3,837 research outputs found
On a class of minimum contrast estimators for Gegenbauer random fields
The article introduces spatial long-range dependent models based on the
fractional difference operators associated with the Gegenbauer polynomials. The
results on consistency and asymptotic normality of a class of minimum contrast
estimators of long-range dependence parameters of the models are obtained. A
methodology to verify assumptions for consistency and asymptotic normality of
minimum contrast estimators is developed. Numerical results are presented to
confirm the theoretical findings.Comment: 23 pages, 8 figure
Developments in the Analysis of Spatial Data
Disregarding spatial dependence can invalidate methods for analyzingcross-sectional and panel data. We discuss ongoing work on developingmethods that allow for, test for, or estimate, spatial dependence. Muchof the stress is on nonparametric and semiparametric methods.
Bounded Influence Approaches to Constrained Mixed Vector Autoregressive Models
The proliferation of many clinical studies obtaining multiple biophysical signals from several individuals repeatedly in time is increasingly recognized, a recognition generating growth in statistical models that analyze cross-sectional time series data. In general, these statistical models try to answer two questions: (i) intra-individual dynamics of the response and its relation to some covariates; and, (ii) how this dynamics can be aggregated consistently in a group. In response to the first question, we propose a covariate-adjusted constrained Vector Autoregressive model, a technique similar to the STARMAX model (Stoffer, JASA 81, 762-772), to describe serial dependence of observations. In this way, the number of parameters to be estimated is kept minimal while offering flexibility for the model to explore higher order dependence. In response to (ii), we use mixed effects analysis that accommodates modelling of heterogeneity among cross-sections arising from covariate effects that vary from one cross-section to another. Although estimation of the model can proceed using standard maximum likelihood techniques, we believed it is advantageous to use bounded influence procedures in the modelling (such as choosing constraints) and parameter estimation so that the effects of outliers can be controlled. In particular, we use M-estimation with a redescending bounding function because its influence function is always bounded. Furthermore, assuming consistency, this influence function is useful to obtain the limiting distribution of the estimates. However, this distribution may not necessarily yield accurate inference in the presence of contamination as the actual asymptotic distribution might have wider tails. This led us to investigate bootstrap approximation techniques. A sampling scheme based on IID innovations is modified to accommodate the cross-sectional structure of the data. Then the M-estimation is applied to each bootstrap sample naively to obtain the asymptotic distribution of the estimates.We apply these strategies to the extracted BOLD activation from several regions of the brain from a group of individuals to describe joint dynamic behavior between these locations. We used simulated data with both innovation and additive outliers to test whether the estimation procedure is accurate despite contamination
Parameter estimation in nonlinear AR–GARCH models
This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. Strong consistency and asymptotic normality of the global Gaussian quasi maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.Nonlinear Autoregression, Generalized Autoregressive Conditional Heteroskedasticity, Nonlinear Time Series Models, Quasi-Maximum Likelihood Estimation, Strong Consistency, Asymptotic Normality
Parameter Estimation in Nonlinear AR-GARCH Models
This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a functional coefficient autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. Strong consistency and asymptotic normality of the global Gaussian quasi maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.AR-GARCH, asymptotic normality, consistency, nonlinear time series, quasi maximum likelihood estimation
Efficient prediction for linear and nonlinear autoregressive models
Conditional expectations given past observations in stationary time series
are usually estimated directly by kernel estimators, or by plugging in kernel
estimators for transition densities. We show that, for linear and nonlinear
autoregressive models driven by independent innovations, appropriate smoothed
and weighted von Mises statistics of residuals estimate conditional
expectations at better parametric rates and are asymptotically efficient. The
proof is based on a uniform stochastic expansion for smoothed and weighted von
Mises processes of residuals. We consider, in particular, estimation of
conditional distribution functions and of conditional quantile functions.Comment: Published at http://dx.doi.org/10.1214/009053606000000812 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Asymptotic spectral theory for nonlinear time series
We consider asymptotic problems in spectral analysis of stationary causal
processes. Limiting distributions of periodograms and smoothed periodogram
spectral density estimates are obtained and applications to the spectral domain
bootstrap are given. Instead of the commonly used strong mixing conditions, in
our asymptotic spectral theory we impose conditions only involving
(conditional) moments, which are easily verifiable for a variety of nonlinear
time series.Comment: Published in at http://dx.doi.org/10.1214/009053606000001479 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Estimating invariant laws of linear processes by U-statistics
Suppose we observe an invertible linear process with independent mean-zero
innovations and with coefficients depending on a finite-dimensional parameter,
and we want to estimate the expectation of some function under the stationary
distribution of the process. The usual estimator would be the empirical
estimator. It can be improved using the fact that the innovations are centered.
We construct an even better estimator using the representation of the
observations as infinite-order moving averages of the innovations. Then the
expectation of the function under the stationary distribution can be written as
the expectation under the distribution of an infinite series in terms of the
innovations, and it can be estimated by a U-statistic of increasing order
(also called an ``infinite-order U-statistic'') in terms of the estimated
innovations. The estimator can be further improved using the fact that the
innovations are centered. This improved estimator is optimal if the
coefficients of the linear process are estimated optimally
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