Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes with jumps

In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck processes with jumps based on continuous observations. We derive likelihood functions by using semimartingale theory. From this we get explicit formulas for estimators. The strong consistence and asymptotic normality of estimators are proved by using the method of stochastic integration.     

Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes

In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck (ROU) processes based on continuous observations. Both the cases with one-sided barrier and two-sided barriers are considered. We derive the explicit formulas for the estimators, and then prove their strong consistency and asymptotic normality. Moreover, the bias and mean square errors are represented in terms of the solutions to some PDEs with homogeneous Neumann boundary conditions. We also illustrate the asymptotic behavior of the estimators through a simulation study.     

Sequential estimation for nonhomogeneous Ornstein-Uhlenbeck processes

In this paper, we propose a stochastic process, which is a class of nonhomogeneous diffusion process from the perspective of the corresponding nonlinear stochastic differential equation. The parameter included in the drift term are estimated by sequential maximum likelihood methodology on the basis of continuous sampling of the process. The sequential estimators are proved to be closed, unbiased, strongly consistent, normally distributed, and optimal in the mean square sense.     

Asymptotic properties of the GMLE in the case 1 interval-censorship model with discrete inspection times

We consider the case 1 interval censorship model in which the survival time has an arbitrary distribution function F₀ and the inspection time has a discrete distribution function G. In such a model one is only able to observe the inspection time and whether the value of the survival time lies before or after the inspection time. We prove the strong consistency of the generalized maximum-likelihood estimate (GMLE) of the distribution function F₀ at the support points of G and its asymptotic normality and efficiency at what we call regular points. We also present a consistent estimate of the asymptotic variance at these points. The first result implies uniform strong consistency on [0, ∞) if F₀ is continuous and the support of G is dense in [0, ∞). For arbitrary F₀ and G, Peto (1973) and Tumbull (1976) conjectured that the convergence for the GMLE is at the usual parametric rate n½ Our asymptotic normality result supports their conjecture under our assumptions. But their conjecture was disproved by Groeneboom and Wellner (1992), who obtained the nonparametric rate ni under smoothness assumptions on the F₀ and G.     

An efficient method for the estimation of multivariate moving averge models

Durbin's (1959) efficient method for the estimation of univariate moving average models is generalized to the vector case. Strong consistency and asymptotic normality of the estimator is proved. A simulation experiment is performed to illustrate the behaviour of the method in finite samples.     

Sequential maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes

The paper studies the properties of a sequential maximum likelihood estimator of the drift parameter in a one dimensional reflected Ornstein-Uhlenbeck process. We observe the process until the observed Fisher information reaches a specified precision level. We derive the explicit formulas for the sequential estimator and its mean squared error. The estimator is shown to be unbiased and uniformly normally distributed. A simulation study is conducted to assess the performance of the estimator compared with the ordinary maximum likelihood estimator.     

Whittle estimation in multivariate CCC-GARCH processes

In this paper, we explore some probabilistic properties and statistical analysis of multivariate constant conditional correlation GARCH (CCC-GARCH for short) model. So, in the first part we give the conditions for the model stationarity and its finite moments up to some orders. In the second part, the Whittle estimator is proposed for the parameters CCC-GARCH model based on a transformation. This Whittle estimator is shown to be consistent when the data have finite 4th moment, and its asymptotic normality is established when the data have finite 8th moment. Finite sample properties of this Whittle estimator are further examined through Monte-Carlo experiments.     

Robust estimation and selection for single-index regression model

In this paper, we consider a single-index regression model for which we propose a robust estimation procedure for the model parameters and an efficient variable selection of relevant predictors. The proposed method is known as the penalized generalized signed-rank procedure. Asymptotic properties of the proposed estimator are established under mild regularity conditions. Extensive Monte Carlo simulation experiments are carried out to study the finite sample performance of the proposed approach. The simulation results demonstrate that the proposed method dominates many of the existing ones in terms of robustness of estimation and efficiency of variable selection. Finally, a real data example is given to illustrate the method.     

Local asymptotic properties for Cox-Ingersoll-Ross process with discrete observations

In this paper, we consider a one-dimensional Cox-Ingersoll-Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process together with the estimation for positive and negative polynomial moments of the CIR process. In this study, we require the same conditions of high frequency and infinite horizon as in the case of ergodic diffusions with globally Lipschitz coefficients studied earlier in the literature. However, in the non-ergodic cases, additional assumptions on the decreasing rate are required due to the fact that the square root diffusion coefficient of the CIR process is not regular enough.     

Pseudo maximum-likelihood estimation of the univariate GARCH (1,1) and asymptotic properties

One provides in this paper the pseudo-likelihood estimator (PMLE) and asymptotic theory for the GARCH (1,1) process. Strong consistency of the pseudo-maximum-likelihood estimator (MLE) is established by appealing to conditions given in Jeantheau (1998) concerning the existence of a stationary and ergodic solution to the multivariate GARCH (p, q) process. One proves the asymptotic normality of the PMLE by appealing to martingales' techniques.     

Asymptotic normality of generalized maximum spacing estimators for multivariate observations

In this paper, the maximum spacing method is considered for multivariate observations. Nearest neighbor balls are used as a multidimensional analogue to univariate spacings. A class of information-type measures is used to generalize the concept of maximum spacing estimators of model parameters. Asymptotic normality of these generalized maximum spacing estimators is proved when the assigned model class is correct, that is, the true density is a member of the model class.     

Simultaneous sparse model selection and coefficient estimation for heavy-tailed autoregressive processes

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator satisfies a strong consistency, O_P(N^?1/2) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator and smoothly clipped average deviation, are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations [Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391–422]. A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.     

Estimation of probability of extinction of a multitype branchihg process

Maximoa likelihood estimation of the probability of ultimata extinction of a possibly age dependentaultitype branching process is studied when independent random samples from off spring distributions are available, In multitype daIton-fatson branching process     

Parameter estimation for fractional Poisson processes

The paper proposes a formal estimation procedure for parameters of the fractional Poisson process (fPp). Such procedures are needed to make the fPp model usable in applied situations. The basic idea of fPp, motivated by experimental data with long memory is to make the standard Poisson model more flexible by permitting non-exponential, heavy-tailed distributions of interarrival times and different scaling properties. We establish the asymptotic normality of our estimators for the two parameters appearing in our fPp model. This fact permits construction of the corresponding confidence intervals. The properties of the estimators are then tested using simulated data.     

The Ornstein-Uhlenbeck Dirichlet process and other time-varying processes for Bayesian nonparametric inference

This paper introduces a new class of time-varying, measure-valued stochastic processes for Bayesian nonparametric inference. The class of priors is constructed by normalising a stochastic process derived from non-Gaussian Ornstein-Uhlenbeck processes and generalises the class of normalised random measures with independent increments from static problems. Some properties of the normalised measure are investigated. A particle filter and MCMC schemes are described for inference. The methods are applied to an example in the modelling of financial data.     

Uniform strong consistency of histogram density estimation for dependent process

Histogram density estimator is very intuitive and easy to compute and has been widely adopted. Especially in today's big data environment, people pay more attention to the computational cost and are more willing to choose estimators with less to compute. And so, many scholars have been interested in the various estimates based on the histogram technique. Under strong mixing process, this article studies the uniform strong consistency of histogram density estimator and the convergence rate. Our conditions on the mixing coefficient and the bin width are very mild.