## Nonparametric Estimation of a Periodic Sequence (2010)

### BibTeX

@MISC{Sun10nonparametricestimation,

author = {Ying Sun and Jeffrey D. Hart and Marc G. Genton},

title = {Nonparametric Estimation of a Periodic Sequence},

year = {2010}

}

### OpenURL

### Abstract

A nonparametric method is proposed for estimating the period and values of a periodic sequence when the data are evenly spaced in time. The period is estimated by a “leave-out-onecycle” version of cross-validation (CV) and complements the periodogram, a typical frequency domain period estimation tool. The CV method is computationally simple and implicitly penalizes multiples of the smallest period, leading to a “virtually ” consistent estimator, which is investigated both theoretically and by simulation. Estimating a period is tantamount to selecting a model, and it is shown that the CV estimator works much better in the period estimation context than it does in other model selection problems. As applications, the CV method is demonstrated on three well-known time series: the sunspots and lynx trapping data, and the El Niño series of sea surface temperatures.

### Citations

2831 |
Estimating the dimension of a model
- Schwarz
- 1978
(Show Context)
Citation Context ...ion 2.1. The usual variations of AIC are also possible. These have the general form C(q) = nlogˆσ 2 q + cn(q + 1), q = 2, . . . , Mn, (3) for some (specified) penalty cn. Bayes information criterion (=-=Schwarz 1978-=-), or BIC, corresponds 5to cn = logn. Another possibility is a Hannan-Quinn-type criterion with cn = loglogn (Hannan and Quinn 1979). Both these methods have cn > 2 for all n large enough, and hence ... |

1748 |
Information theory and an extension of the maximum likelihood principle
- Akaike
- 1973
(Show Context)
Citation Context ...ssment of variance is required to avoid overfitting, i.e., overestimating the period. It is shown that this method of estimating p is asymptotically equivalent to Akaike’s information criterion (AIC, =-=Akaike 1973-=-) when the errors in (1) are assumed to be i.i.d. Gaussian. We show that when p is sufficiently large, our period estimator ˆp is virtually consistent, in the sense that limn→∞ P (ˆp = p) increases to... |

844 | Time series: Theory and methods - Brockwell, Davis - 1987 |

321 |
The determination of the order of an autoregression
- Hannan, Quinn
- 1979
(Show Context)
Citation Context ... . . , Mn, (3) for some (specified) penalty cn. Bayes information criterion (Schwarz 1978), or BIC, corresponds 5to cn = logn. Another possibility is a Hannan-Quinn-type criterion with cn = loglogn (=-=Hannan and Quinn 1979-=-). Both these methods have cn > 2 for all n large enough, and hence have a smaller probability of overestimating the true model dimension than does AIC. In fact, BIC and the Hannan-Quinn criterion pro... |

81 |
Selection of the Order of an Autoregressive Model by Akaike’s Information Criterion
- Shibata
- 1976
(Show Context)
Citation Context ...or example, if the true model is of finite dimension and AIC chooses amongst nested models, then its asymptotic probability of selecting the correct model is 0.71, independent of the model dimension (=-=Shibata 1976-=- and Woodroofe 1982). In spite of its lack of consistency, AIC is still preferred to BIC by some practitioners, since the latter criterion has a higher likelihood of underestimating the model dimensio... |

18 | A survey of statistical work on the MacKenzie River series of annual Canadian lynx trappings for the years 1821– 1934 and a new analysis - Campbell, Walker - 1977 |

17 |
Inadmissibility of the usual estimator for the variance of a normal distribution with unknown
- Stein
- 1964
(Show Context)
Citation Context ...ver the errors have two finite moments, ¯ Yp1, . . . , ¯ Ypp 6are consistent for µ1, . . . , µp, since as n tends to ∞, the number of observed full cycles, kp,i, also tends to ∞. James-Stein theory (=-=Stein 1964-=-) implies that the estimator ¯ Y = ( ¯ Yp1, . . . , ¯ Ypp) is inadmissible when p ≥ 3. A better estimator of a normal mean vector may be obtained by shrinking the sample mean vector towards a specifie... |

15 | Nonparametric estimation of a periodic function - Hall, Reimann, et al. - 2000 |

9 | Consistency of cross-validation when the data are curves - Hart, Wehrly - 1993 |

3 | Using the periodogram to estimate period in nonparametric regression - Hall, Li - 2006 |

2 | Statistical inference for evolving periodic functions - Genton, Hall |

2 | Nonparametric Methods for Estimating Periodic Functions - Hall - 2008 |

2 | Nonparametric methods for deconvolving multiperiodic functions - Hall, Yin - 2003 |

1 | Interdecadal changes in the ENOS-Monsoon system - Torrence, Webster - 1999 |