Discriminant analysis for locally stationary processes

In this paper, we discuss discriminant analysis for locally stationary processes, which constitute a class of non-stationary processes. Consider the case where a locally stationary process (X-t,X-T} belongs to one of two categories described by two hypotheses pi(1) and pi(2). Here T is the length of the observed stretch. These hypotheses specify that {X-t,X-T} has time-varying spectral densities f(u,lambda) and g(u,lambda) under pi(1) and pi(2), respectively. Although Gaussianity of {X-t,X-T} is not assumed, we use a classification criterion D(f : g), which is an approximation of the Gaussian likelihood ratio for {X-t,X-T) between pi(1) and pi(2). Then it is shown that D(f : g) is consistent, i.e., the misclassification probabilities based on D(f : g) converge to zero as T-->infinity. Next, in the case when g(u,lambda) is contiguous to f (u,lambda), we evaluate the misclassification probabilities, and discuss non-Gaussian robustness of D(f : g). Because the spectra depend on time, the features of non-Gaussian robustness are different from those for stationary processes. It is also interesting to investigate the behavior of D(f : g) with respect to infinitesimal perturbations of the spectra. Introducing an influence function of D(f : g), we illuminate its infinitesimal behavior. Some numerical studies are given. (C) 2003 Elsevier Inc. All rights reserved.