## On Asymptotic Distribution of Realized Volatility

A central limit theorem for realized volatility based on a general stochastic sampling scheme is proved. The asymptotic distribution depends on the sampling scheme, which is written explicitly in terms of asymptotic skewness and kurtosis of returns. The conditions for the central limit theorem to hold are examined for several concrete examples of schemes.

Key words: High frequency data, volatility, stable convergence.

## Estimation of Second-characteristic Matrix Based on Realized Multipower Variations

Realized multipower variation (MPV for short) defined for stochastic processes discretely observed at high frequency is a generalization of the classical realized quadratic variation. In this note, targeted at a class of multidimensional semimartingales with jumps, we consider first-order inference for the second-characteristic matrix, especially its off-diagonal elements. First we review some prior results concerning asymptotic behavior of MPV, and then formulate a stable central limit theorem for a consistent estimator. In consequence, we can readily construct confidence regions of the estimator as in the one-dimensional case.

Key words: Stable central limit theorem, multidimensional semimartingale with finite number of jumps, realize multipower variation, estimation of second characteristic.

## Financial Models of High-frequency Data Using Time Change: A Review

This article reviews approaches to financial modelling based on the method of time change in the field of high-frequency data analysis.

Key words: High-frequency data, time change.

## Estimation of a Drift Parameter for a Small Diffusion Process

We consider the estimation of an unknown drift parameter for a one-dimensional diffusion process with a small perturbed parameter $\varepsilon$. First the maximum likelihood estimator for continuously observed data is surveyed and then we explain that for discrete time observations at $n$ regularly spaced time points $k/n$, $k=0,1,\ldots,n$, the maximum contrast estimator obtained from the contrast function based on the Euler-Maruyama approximation, which is equivalent to the locally Gaussian approximation to the transition density, has an asymptotic efficiency under $(\varepsilon n)^{-1} =o(1)$ as $\varepsilon \rightarrow 0$ and $n \rightarrow \infty$. Next, a martingale estimating function with both eigenfunction and eigenvalue based on the infinitesimal generator of the diffusion is proposed, and asymptotic properties of an M-estimator obtained from the martingale estimating function is shown under the general condition that $\varepsilon \rightarrow 0$ and $n \rightarrow \infty$. However, the proposed martingale estimating function does not generally have an explicit form. In order to generalize the estimating function, we treat an approximate martingale estimating function and asymptotic properties of an M-estimator derived from the approximate martingale estimating function are stated as $\varepsilon \rightarrow 0$ and $n \rightarrow \infty$.

Key words: Asymptotic efficiency, diffusion process with small perturbed parameter, discrete time observations, eigenfunction, martingale estimating function.

## Nonparametric Goodness of Fit Tests for Diffusion Processes

It is well known that the Kolmogorov-Smirnov statistics for the goodness of fit test problem for i.i.d. data is asymptotically distribution free. However, in ergodic diffusion process models, a natural Kolmogorov-Smirnov type test based on the empirical distribution function is not asymptotically distribution free. In this paper, we review some recent results on nonparametric goodness of fit tests in diffusion process models, based on an idea of score marked empirical process. We see that our tests are asymptotically distribution free and consistent. We consider four cases consist of small or ergodic diffusions, and, continuously or discretely observed cases. We also mention some results given by Dachian and Kutoyants (2008).

Key words: Diffusion process, goodness of fit, asymptotically distribution free, Donsker's invariance principle, martingale, random field.

## Threshold Estimation for Jump-type Stochastic Processes from Discrete Observations

We consider a stochastic process that satisfies a stochastic differential equation with jumps that includes some unknowns. We observed the process at discrete time points. In application, it is important to estimate unknown parameters or some functionals of the process from discrete samples. The recent development of statistical inference for such discretely observed models is remarkable and many interesting results have been obtained by many authors. This paper focuses on overviewing the threshold estimation method, which responds flexibly to many types of estimation problems. The essence of the method is to employ an asymptotic filter that detects sampling intervals where a jump has occurred. The filter judges that there is a jump in the interval if the corresponding increment of neighboring data is larger than a threshold that is meaningfully determined by observers. The error rate of the judgement decreases as the sampling frequency increases if a suitable threshold is used. The purpose of this paper is to provide an easy and intuitively understandable explanation of the theory with some recent asymptotic results and to introduce some practical issues to be investigated in the future.

Key words: Jump-type stochastic processes, discrete observations, threshold estimation, asymptotic inference, selecting threshold.

## Semiparametric Inference in Survival Analysis and Related Topics

Inference procedures for semiparametric non-proportional hazard regression models such as an additive hazards model, a proportional odds model and an accelerated failure time model have been developed. While the maximum partial likelihood method was not applied successfully to these models, inference procedures have been developed for them in a unified way based on martingale estimating equations. Furthermore, regression diagnostics methods have been developed in a unified way based on martingale residuals. This paper reviews recent developments of semiparametric failure time regression models and some related topics.

Key words: Cox proportional hazards model, accelerated failure time model, additive hazards model, empirical process, linear transformation model, martingale.

## Stochastic Models for Analysis of Household Transmission Data: Examining Human-made Transmission Experiments

To date, various epidemiologic models have been proposed for analyzing infectious disease data, most notably, based on branching processes, birth-death processes and spatially-structured contact processes. To appropriately quantify the transmission dynamics using the observed data, the model has to capture crucial elements of the heterogeneous patterns of transmission. Household transmission plays a key role in characterizing the most important aspects of heterogeneities for many directly transmitted diseases. Household transmission can be deemed a human-made experiment where all family members experience close contact. This article reviews essential pre-requisites of the data analysis and several applications of household transmission models to the data, specifying each of the epidemiologic assumptions and offering detailed insights into the actual mechanisms of household transmission. In particular, it focuses on two topics: a probability model for the analysis of final size distribution in households and another transmission model for deriving the household reproduction number, a threshold parameter of an epidemic. Whenever a household transmission model is employed to estimate key epidemiologic parameters, it is critical to assess the validaty and reality of underlying assumptions with respect to disease and data.

Key words: Infectious diseases, stochastic process, epidemiology, epidemic, household, model.

## Eulerian Numbers in Modified Bucket Sorting and Its Related Distribution Theories

This paper discusses the discrete probability distribution function induced from modified bucket sorting. The distribution is related to the unique sequence called “Eulerian Numbers” proposed by Euler. The sequence number obtained by modified bucket sorting is expressed as a recurrence formula. It is mathematically equivalent to the recurrence formula expression of Eulerian numbers. However, there are sensitive differences between a derivation process of our proposal and Eulerian numbers expressed as a number of permutation accents. The recurrence relation for the moments of the distribution is also given. The mean, the variance and higher order cumulants are derived using the relation. Interesting results of the relationship among the proposed, normal and uniform distributions are shown by asymptotic expansion. The efficiency of the proposed method is illustrated through numerical experiments.

Key words: Asymptotic expansion, cumulant, Eulerian distribution, normal distribution, uniform distribution.

## Statistical Models for Earthquake Clustering and Declustering

This review paper summarizes the statistical methods associated with modeling earthquake clusters and declustering. Earthquake clusters are well described by the epidemic-type aftershocks sequence (ETAS) model. In this model, each earthquake, whether it is from the background or triggered by a previous earthquake, triggers its own child events independently according to some probability rules. The stochastic declustering method is developed by making use of the additive property of this model. With this method, each earthquake in the catalog can be identified to be a background event or be triggered from a particular previous event in probabilities. These estimated probabilities can be used to test hypotheses associated with seismicity clustering or background.

Key words: Earthquake, ETAS model, cluster, declustering, second-order residual analysis.