TY - JOUR

T1 - Discriminant analysis for locally stationary processes

AU - Sakiyama, Kenji

AU - Taniguchi, Masanobu

PY - 2004/8

Y1 - 2004/8

N2 - In this paper, we discuss discriminant analysis for locally stationary processes, which constitute a class of non-stationary processes. Consider the case where a locally stationary process {Xt,T} belongs to one of two categories described by two hypotheses π1 and π2. Here T is the length of the observed stretch. These hypotheses specify that {Xt,T} has time-varying spectral densities f(u,λ) and g(u,λ) under π1 and π2, respectively. Although Gaussianity of {Xt,T} is not assumed, we use a classification criterion D(f:g), which is an approximation of the Gaussian likelihood ratio for {Xt,T} between π1 and π2. Then it is shown that D(f:g) is consistent, i.e., the misclassification probabilities based on D(f:g) converge to zero as T→∞. Next, in the case when g(u,λ) is contiguous to f(u,λ), we evaluate the misclassification probabilities, and discuss non-Gaussian robustness of D(f:g). Because the spectra depend on time, the features of non-Gaussian robustness are different from those for stationary processes. It is also interesting to investigate the behavior of D(f:g) with respect to infinitesimal perturbations of the spectra. Introducing an influence function of D(f:g), we illuminate its infinitesimal behavior. Some numerical studies are given.

AB - In this paper, we discuss discriminant analysis for locally stationary processes, which constitute a class of non-stationary processes. Consider the case where a locally stationary process {Xt,T} belongs to one of two categories described by two hypotheses π1 and π2. Here T is the length of the observed stretch. These hypotheses specify that {Xt,T} has time-varying spectral densities f(u,λ) and g(u,λ) under π1 and π2, respectively. Although Gaussianity of {Xt,T} is not assumed, we use a classification criterion D(f:g), which is an approximation of the Gaussian likelihood ratio for {Xt,T} between π1 and π2. Then it is shown that D(f:g) is consistent, i.e., the misclassification probabilities based on D(f:g) converge to zero as T→∞. Next, in the case when g(u,λ) is contiguous to f(u,λ), we evaluate the misclassification probabilities, and discuss non-Gaussian robustness of D(f:g). Because the spectra depend on time, the features of non-Gaussian robustness are different from those for stationary processes. It is also interesting to investigate the behavior of D(f:g) with respect to infinitesimal perturbations of the spectra. Introducing an influence function of D(f:g), we illuminate its infinitesimal behavior. Some numerical studies are given.

KW - Classification criterion

KW - Influence function

KW - Least favorable spectral density

KW - Locally stationary vector process

KW - Misclassification probability

KW - Non-Gaussian robust

KW - Time-varying spectral density matrix

UR - http://www.scopus.com/inward/record.url?scp=3042640076&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042640076&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2003.08.002

DO - 10.1016/j.jmva.2003.08.002

M3 - Article

AN - SCOPUS:3042640076

SN - 0047-259X

VL - 90

SP - 282

EP - 300

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

IS - 2

ER -