Parameter estimation for multivariate diffusion processes with the time inhomogeneously positive semidefinite diffusion matrix

Statistical inference for the diffusion coefficients of multivariate diffusion processes has been well established in recent years; however, it is not the case for the drift coefficients. Furthermore, most existing estimation methods for the drift coefficients are proposed under the assumption that the diffusion matrix is positive definite and time homogeneous. In this article, we put forward two estimation approaches for estimating the drift coefficients of the multivariate diffusion models with the time inhomogeneously positive semidefinite diffusion matrix. They are maximum likelihood estimation methods based on both the martingale representation theorem and conditional characteristic functions and the generalized method of moments based on conditional characteristic functions, respectively. Consistency and asymptotic normality of the generalized method of moments estimation are also proved in this article. Simulation results demonstrate that these methods work well.     

Central Limit Theorems of Local Polynomial Threshold Estimator for Diffusion Processes with Jumps

Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the non‐parametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the σ ‐field generated by diffusion processes is destroyed, so the classical central limit theorem for martingale difference sequences cannot work. In high‐frequency data, we proved the central limit theorems of local polynomial threshold estimators for the volatility function in diffusion processes with jumps by Jacod's stable convergence theorem. We believe that our proof procedure for local polynomial threshold estimators provides a new method in this field, especially in the local linear case.     

Nonparametric Methods in Continuous Time Model Specification

Some popular parametric diffusion processes have been assumed as such underlying diffusion processes. This paper considers an important case where both the drift and volatility functions of the underlying diffusion process are unknown functions of the underlying process, and then proposes using two novel testing procedures for the parametric specification of both the drift and diffusion functions. The finite-sample properties of the proposed tests are assessed through using data generated from four popular parametric models. In our implementation, we suggest using a simulated critical value for each case in addition to the use of an asymptotic critical value. Our detailed studies show that there is little size distortion when using a simulated critical value while the proposed tests have some size distortions when using an asymptotic critical value in each case.     

Nonparametric Methods in Continuous Time Model Specification

Some popular parametric diffusion processes have been assumed as such underlying diffusion processes. This paper considers an important case where both the drift and volatility functions of the underlying diffusion process are unknown functions of the underlying process, and then proposes using two novel testing procedures for the parametric specification of both the drift and diffusion functions. The finite-sample properties of the proposed tests are assessed through using data generated from four popular parametric models. In our implementation, we suggest using a simulated critical value for each case in addition to the use of an asymptotic critical value. Our detailed studies show that there is little size distortion when using a simulated critical value while the proposed tests have some size distortions when using an asymptotic critical value in each case.     

Making wald tests work for cointegrated VAR systems

Wald tests of restrictions on the coefficients of vector autoregressive (VAR) processes are known to have nonstandard asymptotic properties for 1(1) and cointegrated systems of variables. A simple device is proposed which guarantees that Wald tests have asymptotic X2-distributions under general conditions. If the true generation process is a VAR(p) it is proposed to fit a VAR(p+l) to the data and perform a Wald test on the coefficients of the first p lags only. The power properties of the modified tests are studied both analytically and numerically by means of simple illustrative examples.     

Closed-form likelihood estimation for one type of affine point processes

ABSTRACT

This paper presents a closed-form likelihood approximation for one type of affine point processes widely used in financial credit risk models. We proceed by first conjecturing the concrete series form of the transition density, verifying our postulation and then establishing the related coefficients by means of Kolmogorov equations. The asymptotic properties of the maximum-likelihood estimators (MLEs) are given in the end.     

Marginal Means/Rates Models for Multiple Type Recurrent Event Data

Recurrent events are frequently observed in biomedical studies, and often more than one type of event is of interest. Follow-up time may be censored due to loss to follow-up or administrative censoring. We propose a class of semi-parametric marginal means/rates models, with a general relative risk form, for assessing the effect of covariates on the censored event processes of interest. We formulate estimating equations for the model parameters, and examine asymptotic properties of the parameter estimators. Finite sample properties of the regression coefficients are examined through simulations. The proposed methods are applied to a retrospective cohort study of risk factors for preschool asthma.     

Forward Moving Average Representation in Multivariate MA(1) Processes

Forward-moving average coefficients are in general different from their corresponding backward-moving average coefficients in multivariate stationary time series. There is lack of practical methods to derive forward-moving average coefficients from the backward ones. In this article, we establish a new practical approach for obtaining the forward-moving average coefficients for multivariate moving average processes of order one.     

Parameter Estimation for a Discretely Observed Integrated Diffusion Process

Abstract.  We consider the estimation of unknown parameters in the drift and diffusion coefficients of a one-dimensional ergodic diffusion X when the observation is a discrete sampling of the integral of X at times i Δ , i  =  1 ,…, n . Assuming that the sampling interval tends to 0 while the total length time interval tends to infinity, we first prove limit theorems for functionals associated with our observations. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent and asymptotically Gaussian with different rates for drift and diffusion coefficient parameters.     

Discretely Observed Diffusions: Approximation of the Continuous-time Score Function

We discuss parameter estimation for discretely observed, ergodic diffusion processes where the diffusion coefficient does not depend on the parameter. We propose using an approximation of the continuous-time score function as an estimating function. The estimating function can be expressed in simple terms through the drift and the diffusion coefficient and is thus easy to calculate. Simulation studies show that the method performs well.     

Central limit theorems of range-based estimators for diffusion models

Abstract

In this article, we consider non parametric range-based estimation procedure for diffusion processes and propose a instantaneous volatility estimator. Under some weak conditions, we certify that the proposed estimator has convergence in probability. Adding some necessary conditions, we prove a central limit theorem. By inference, we reach a conclusion that, with high frequency data in hand, the proposed estimator is more precise than those pure realized instantaneous volatility ones. Numerical simulation illustrates the finite sample properties of the proposed estimator.     

Semiparametric partially linear varying coefficient models with panel count data

This paper studies semiparametric regression analysis of panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. To explore the nonlinear interactions between covariates, we propose a class of partially linear models with possibly varying coefficients for the mean function of the counting processes with panel count data. The functional coefficients are estimated by B-spline function approximations. The estimation procedures are based on maximum pseudo-likelihood and likelihood approaches and they are easy to implement. The asymptotic properties of the resulting estimators are established, and their finite-sample performance is assessed by Monte Carlo simulation studies. We also demonstrate the value of the proposed method by the analysis of a cancer data set, where the new modeling approach provides more comprehensive information than the usual proportional mean model.     

Asymptotic Properties of QML Estimators for VARMA Models with Time‐dependent Coefficients

This paper is about vector autoregressive‐moving average models with time‐dependent coefficients to represent non‐stationary time series. Contrary to other papers in the univariate case, the coefficients depend on time but not on the series' length n. Under appropriate assumptions, it is shown that a Gaussian quasi‐maximum likelihood estimator is almost surely consistent and asymptotically normal. The theoretical results are illustrated by means of two examples of bivariate processes. It is shown that the assumptions underlying the theoretical results apply. In the second example, the innovations are marginally heteroscedastic with a correlation ranging from ?0.8 to 0.8. In the two examples, the asymptotic information matrix is obtained in the Gaussian case. Finally, the finite‐sample behaviour is checked via a Monte Carlo simulation study for n from 25 to 400. The results confirm the validity of the asymptotic properties even for short series and the asymptotic information matrix deduced from the theory.     

A pathwise inference method for the parameters of diffusion terms

We consider inference of the parameters of the diffusion term for continuous time stochastic processes with a power-type dependence of the diffusion coefficient from the underlying process such as Cox–Ingersoll–Ross, CKLS, and similar processes. We suggest some original pathwise estimates for this coefficient and for the power index based on an analysis of an auxiliary continuous time complex-valued process generated by the underlying real-valued process. These estimates do not rely on the distribution of the underlying process and on a particular choice of the drift. Some numerical experiments are used to illustrate the feasibility of the suggested method.     

Stochastic approximation methods in linear regression models (with consideration of errors in the regressors) 1

Usually, the dependence in stationary processes is described by a set of coefficients. In this paper, a measure of dependence is proposed which can be used instead of the autocorrelation function, and another measure for the dependence between two processes instead of cross-correlation function and coherence coefficients. In the end, an improvement of extrapolation of a process is investigated which is caused by the knowledge of another related process.     

An empirical analysis of the volatility of the Japanese stock price index: a non-parametric approach

This paper presents an empirical analysis of stochastic features of volatility in the Japanese stock price index, or TOPIX, using high-frequency data sampled every 5 min. The process of TOPIX is modeled by a stochastic differential equation with the time-homogeneous drift and diffusion coefficients. To avoid the risk of misspecification for the volatility function, which is defined by the squared diffusion coefficient, the local polynomial model is applied to the data, and then produced the estimates of the volatility function together with their confidence intervals. The result of the estimation suggests that the volatility function shows similar patterns for one period, but drastically changes for another.     

Robustness and monotonicity properties of generalized correlation coefficients

A new class of generalized correlation coefficients that contains the Pearson and Kendall statistics as special cases was defined by Chinchilli et al. (2005) and applied to the estimation of correlations coefficients within the context of 2×2 cross-over designs for clinical trials. In this paper, we determine the infinitesimal robustness and local stability properties of these generalized correlation coefficients by deriving their corresponding influence functions. For cases in which the population distribution is a bivariate normal or a mixture of bivariate normal distributions we obtain explicit formulas, and establish monotonicity and sign-reverse rule properties of the generalized correlation coefficients.     

Efficient and flexible model-based clustering of jumps in diffusion processes

Jump–diffusion processes involving diffusion processes with discontinuous movements, called jumps, are widely used to model time-series data that commonly exhibit discontinuity in their sample paths. The existing jump–diffusion models have been recently extended to multivariate time-series data. The models are, however, still limited by a single parametric jump-size distribution that is common across different subjects. Such strong parametric assumptions for the shape and structure of a jump-size distribution may be too restrictive and unrealistic for multiple subjects with different characteristics. This paper thus proposes an efficient Bayesian nonparametric method to flexibly model a jump-size distribution while borrowing information across subjects in a clustering procedure using a nested Dirichlet process. For efficient posterior computation, a partially collapsed Gibbs sampler is devised to fit the proposed model. The proposed methodology is illustrated through a simulation study and an application to daily stock price data for companies in the S&P 100 index from June 2007 to June 2017.     

The Diffusion of Innovations: A Methodological Reappraisal

Studies of diffusion have traditionally relied on specific distributions, primarily the logistic, to characterize and estimate those processes. We argue that such an approach gives rise to serious problems of comparability and interpretation and may result in large biases in the estimates of the parameters of interest. We propose instead the Gini's expected mean difference as a measure of diffusion speed. We discuss its advantages over the traditional approach, present a nonparametric estimation procedure, and tackle with it the problems of truncated processes and of intergroup comparisons. We also elaborate on the use of the hazard rate and suggest various extensions. The diffusion of computed tomography scanners is presented as an illustration.     

Estimation for change point of discretely observed ergodic diffusion processes

We treat the change point problem in ergodic diffusion processes from discrete observations. Tonaki et al. (2021a) proposed adaptive tests for detecting changes in the diffusion and drift parameters in ergodic diffusion process models. When any change in the diffusion or drift parameter is detected by this or any other method, the next question to consider is where the change point is located. Therefore, we propose the method to estimate the change point of the parameter for two cases: the case where there is a change in the diffusion parameter, and the case where there is no change in the diffusion parameter but a change in the drift parameter. Furthermore, we present rates of convergence and distributional results of the change point estimators. Some examples and simulation results are also given.