Prediction Problems for Square-Transformed Stationary Processes

This paper discusses the prediction problems for square-transformed process, Y _t = X _t ², where X _t is a stationary process with spectral density g(). The square-transformation is important in prediction of the volatility of ARCH models. First, we evaluate the mean square prediction error for square-transformed process when the predictor is constructed from the true spectral density g(). However, it is often that the true structure g() is not completely specified. Hence, we consider the problem of misspecified prediction when a conjectured spectral density f (), , is fitted to g(). Then, constructing the best linear predictor based on f (), we can evaluate the prediction error for square-transformed process. Also, we consider a bias adjusted prediction problem for the above two cases. Furthermore, we may suppose that X _t is a non-Gaussian process. Then, we evaluate the mean square prediction errors when the best linear predictor is constructed by the true spectral density g() and the conjectured spectral density f (), respectively. Since is usually unknown we estimate it by a quasi-MLE . The second-order asymptotic approximations of the mean square errors of the predictors based on g() and f () are given. Finally, we provide some numerical examples, which show some unexpected features.     

Nonparametric Density Estimation for a Long-Range Dependent Linear Process

We estimate the marginal density function of a long-range dependent linear process by the kernel estimator. We assume the innovations are i.i.d. Then it is known that the term of the sample mean is dominant in the MISE of the kernel density estimator when the dependence is beyond some level which depends on the bandwidth and that the MISE has asymptotically the same form as for i.i.d. observations when the dependence is below the level. We call the latter the case where the dependence is not very strong and focus on it in this paper. We show that the asymptotic distribution of the kernel density estimator is the same as for i.i.d. observations and the effect of long-range dependence does not appear. In addition we describe some results for weakly dependent linear processes.     

Asymptotic expansions of the distributions of some test statistics for Gaussian ARMA processes

Let {X_t} be a Gaussian ARMA process with spectral density f_θ(λ), where θ is an unknown parameter. The problem considered is that of testing a simple hypothesis H:θ = θ₀ against the alternative A:θ ≠ θ₀. For this problem we propose a class of tests , which contains the likelihood ratio (LR), Wald (W), modified Wald (MW) and Rao (R) tests as special cases. Then we derive the χ² type asymptotic expansion of the distribution of T up to order n⁻¹, where n is the sample size. Also we derive the χ² type asymptotic expansion of the distribution of T under the sequence of alternatives A_n: θ = θ₀ + /√n, ε > 0. Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions.