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 {Xt,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 {Xt,T} has time-varying spectral densities f(u,[lambda]) and g(u,[lambda]) under [pi]1 and [pi]2, respectively. Although Gaussianity of {Xt,T} is not assumed, we use a classification criterion D( f:g), which is an approximation of the Gaussian likelihood ratio for {Xt,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.