### Abstract

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

Pages (from-to) | 445-456 |

Number of pages | 12 |

Journal | Acta Mathematicae Applicatae Sinica-English Series |

Volume | 25 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 |

### Fingerprint

### Keywords

- dimension reduction
- Kullback-Leibler discrepancy
- Shannon's entropy
- nonparametric regression
- local linear regression
- Conditional density function

### Cite this

*Acta Mathematicae Applicatae Sinica-English Series*,

*25*(3), 445-456. https://doi.org/10.1007/s10255-008-8815-1

**Approximating conditional density functions using dimension reduction.** / Fan, J Q; Peng, L; Yao, Q W; Zhang, Wen Yang.

Research output: Contribution to journal › Article

*Acta Mathematicae Applicatae Sinica-English Series*, vol. 25, no. 3, pp. 445-456. https://doi.org/10.1007/s10255-008-8815-1

}

TY - JOUR

T1 - Approximating conditional density functions using dimension reduction

AU - Fan, J Q

AU - Peng, L

AU - Yao, Q W

AU - Zhang, Wen Yang

PY - 2009

Y1 - 2009

N2 - We propose to approximate the conditional density function of a random variable Y given a dependent random d-vector X by that of Y given theta X-tau, where the unit vector theta is selected such that the average Kullback-Leibler discrepancy distance between the two conditional density functions obtains the minimum. Our approach is nonparametric as far as the estimation of the conditional density functions is concerned. We have shown that this nonparametric estimator is asymptotically adaptive to the unknown index theta in the sense that the first order asymptotic mean squared error of the estimator is the same as that when theta was known. The proposed method is illustrated using both simulated and real-data examples.

AB - We propose to approximate the conditional density function of a random variable Y given a dependent random d-vector X by that of Y given theta X-tau, where the unit vector theta is selected such that the average Kullback-Leibler discrepancy distance between the two conditional density functions obtains the minimum. Our approach is nonparametric as far as the estimation of the conditional density functions is concerned. We have shown that this nonparametric estimator is asymptotically adaptive to the unknown index theta in the sense that the first order asymptotic mean squared error of the estimator is the same as that when theta was known. The proposed method is illustrated using both simulated and real-data examples.

KW - dimension reduction

KW - Kullback-Leibler discrepancy

KW - Shannon's entropy

KW - nonparametric regression

KW - local linear regression

KW - Conditional density function

UR - http://www.scopus.com/inward/record.url?scp=66749126022&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/s10255-008-8815-1

U2 - 10.1007/s10255-008-8815-1

DO - 10.1007/s10255-008-8815-1

M3 - Article

VL - 25

SP - 445

EP - 456

JO - Acta Mathematicae Applicatae Sinica-English Series

JF - Acta Mathematicae Applicatae Sinica-English Series

SN - 0168-9673

IS - 3

ER -