Parameter estimation in two-type continuous-state branching processes with immigration

We study the parameter estimation of two-type continuous-state branching processes with immigration based on low frequency observations at equidistant time points. The ergodicity of the processes is proved. The estimators are based on the minimization of a sum of squared deviation about conditional expectations. We also establish the strong consistency and central limit theorems of the conditional least squares estimators and the weighted conditional least squares estimators of the drift and diffusion coefficients based on low frequency observations.     

Estimating Some Characteristics of the Conditional Distribution in Nonparametric Functional Models

This paper deals with a scalar response conditioned by a functional random variable. The main goal is to estimate nonparametrically some characteristics of this conditional distribution. Kernel type estimators for the conditional cumulative distribution function and the successive derivatives of the conditional density are introduced. Asymptotic properties are stated for each of these estimates, and they are applied to the estimations of the conditional mode and conditional quantiles. Our asymptotic results highlightes the importance of the concentration properties on small balls of the probability measure of the underlying functional variable. So, a special section is devoted to show how our results behave in several situations when the functional variable is a continuous time process, with special attention to diffusion processes and Gaussian processes. Even if the main purpose of our paper is theoretical, an application to some chemiometrical data set coming from food industry is presented in a short final section. This example illustrates the easy implementation of our method as well as its good behaviour for finite sample sizes.     

Kernel estimation and interpolation for time series containing missing observations

Summary Kernel estimators of conditional expectations are adapted for use in the analysis of stationary time series containing missing observations. Estimators of conditional expectations at fixed points are shown to have an asymptotic distribution with a relatively simple variance-covariance structure. The kernel method is also used to interpolate missing observations, and is shown to converge in probability to the least squares predictor. The results are established under the strong mixing condition and moment conditions, and the methods are applied to a real data set.     

基于神经元网络的条件分位数估计的收敛速度

本文我们给出了基于神经元网络的随机过程的条件分位数的均方收敛速度．无论是在独立同分布情况下还是在平稳混合(α-混合β-混合)的情况下，我们都给出了相应的结果．结果与基于神经元网络的回归估计的收敛速度相同．采用的技术同Zhang(1998)一致．     

Weighted local polynomial estimations of a non-parametric function with censoring indicators missing at random and their applications

In this paper, we consider the weighted local polynomial calibration estimation and imputation estimation of a non-parametric function when the data are right censored and the censoring indicators are missing at random, and establish the asymptotic normality of these estimators. As their applications, we derive the weighted local linear calibration estimators and imputation estimations of the conditional distribution function, the conditional density function and the conditional quantile function, and investigate the asymptotic normality of these estimators. Finally, the simulation studies are conducted to illustrate the finite sample performance of the estimators.     

Aggregation of spectral density estimators

Given stationary time series data, we study the problem of finding the best linear combination of a set of lag window spectral density estimators with respect to the mean squared risk. We present an aggregation procedure and prove a sharp oracle inequality for its risk. We also provide simulations demonstrating the performance of our aggregation procedure, given Bartlett and other estimators of varying bandwidths as input. This extends work by P. Rigollet and A. Tsybakov on aggregation of density estimators.     

Inferences in semi-parametric dynamic mixed models for longitudinal count data

This paper considers a semi-parametric mixed model for longitudinal counts under the assumption that for conditional on a common random effect over time the repeated count responses of an individual follow a Poisson AR(1) (auto-regressive order 1) non-stationary correlation structure. A step-by-step estimation approach is developed which provides consistent estimators for the non-parametric function, regression parameters, variance of the random effects, and auto-correlation structure of the model. Proofs for the consistency properties of the estimators along with their convergence rates are derived. A simulation study is conducted to examine first the estimation effects on parameters when the non-parametric function is ignored, and then an overall estimation study is carried out in the presence of the non-parametric function by including its estimation as well.     

Gaugeability and conditional gaugeability

New Kato classes are introduced for general transient Borel right processes, for which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Green-tight measures in the classical Brownian motion case. However, the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.

     

Regularity and Minimality of Infinite Variance Processes

For stationary processes with infinite variance the notions of covariance and spectrum are not defined. We characterize regularity and minimality of such processes in the spirit of some classical results for second-order processes, namely values of the process forming conditional basis for their spans. Several open problems are discussed.     

EPSTA: The coincidence of time-stationary and customer-stationary distributions

In the first part of this paper we present an overview of relationships between time- and customer-stationary distributions of queueing processes. These have been proved by using the properties of random marked point processes, stochastic processes with embedded point processes, Palm distributions and an intensity conservation principle. In the second part a necessary and sufficient condition is established for the coincidence of the two types of stationary distributions, using conditional intensities. We also formulate the property of EPSTA that includes PASTA and ASTA as particular cases. A further result concerns the conditional EPSTA property. Applications to particular queueing systems are considered.     

Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density

In this paper, we study the non-parametric estimation of the invariant density of some ergodic hamiltonian systems, using kernel estimators. The main result is a central limit theorem for such estimators under partial observation (only the positions are observed). The main tools are mixing estimates and refined covariance inequalities, the main difficulty being the strong degeneracy of such processes. This is the first paper of a series of at least two, devoted to the estimation of the characteristics of such processes: invariant density, drift term, volatility.     

Nonparametric quantile regression with heavy-tailed and strongly dependent errors

We consider nonparametric estimation of the conditional qth quantile for stationary time series. We deal with stationary time series with strong time dependence and heavy tails under the setting of random design. We estimate the conditional qth quantile by local linear regression and investigate the asymptotic properties. It is shown that the asymptotic properties are affected by both the time dependence and the tail index of the errors. The results of a small simulation study are also given.     

Nonparametric variance function estimation with missing data

In this paper, a fixed design regression model where the errors follow a strictly stationary process is considered. In this model the conditional mean function and the conditional variance function are unknown curves. Correlated errors when observations are missing in the response variable are assumed. Four nonparametric estimators of the conditional variance function based on local polynomial fitting are proposed. Expressions of the asymptotic bias and variance of these estimators are obtained. A simulation study illustrates the behavior of the proposed estimators.     

含附加信息时条件分位数的估计及其渐近性质

本文利用经验似然方法给出了含附加信息时条件分位数的一类新估计,在一定的正则条件下证明了估计的渐近正态性且渐近方差小于或等于通常的条件分位数核估计的渐近方差．     

Quasi-likelihood estimation of average treatment effects based on model information

In this paper, the estimation of average treatment effects is considered when we have the model information of the conditional mean and conditional variance for the responses given the covariates. The quasi-likelihood method adapted to treatment effects data is developed to estimate the parameters in the conditional mean and conditional variance models. Based on the model information, we define three estimators by imputation, regression and inverse probability weighted methods. All the estimators are shown asymptotically normal. Our simulation results show that by using the model information, the substantial efficiency gains are obtained which are comparable with the existing estimators.     

Large deviation for stationary processes

A quasi-local variational characterization of the entropy for stationary processes is given. This is used to establish upper and lower large deviation estimates for arbitrary stationary processes. The upper and lower rate functions are shown to coincide for all quasi-local stationary processes. The contents of the paper is the following: 1. Introduction; 2. Notations; 3. Relative entropy of conditional expectations; 4. Relative entropy of a stationary process with respect to a covariant family of conditional expectations; 5. The role of locality and quasi-locality properties; 6. Large deviation upper estimate; 7. The Lower estimate; 8. The variational principle.     

On the consistency and finite-sample properties of nonparametric kernel time series regression,autoregression and density estimators

Summary Kernel estimators of conditional expectations and joint probability densities are studied in the context of a vector-valued stationary time series. Weak consistency is established under minimal moment conditions and under a hierarchy of weak dependence and bandwidth conditions. Prompted by these conditions, some finite-sample theory explores the effect of serial dependence on variability of estimators, and its implications for choice of bandwidth. This research was supported by the ESRC.     

On tail behavior of nonlinear autoregressive functional conditional heteroscedastic model with heavy-tailed innovations

We study the tail probability of the stationary distribution of nonparametric non- linear autoregressive functional conditional heteroscedastic (NARFCH) model with heavy- tailed innovations.Our result shows that the tail of the stationary marginal distribution of an NARFCH series is heavily dependent on its conditional variance.When the innovations are heavy-tailed,the tail of the stationary marginal distribution of the series will become heavier or thinner than that of its innovations.We give some specific formulas to show how the increment or decrement of tail heaviness depends on the assumption on the con- ditional variance function.Some examples are given.     

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??In this paper, semiparametric estimation of a regression function in the third order partially linear autoregressive model with first order autoregressive errors is mainly studied. We suppose that the regression function has a parametric framework, and use the conditional least squares method to obtain the parameter estimators. Then semiparametric estimators of the regression function can be given by combining with the nonparametric kernel function adjustment. Furthermore, under certain conditions, the consistency of the estimators is proved. Finally, simulation research is presented to evaluate the effectiveness of the proposed method.     

Estimating Functions for Nonlinear Time Series Models

This paper discusses the problem of estimation for two classes of nonlinear models, namely random coefficient autoregressive (RCA) and autoregressive conditional heteroskedasticity (ARCH) models. For the RCA model, first assuming that the nuisance parameters are known we construct an estimator for parameters of interest based on Godambe's asymptotically optimal estimating function. Then, using the conditional least squares (CLS) estimator given by Tjøstheim (1986, Stochastic Process. Appl., 21, 251–273) and classical moment estimators for the nuisance parameters, we propose an estimated version of this estimator. These results are extended to the case of vector parameter. Next, we turn to discuss the problem of estimating the ARCH model with unknown parameter vector. We construct an estimator for parameters of interest based on Godambe's optimal estimator allowing that a part of the estimator depends on unknown parameters. Then, substituting the CLS estimators for the unknown parameters, the estimated version is proposed. Comparisons between the CLS and estimated optimal estimator of the RCA model and between the CLS and estimated version of the ARCH model are given via simulation studies.