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” f̃(λ) and g̃(λ) under π1 and π2, respectively. In this paper, we suggest a classification statistic In(f̃,g̃) based on the normalized power transfer functions. We show that In(f̃,g̃) is a consistent classification criterion in the sense that the misclassification probabilities converge to zero as the sample size tends to infinity. When g̃(λ) is contiguous to f̃(λ), we also evaluate the goodness of fit of In(f̃,g̃) 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.