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Nonparametric approach for discriminant analysis in time series. (English) Zbl 0873.62036

Summary: We consider the case where a stationary process \(\{X(t)\}\) belongs to one of two categories described by two hypotheses \(\text{II}_1\), and \(\text{II}_2\). These hypotheses specify that \(\{X(t)\}\) has spectral densities \(f_1(\lambda)\) and \(f_2(\lambda)\) under \(\text{II}_1\) and \(\text{II}_2\), respectively. It is known that the log-likelihood ratio based on \(X_n=[X(1),\dots,X(n)]\) gives the optimal classification. Here we propose a new discriminant statistic \(B_\alpha=e_\alpha(\widehat{f}_n,f_2)-e_\alpha(\widehat{f}_n,f_1)\), where \(e_\alpha(f_1,f_2)\) is the \(\alpha\)-entropy of \(f_1(\lambda)\) with respect to \(f_2(\lambda)\) and \(\widehat{f}_n(\lambda)\) is a nonparametric spectral estimator based on \(X_n\). Then it is shown that the misclassification probabilities of \(B_\alpha\) are asymptotically equivalent to those of \(I(f_1,f_2)\), an approximation of Gaussian log-likelihood ratio which is useful for discriminant analysis in time series. Furthermore \(B_\alpha\) is shown to have peak robustness with respect to the spectral density. However \(I(f_1,f_2)\) does not have such property. Finally, simulation studies are given to confirm the theoretical results.

MSC:

62G05 Nonparametric estimation
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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