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**Consistent cross-validatory model-selection for dependent data: hv-block cross-validation.**
*(English)*
Zbl 1011.62118

Summary: This paper considers the impact of J. Shao’s [J. Am. Stat. Assoc. 88, No. 422, 486-494 (1993; Zbl 0773.62051)] recent results regarding the asymptotic inconsistency of model selection via leave-one-out cross-validation on \(h\)-block cross-validation, a cross-validatory method for dependent data proposed by P. Burman, E. Chow and D. Nolan [Biometrika 81, No. 2, 351-358 (1994)]. It is show that \(h\)-block cross-validation is inconsistent in the sense of Shao and therefore is not asymptotically optimal. A modification of the \(h\)-block method, dubbed ‘hv-block’ cross-validation, is proposed which is asymptotically optimal. The proposed approach is consistent for general stationary observations in the sense that the probability of selecting the model with the best predictive ability converges to 1 as the total number of observations approaches infinity.

This extends existing results and yields a new approach which contains leave-one-out cross-validation, leave-\(n_v\)-out cross-validation, and \(h\)-block cross-validation as special cases. Applications are considered.

This extends existing results and yields a new approach which contains leave-one-out cross-validation, leave-\(n_v\)-out cross-validation, and \(h\)-block cross-validation as special cases. Applications are considered.

### Citations:

Zbl 0773.62051
Full Text:
DOI

### References:

[1] | Akaike, H.: A new look at the statistical model identification. IEEE transactions in automatic control 19, No. 6, 716-723 (1974) · Zbl 0314.62039 |

[2] | Allen, D. M.: The relationship between variable selection and data augmentation and a method for prediction. Technometrics 16, 125-127 (1974) · Zbl 0286.62044 |

[3] | Breiman, L.; Friedman, J.; Olshen, R.; Stone, C.: Classification and regression trees. (1984) · Zbl 0541.62042 |

[4] | Burman, P.: A comparative study of ordinary cross-validation, v-fold cross-validation, and the repeated learning-testing methods. Biometrika 76, 503-514 (1989) · Zbl 0677.62065 |

[5] | Burman, P.: Estimation of optimal transformations using v-fold cross-validation and repeated learning-testing methods. Sankhyā 52, 314-345 (1990) · Zbl 0745.62073 |

[6] | Burman, P.; Chow, E.; Nolan, D.: A cross-validatory method for dependent data. Biometrika 84, 351-358 (1994) · Zbl 0825.62669 |

[7] | Burman, P.; Nolan, D.: Data dependent estimation of prediction functions. Journal of time series analysis 13, 189-207 (1992) · Zbl 0754.62018 |

[8] | Cootner, P.: The random character of stock market prices. (1964) |

[9] | Fama, E. F.: The behavior of stock market prices. Journal of business 38, 34-105 (1965) · Zbl 0129.11903 |

[10] | Geisser, S.: A predictive sample reuse method with application. Journal of the American statistical association 70, 320-328 (1975) · Zbl 0321.62077 |

[11] | Györfi, L.; Härdle, W.; Sarda, P.; Vieu, P.: Nonparametric curve estimation from time series. (1989) · Zbl 0697.62038 |

[12] | Meese, R.; Rogoff, K. D.: Empirical exchange rate models of the seventies: do they fit out of sample?. Journal of international economics 14, 3-24 (1983) |

[13] | Racine, J.: Feasible cross-validatory model selection for general stationary processes. Journal of applied econometrics 12, 169-179 (1997) |

[14] | Shao, J.: Linear model selection by cross-validation. Journal of the American statistical association 88, No. 422, 486-495 (1993) · Zbl 0773.62051 |

[15] | Shao, J.: Bootstrap model selection. Journal of the American statistical association 91, No. 434, 655-665 (1996) · Zbl 0869.62030 |

[16] | Stone, M.: Cross-validatory choice and assessment of statistical predictions (with discussion). Journal of the royal statistical society 36, 111-147 (1974) · Zbl 0308.62063 |

[17] | Stone, M.: Cross-validation: a review. Mathematics of operations statistics 9, 127-139 (1977) · Zbl 0392.62052 |

[18] | Whaba, G.; Wold, S.: A completely automatic French curve: Fitting spline functions by cross-validation. Communications in statistics 4, 1-17 (1975) · Zbl 0305.62043 |

[19] | Zhang, P.: Model selection via multifold cross-validation. Annals of statistics 21, 299-313 (1993) · Zbl 0770.62053 |

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