Nonparametric approach for discriminant analysis in time series

In this paper, we shall consider the case where a stationary process {X(t)} belongs to one of two categories described by two hypotheses II1 and II2. These hypotheses specify that {X(t)} has spectral densities f1(λ) and f2(λ) under II1 and II2, respectively. It is known that the log-likelihood ratio based on Xn = [X(l),.,X(n)]' gives the optimal classification. Here we propose a new discriminant statistic Bα = eα(n, f2) – eα(n, f1), where eα(f1, f2) is the α-entropy of (A) with respect to f2(λ) and2(λ) is a nonparametric spectral estimator based on Xn. Then it is shown that the misclassification probabilities of Bα are asymptotically equivalent to those of I(f1, f2), an approximation of Gaussian log-likelihood ratio which is useful for discriminant analysis in time series. Furthermore Bα is shown to have peak robustness with respect to the spectral density. However I(f1, f2) does not have such property. Finally, simulation studies are given to confirm the theoretical results.