A bispectral-based test for Gaussianity and linearity of a time series

Vita.The prim ary focus of this dissertation is to improve a current m ethod for testing if a tim e series is Gaussian or linear and to compare the improvement to the existing methods. Traditionally, the analysis of a time series has relied upon two assumptions - th a t the tim e series is stationary and linear. Under these assumptions, methods are abundant and their theory is well developed. In the past twenty years, methods for nonlinear tim e series analysis have emerged making it possible to analyze a time series as a nonlinear process. W ith these capabilities, it is now desirable to have reliable m ethods for testing th e linearity of a time series. The m ost popular test for Gaussianity and linearity of a time series (Hinich, 1982) is nonparam etric and uses the asymptotic normality of the interquartile range of a set of frequency domain quantities which under linearity are asymptotically independent identically distributed noncentral chi-square random variables with two degrees of freedom. We show th at asymptotic normality is a poor approximation, even for very large sample sizes, and give saddlepoint approximations to the null distribution of the test statistic. We motivate the saddlepoint approximation by considering the difference of two order statistics from a central chi-square sample, in which case we derive an exact probability density function and saddlepoint approximation. In the noncentral case we give an approximate saddlepoint approximation based on a m ethod due to Wang (1992). In both cases we show that the saddlepoint approximation is much better than the norm al approximation, especially for realistic sample sizes. Furthermore, since saddlepoint approximations are computationally intensive, we present a Mathematica program that calculates saddlepoint approximations of the density and distribution function of the difference of two order statistics. The parent distribution can b e any of the distributions included in Mathematicals ContinuousDistributions package. A secondary purpose of this dissertation is the introduction of a nonlinear time series analysis package in S. Although software is available for the analysis of stationary, linear tim e series, e.g. Newton (1988) and Brockwell and Davis (1991), the availability of interactive nonlinear time series analysis packages is limited