Joint Asymptotic Normality Research Articles

Multiple local whittle estimation in stationary systems

Moving from univariate to bivariate jointly dependent long-memory time series introduces a phase parameter $(\gamma)$, at the frequency of principal interest, zero; for short-memory series $\gamma=0$ automatically. The latter case has also been stressed under long memory, along with the ``fractional differencing'' case $\gamma=(\delta_2-\delta_1)\pi /2$, where $\delta_1$, $\delta_2$ are the memory parameters of the two series. We develop time domain conditions under which these are and are not relevant, and relate the consequent properties of cross-autocovariances to ones of the (possibly bilateral) moving average representation which, with martingale difference innovations of arbitrary dimension, is used in asymptotic theory for local Whittle parameter estimates depending on a single smoothing number. Incorporating also a regression parameter $(\beta)$ which, when nonzero, indicates cointegration, the consistency proof of these implicitly defined estimates is nonstandard due to the $\beta$ estimate converging faster than the others. We also establish joint asymptotic normality of the estimates, and indicate how this outcome can apply in statistical inference on several questions of interest. Issues of implemention are discussed, along with implications of knowing $\beta$ and of correct or incorrect specification of $\gamma$, and possible extensions to higher-dimensional systems and nonstationary series.

Multivariate Signal Parameter Estimation Under Dependent Noise From 1-Bit Dithered Quantized Data

Motivated by applications in sensor networks and communications, we consider multivariate signal parameter estimation when only dithered 1-bit quantized samples are available. The observation noise is taken to be a stationary, strongly mixing process, which covers a wide range of processes including autoregressive moving average (ARMA) models. The noise is allowed to be Gaussian or to have a heavy-tail (with possibly infinite variance). An estimate of the signal parameters is proposed and is shown to be weakly consistent. Joint asymptotic normality of the parameters estimate is also established and the asymptotic mean and covariance matrices are identified.

Asymptotic properties of local polynomial regression with missing data and correlated errors

The main objective of this work is the nonparametric estimation of the regression function with correlated errors when observations are missing in the response variable. Two nonparametric estimators of the regression function are proposed. The asymptotic properties of these estimators are studied; expresions for the bias and the variance are obtained and the joint asymptotic normality is established. A simulation study is also included.

Semiparametric Regression Estimation in Copula Models

We consider some methods of semiparametric regression estimation in multivariate models when the common distribution function is represented using a copula and the marginals satisfy a generalized regression model using a transfer functional. Sufficient conditions for consistency and joint asymptotic normality of the finite-dimensional parameters are obtained.

Testing for small bias of tail index estimators

We determine the joint asymptotic normality of kernel and weighted least-squares estimators of the upper tail index of a regularly varying distribution when each estimator is a bivariate function of two parameters: the tuning parameter is motivated by possible underlying second-order behavior in regular variation, while no such behavior is assumed, and the fraction parameter determines that upper portion of the sample on which the estimator is based. Under the hypothesis that the scaled asymptotic biases of the estimators vanish uniformly in the parameter points considered, these results imply joint asymptotic normality for deviations of ratios of the estimators from 1, which in turn yield asymptotic chi-square tests for checking the small-bias hypothesis, equivalent to the constructibility of asymptotic confidence intervals. The test procedure suggests adaptive choices of the tuning and fraction parameters: data-driven (t)estimators.

Fixed design regression quantiles for time series

This paper studies nonparametric estimation of regression quantiles under the fixed design model. We suppose that the error random variables are coming from a strictly stationary stochastic process satisfying the strong mixing condition. The joint asymptotic normality for the estimators of several quantiles is given. The same property is established for the regression quantile estimator at different fixed design points.

Local polynomial fitting under association

We consider the estimation of multivariate regression functions r( x 1,…, x d ) and their partial derivatives up to a total order p⩾1 using high-order local polynomial fitting. The processes { Y i , X i } are assumed to be (jointly) associated. Joint asymptotic normality is established for the estimates of the regression function r and all its partial derivatives up to the total order p. Expressions for the bias and variance/covariance matrix (of the asymptotic distribution) are given.

Local polynomial regression smoothers with AR-error structure

Consider the fixed regression model with random observation error that follows an AR(1) correlation structure. In this paper, we study the nonparametric estimation of the regression function and its derivatives using a modified version of estimators obtained by weighted local polynomial fitting. The asymptotic properties of the proposed estimators are studied: expressions for the bias and the variance/covariance matrix of the estimators are obtained and the joint asymptotic normality is established. In a simulation study, a better behavior of the Mean Integrated Squared Error of the proposed regression estimator with respect to that of the classical local polynomial estimator is observed when the correlation of the observations is large.

LOCAL POLYNOMIAL REGRESSION ESTIMATION WITH CORRELATED ERRORS

In this paper, we study the nonparametric estimation of the regression function and its derivatives using weighted local polynomial fitting. Consider the fixed regression model and suppose that the random observation error is coming from a strictly stationary stochastic process. Expressions for the bias and the variance array of the estimators of the regression function and its derivatives are obtained and joint asymptotic normality is established. The influence of the dependence of the data is observed in the expression of the variance. We also propose a variable bandwidth selection procedure. A simulation study and an analysis with real economic data illustrate the proposed selection method.

Multivariate regression estimation of continuous-time processes from sampled data: local polynomial fitting approach

Let (Y,X)={Y(t),X(t),-/spl infin/<t</spl infin/} be real-valued continuous-time jointly stationary processes and let (t/sub j/) be a renewal point processes on (0,/spl infin/), with a finite mean rate, independent of (Y,X). We consider the estimation of regression function r(x/sub 0/, x/sub 1/,...,x/sub m-1/; /spl tau//sub 1/,...,/spl tau//sub m/) of /spl psi/(Y(/spl tau//sub m/)) given (X(0)=x/sub 0/, X(/spl tau//sub 1/)=x/sub 1/,...,X(/spl tau//sub m-1/)=/sub x-1/) for arbitrary lags 0</spl tau//sub 1/<...< /spl tau//sub m/ on the basis of the discrete-time observations {Y(t/sub j/),X(t/sub j/),t/sub j/)/sub j=1//sup n/. We estimate the regression function and all its partial derivatives up to a total order p/spl ges/1 using high-order local polynomial fitting. We establish the weak consistency of such estimates along with rates of convergence. We also establish the joint asymptotic normality of the estimates for the regression function and all its partial derivatives up to a total order p/spl ges/1 and provide explicit expressions for the bias and covariance matrix (of the asymptotically normal distribution).

Estimation and test of linearity for a class of additive nonlinear models

This paper deals with the estimation and the test for linearity of models belonging to a class of additive nonlinear ones. We prove the joint asymptotic normality for a kernel estimator and provide a test for linearity of each function defining the model.

Multivariate regression estimation: Local polynomial fitting for time series

Abstract We consider the estimation of the multivariate regression function m(x1, …, xd) = E[ψ(Yd)|X1 = x1, …, Xd = xd], and its partial derivatives, for stationary random processes Yi, Xi using local higher-order polynomial fitting. Particular cases of ψ yield estimation of the conditional mean, conditional moments and conditional distributions. Joint asymptotic normality is established for estimates of the regression function and its partial derivatives for strongly mixing and ϱ-mixing processes. Expressions for the bias and variance/covariance matrix (of the asymptotically normal distribution) for these estimators are given.

Local Polynomial Estimation of Regression Functions for Mixing Processes

Local polynomial fitting has many exciting statistical properties which where established under i.i.d. setting. However, the need for non‐linea r time series modeling, constructing predictive intervals, understanding divergence of non‐linear time series requires the development of the theory of local polynomial fitting for dependent data. In this paper, we study the problem of estimating conditional mean functions and their derivatives via a local polynomial fit. The functions include conditional moments, conditional distribution as well as conditional density functions. Joint asymptotic normality for derivative estimation is established for both strongly mixing and ρ‐mixing processes.

A CENTRAL LIMIT THEOREM FOR m(n) AUTOCOVARIANCES

Many of the fundamental results in time series analysis depend on the joint asymptotic normality of a fixed number m of the sample autocovariances. However, in practice, the m is often chosen after the number of observations, n, is known, with m then treated as fixed. In this paper a Berry‐Esseen type result is proved for m(n) autocovariances for m growing at a certain rate.

Multivariate regression estimation local polynomial fitting for time series

We consider the estimation of the multivariate regression function m( x 1, …, x d ) = E[ ψ( Y d )| X 1 = x 1, …, X d = x d ], and its partial derivatives, for stationary random processes Y i , X i using local higher-order polynomial fitting. Particular cases of ψ yield estimation of the conditional mean, conditional moments and conditional distributions. Joint asymptotic normality is established for estimates of the regression function and its partial derivatives for strongly mixing and ϱ-mixing processes. Expressions for the bias and variance/covariance matrix (of the asymptotically normal distribution) for these estimators are given.

Multivariate Locally Weighted Polynomial Fitting and Partial Derivative Estimation

Nonparametric regression estimator based on locally weighted least squares fitting has been studied by Fan and Ruppert and Wand. The latter paper also studies, in the univariate case, nonparametric derivative estimators given by a locally weighted polynomial fitting. Compared with traditional kernel estimators, these estimators are often of simpler form and possess some better properties. In this paper, we develop current work on locally weighted regression and generalize locally weighted polynomial fitting to the estimation of partial derivatives in a multivariate regression context. Specifically, for both the regression and partial derivative estimators we prove joint asymptotic normality and derive explicit asymptotic expansions for their conditional bias and conditional convariance matrix (given observations of predictor variables) in each of the two important cases of local linear fit and local quadratic fit.

Estimation of moments under koziol-green model for random censorship

The Koziol-Green model for random censorship is considered (the model of two competing risks with proportional hazards) and an estimate of the rth moment under this model is proposed. Some asymptotic properties of the suggested estimate are studied. The estimate is shown to be consistent and asymptotically normally distributed. Moreover, the joint asymptotic normality of estimated moments is proved. The results are applied to the estimated variance.

Joint asymptotic normality of kernel estimates under dependence conditions, with application to hazard rate

et Xtt =…, ‐1, 0, 1…be R d -valued (d≧1) random variables forming a strictly stationary sequence satisfying a weak dependence condition, and let f be the probability density function of the X's. On the basis of n successive random variables, recursive and nonrecursive kernel estimates of f are considered and the joint asymptotic normality of the proposed estimates, evaluated at a finite number of distinct continuity points of f, is established. By means of an example, it is demonstrated that the recursive estimate may be superior to the nonrecursive one, in the sense of reducing the variance of the asymptotic distribution. The results are applied to the problem of estimating the hazard rate and in establishing the joint asymptotic normality of the proposed estimates. The above mentioned variance reduction possibility carries over to the hazard rate estimation problem.

Density estimation for point processes

A general nonparametric density estimation problem is considered in which the data is generated by a spatial point process. Several practical problems are special cases of it, including those of estimating the common probability density of a sequence of random vectors and estimating the product density of a stationary multivariate point process. Kernel and k-nearest neighbor estimators are defined and in each case the joint asymptotic normality and consistency of the estimates of the density at a given finite number of points is derived.

Estimation for a class of simple queueing networks

Maximum likelihood estimators of the parameters of an open Jackson network are derived, and their joint asymptotic normality is established. The problem of estimation for tandem queues is discussed as a special case of the Jackson system. These results are valid when the system is not necessarily in equilibrium.