Discriminant analysis for dynamics of stable processes

We consider the problem of classifying an α-stable linear process into two categories described by two hypotheses π₁ and π₂. These hypotheses are specified by the "normalized power transfer functions" over(f, ̃) (λ) and over(g, ̃) (λ) under π₁ and π₂, respectively. In this paper, we suggest a classification statistic I_n (over(f, ̃), over(g, ̃)) based on the normalized power transfer functions. We show that I_n (over(f, ̃), over(g, ̃)) is a consistent classification criterion in the sense that the misclassification probabilities converge to zero as the sample size tends to infinity. When over(g, ̃) (λ) is contiguous to over(f, ̃) (λ), we also evaluate the goodness of fit of I_n (over(f, ̃), over(g, ̃)) in terms of the misclassification probabilities. Our results have potential applications in various fields, e.g., credit rating in finance, and so on. Several numerical examples will be given.