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,T} belongs to one of two categories described by two hypotheses π₁ and π₂. Here T is the length of the observed stretch. These hypotheses specify that {X_t,T} has time-varying spectral densities f(u,λ) and g(u,λ) under π₁ and π₂, respectively. Although Gaussianity of {X_t,T} is not assumed, we use a classification criterion D(f:g), which is an approximation of the Gaussian likelihood ratio for {X_t,T} between π₁ and π₂. 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→∞. Next, in the case when g(u,λ) is contiguous to f(u,λ), 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.