodic diffusions. Our test is based on the discrete-time observation of the processes, and the diffusion coefficient is a nuisance function which is estimated in some sense in our testing procedure.We prove that the limit distribution of our test is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free.We also show that our test is consistent under any fixed alternatives..
23. Reiichiro Kawai, Hiroki Masuda, On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling, Statistics and Probability Letters, 10.1016/j.spl.2010.12.011, 81, 4, 460-469, 2011.04, We discuss the local asymptotic behavior of the likelihood function associated with all the four characterizing parameters (α, β, δ, μ) of the Meixner Lévy process under high-frequency sampling scheme. We derive the optimal rate of convergence for each parameter and the Fisher information matrix in a closed form. The skewness parameter β exhibits a slower rate alone, relative to the other three parameters free of sampling rate. An unusual aspect is that the Fisher information matrix is constantly singular for full joint estimation of the four parameters. This is a particular phenomenon in the regular high-frequency sampling setting and is of essentially different nature from low-frequency sampling. As soon as either α or δ is fixed, the Fisher information matrix becomes diagonal, implying that the corresponding maximum likelihood estimators are asymptotically orthogonal..
24. Reiichiro Kawai, Hiroki Masuda, On simulation of tempered stable random variates, Journal of Computational and Applied Mathematics, 10.1016/j.cam.2010.12.014, 235, 8, 2873-2887, 2011.02, Various simulation methods for tempered stable random variates with stability index greater than one are investigated with a view towards practical implementation, in particular cases of very small scale parameter, which correspond to increments of a tempered stable Lvy process with a very short stepsize. Methods under consideration are based on acceptancerejection sampling, a Gaussian approximation of a small jump component, and infinite shot noise series representations. Numerical results are presented to discuss advantages, limitations and trade-off issues between approximation error and required computing effort. With a given computing budget, an approximative acceptancerejection sampling technique Baeumer and Meerschaert (2009) [11] is both most efficient and handiest in the case of very small scale parameter and moreover, any desired level of accuracy may be attained with a small amount of additional computing effort..
25. 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 √ nh1−1/β n, where n denotes sample size and hn > 0 the sampling mesh satisfying that hn → 0 and nhn → ∞. 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 √ nhn, 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..
26. Sangyeol Lee, Hiroki Masuda, Jarque-Bera normality test for the driving Lévy process of a discretely observed univariate SDE, Statistical Inference for Stochastic Processes, 10.1007/s11203-010-9043-x, 13, 2, 147-161, 2010.06, In this paper, we study the Jarque-Bera test for a class of univariate parametric stochastic differential equations (SDE) dXt = b(Xt, α)dt + dZt, constructed based on the sample observed at discrete time points tin = ihn, i = 1, 2,..., n, where Z is a nondegenerate Lévy process with finite moments and h is a sequence of positive real numbers with nhn → ∞ and nhn2 → 0 as n → ∞. It is shown that under proper conditions, the Jarque-Bera test statistic based on the Euler residuals can be used to test for the normality of the unobserved Z and the proposed test is consistent against the presence of any nontrivial jump components. Our result indicates that the Jarque-Bera test is easy to implement and asymptotically distribution-free with no fine-tuning parameters. Simulation results to validate the test are given for illustration..
27. Hiroki Masuda, Erratum to
"Ergodicity and exponential β -mixing bound for multidimensional diffusions with jumps" [Stochastic Process. Appl. 117 (2007) 35-56] (DOI:10.1016/j.spa.2006.04.010), Stochastic Processes and their Applications, 10.1016/j.spa.2008.02.010, 119, 2, 676-678, 2009.02, Hiroki Masuda, Graduate School of Mathematics, Kyushu University, presented an erratum on ergodicity and exponential β-mixing bound for multidimensional diffusions with jumps. He clarified that in assumption 2(a), a bounded transition density of Yu△ was supposed to exist, however, X△ was actually needed. In the proof of Lemma 2.4(i) q ∈ (0, 1) from the beginning, although the condition Lemma 2.4(i) contains q possibly lying in [1,2); this point does not affect Lemma 2.4(ii), while the assertion Af(x) ≥ Gf(x) + 0(1) fails. This point is needed to be modified. Moreover, the proof of Lemma 2.5 contains some mistakes in estimating J* f(x): what should be J* f(x)..
28. Hiroki Masuda, Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling, Annals of the Institute of Statistical Mathematics, 10.1007/s10463-007-0131-7, 61, 1, 181-195, 2009.03, We study joint efficient estimation of two parameters dominating either the inverse-Gaussian or gamma subordinator, based on discrete observations sampled at (ti
n)ti=n
n satisfying h n : max i≤n(ti
n) - t i-n
n) → 0 as n → ∞. Under the condition that Tn:=tn
n → ∞ as n → ∞ we have two kinds of optimal rates, √n and √Tn . Moreover, as in estimation of diffusion coefficient of a Wiener process the √n -consistent component of the estimator is effectively workable even when Tn does not tend to infinity. Simulation experiments are given under several h n's behaviors..
29. 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.
30. Hiroki Masuda, On stability of diffusions with compound-Poisson jumps, The Bulletin of Informatics and Cybernetics, 40巻, pp.61-74., 2008.12.
31. 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 (Xt) 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..
32. Hiroki Masuda, Classical method of moments for partially and discretely observed ergodic models, Statistical Inference for Stochastic Processes, 10.1023/B:SISP.0000049120.83388.89, 8, 1, 25-50, 2005.05, We discuss the method of moments for a partially and discretely observed model driven by a time-homogeneous Lévy process. We suppose that the unobserved process is an ε-Markov process and that the data, which comes from another process, are available only at regularly spaced time points. Stochastic differential equations are particularly treated among many other possible models. Some illustrative examples are presented with simulations..
33. 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.
34. 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.
35. Hiroki Masuda, Nakahiro Yoshida, An application of the double Edgeworth expansion to a filtering model with Gaussian limit, Statistics and Probability Letters, 10.1016/j.spl.2004.08.002, 70, 1, 37-48, 70, no.1, 37-48., 2004.01.
36. Hiroki Masuda, On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process, Bernoulli, 10.3150/bj/1077544605, 10, 1, 97-120, 2004.02, We prove the following probabilistic properties of a multidimensional Ornstein-Uhlenbeck process driven by a general Lévy process, under mild regularity conditions: the strong Feller property; the existence of a smooth transition density; and the exponential β-mixing property. As a class of possible invariant distributions of an Ornstein-Uhlenbeck process, we also discuss centred and non-skewed multidimensional generalized hyperbolic distributions..