We consider the testing problem for the sphericity hypothesis regarding the covariance matrix based on high-dimensional time series, under the assumption that the sample size n and the dimension p satisfy Limn,p→∞ p/n = c ∈ (0, ∞). Recently, several studies on test statistics for sphericity of independent and identically distributed p-dimensional random variables have been carried out under the assumption that both n and p diverge to infinity. A test statistic for sphericity has been proved to be well behaved even when p&gt
n. We investigate the test statistic under situations of high-dimensional time series. The asymptotic null distribution of the test statistic is shown to be standard normal distribution when the observations come from Gaussian stationary processes. In the simulation study, we illustrate the properties of the test statistic for several time series models. We apply the test to a problem of portfolio selection in our empirical study.

A new class of disparities from the point of view of prediction problem is proposed for minimum contrast estimation of spectral densities of stationary processes. We investigate asymptotic properties of the minimum contrast estimators based on the new disparities for stationary processes with both finite and infinite variance innovations. The relative efficiency and the robustness against randomly missing observations are shown in our numerical simulations. (C) 2017 Elsevier B.V. All rights reserved.

Interpolation is an important issue for a variety fields of statistics (e.g., missing data analysis). In time series analysis, the best interpolator for missing points problem has been investigated in several ways. In this paper, the asymptotics of a contrast function estimator defined by pseudo interpolation error for stationary process are investigated. We estimate parameters of the process by minimizing the pseudo interpolation error written in terms of a fitted parametric spectral density and the periodogram based on observed stretch. The estimator has the consistency and asymptotical normality. Although the criterion for the interpolation problem is known as the best in the sense of smallest mean square error for past and future extrapolation, it is shown that the estimator is asymptotically inefficient in general parameter estimation, which leads to an unexpected result.

Empirical likelihood approach is one of non-parametric statistical methods, which is applied to the hypothesis testing or construction of confidence regions for pivotal unknown quantities. This method has been applied to the case of independent identically distributed random variables and second order stationary processes. In recent years, we observe heavy-tailed data in many fields. To model such data suitably, we consider symmetric scalar and multivariate alpha-stable linear processes generated by infinite variance innovation sequence. We use a Whittle likelihood type estimating function in the empirical likelihood ratio function and derive the asymptotic distribution of the empirical likelihood ratio statistic for alpha-stable linear processes. With the empirical likelihood statistic approach, the theory of estimation and testing for second order stationary processes is nicely extended to heavy-tailed data analyses, not straightforward, and applicable to a lot of financial statistical analyses.

The precise large deviations for the partial sums and random sums of negatively associated random variables with consistently varying tails are derived. The results obtained are similar to those of independent and identically distributed random variables. (c) 2006 Elsevier B.V. All rights reserved.