### Abstract

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 {X_{t,T}} belongs to one of two categories described by two hypotheses π_{1} and π_{2}. Here T is the length of the observed stretch. These hypotheses specify that {X_{t,T}} has time-varying spectral densities f(u,λ) and g(u,λ) under π_{1} and π_{2}, respectively. Although Gaussianity of {X_{t,T}} is not assumed, we use a classification criterion D(f:g), which is an approximation of the Gaussian likelihood ratio for {X_{t,T}} between π_{1} and π_{2}. 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.

Original language | English |
---|---|

Pages (from-to) | 282-300 |

Number of pages | 19 |

Journal | Journal of Multivariate Analysis |

Volume | 90 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2004 Aug |

### Fingerprint

### Keywords

- Classification criterion
- Influence function
- Least favorable spectral density
- Locally stationary vector process
- Misclassification probability
- Non-Gaussian robust
- Time-varying spectral density matrix

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Numerical Analysis
- Statistics and Probability

### Cite this

*Journal of Multivariate Analysis*,

*90*(2), 282-300. https://doi.org/10.1016/j.jmva.2003.08.002

**Discriminant analysis for locally stationary processes.** / Sakiyama, Kenji; Taniguchi, Masanobu.

Research output: Contribution to journal › Article

*Journal of Multivariate Analysis*, vol. 90, no. 2, pp. 282-300. https://doi.org/10.1016/j.jmva.2003.08.002

}

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

VL - 90

SP - 282

EP - 300

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 2

ER -