On the rate of convergence of recursive kernel estimates of probability densities

Recursive estimates f_n^r(x)of the rth derivative f^r(x)(r=0,1)of the univariate probability density f(x) for strictly stationary processes {X_j,} are considered. The asymptotic variance-covariance of f_n^r(x)is established for stationary triangular arrays of random variables satisfying various asymptotic independence-uncorrelatedness conditions.     

Semiparametric Bayesian Inference for Time Series with Mixed Spectra

A Bayesian analysis is presented of a time series which is the sum of a stationary component with a smooth spectral density and a deterministic component consisting of a linear combination of a trend and periodic terms. The periodic terms may have known or unknown frequencies. The advantage of our approach is that different features of the data—such as the regression parameters, the spectral density, unknown frequencies and missing observations—are combined in a hierarchical Bayesian framework and estimated simultaneously. A Bayesian test to detect deterministic components in the data is also constructed. By using an asymptotic approximation to the likelihood, the computation is carried out efficiently using the Markov chain Monte Carlo method in O ( Mn ) operations, where n is the sample size and M is the number of iterations. We show empirically that our approach works well on real and simulated samples.     

A comparison of equally spaced designs with different correlation structures in one and more dimensions

Equally spaced designs are compared using the generalized variance as a measure of efficiency. Results for polynomial models are derived on the increased efficiency arising from increasing the number of design points when the regions are fixed and when the regions are expanded. The effects of dependence among the observations on these results are studied by considering a particular family of stationary correlated error structures.     

On the invertibility conditions for moving average processes

Conditions for the general Moving Average process, of order q, to be invertible or borderline non-invertible are deduced. These are termed the acceptability conditions. It turns out that they depend on the magnitude of the final moving average parameter, θ _q . If ‖θ _q ‖ >1, the process is not acceptable. Should ‖θ _q ‖ = 1, the conditions, for any particular q, follow simply - if use is made of the remainder theorem. When ‖θ_q‖< 1, an appeal is made to ROUCH* E'S theorem, to establish the conditions. Analogous stationarity results immediately follow for autoregressive processes.     

Monitoring Variance Change in Infinite Order Moving Average Processes and Nonstationary Autoregressive Processes

This article considers the sequential monitoring problem of variance change in stationary and non stationary time series. We suggest a CUSUM of squares procedure to detect variance change in infinite order moving average processes, and a residual CUSUM of squares procedure to detect variance change in non stationary autoregressive processes. Moreover, we introduce a bandwidth parameter to improve the monitoring power when change point does not occur at the early stage of monitoring. It is shown that both procedures have the same null distribution. The procedures are illustrated via a simulation study and an investigation of daily Mexico/US exchange rates.     

Posterior consistency for the spectral density of non-Gaussian stationary time series

Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation involving a nonparametric correction of a parametric likelihood has been proposed in the literature with a proof of posterior consistency for spectral density estimation in combination with the Bernstein–Dirichlet process prior for Gaussian time series. In this article, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood is also extended to non-Gaussian time series. Small sample properties of this approach are illustrated with several examples of non-Gaussian time series.     

Noisy Stochastic Games

This paper establishes existence of a stationary Markov perfect equilibrium in general stochastic games with noise—a component of the state that is nonatomically distributed and not directly affected by the previous period's state and actions. Noise may be simply a payoff‐irrelevant public randomization device, delivering known results on the existence of correlated equilibrium as a special case. More generally, noise can take the form of shocks that enter into players' stage payoffs and the transition probability on states. The existence result is applied to a model of industry dynamics and to a model of dynamic electoral competition.