HIGHER ORDER ASYMPTOTIC THEORY FOR DISCRIMINANT ANALYSIS IN GAUSSIAN STATIONARY PROCESSES

抄録

This paper discusses the problem of classifying an observed stretch X=(x₁, …, x_r) into Π₁ or Π₂, where Π_i is a Gaussian stationary process with zero mean and spectral density f_θi(λ). We propose a new discriminant statistic based on some estimator θ=θ(X) of a spectral parameter. The statistic D[θ, W] is motivated by a spectral measure with divergence function W. Most of the work presented is devoted to higher order asymptotic theory when θ₂ is contiguous to θ₁, in order to study the asymptotic difference between different D[θ, W]. In particular, it is shown that for any choice of W, D[θ, W] has the same second order averaged risk as the optimal likelihood ratio (LR) if θ belongs to an appropriate class of asymptotically efficient estimators, and the third order term of the averaged risk is minimized by the (bias-adjusted) maximum likelihood estimator (MLE). We also examine the case of the rule based on the MLE without bias adjustment.