### Abstract

The problem of estimating the center of a symmetric distribution is well studied and many nonparametric procedures are available. It often serves as the test problem for many nonparametric estimation procedures, and stimulated the development of efficient nonparametric estimation theory. We use this familiar setting to illustrate a novel use of empirical likelihood method for estimation and testing. Empirical likelihood is a general nonparametric inference method, see Owen [Owen, A. (2001). Empirical Likelihood. London: Chapman and Hall]. However, for symmetric location problem (and some other problems) empirical likelihood has difficulties. Owen (2001) call them "challenges for the empirical likelihood". We propose and study a way to use the empirical likelihood with such problems by modifying the parameter space. We illustrate this approach by applying it to the symmetric location problem. We show that the usual asymptotic theory of empirical likelihood still holds and the asymptotic efficiency of the so obtained empirical NPMLE of location is studied.

Original language | English (US) |
---|---|

Pages (from-to) | 2233-2243 |

Number of pages | 11 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 33 |

Issue number | 9 SPEC.ISS. |

DOIs | |

State | Published - Sep 2004 |

Externally published | Yes |

### Fingerprint

### Keywords

- Asymptotic chi-square distribution
- Many constraints of symmetry
- Nonparametric information bound

### ASJC Scopus subject areas

- Statistics and Probability
- Safety, Risk, Reliability and Quality

### Cite this

*Communications in Statistics - Theory and Methods*,

*33*(9 SPEC.ISS.), 2233-2243. https://doi.org/10.1081/STA-200026602

**Symmetric Location Estimation /Testing by Empirical Likelihood.** / Kim, Kyoungmi; Zhou, Mai.

Research output: Contribution to journal › Article

*Communications in Statistics - Theory and Methods*, vol. 33, no. 9 SPEC.ISS., pp. 2233-2243. https://doi.org/10.1081/STA-200026602

}

TY - JOUR

T1 - Symmetric Location Estimation /Testing by Empirical Likelihood

AU - Kim, Kyoungmi

AU - Zhou, Mai

PY - 2004/9

Y1 - 2004/9

N2 - The problem of estimating the center of a symmetric distribution is well studied and many nonparametric procedures are available. It often serves as the test problem for many nonparametric estimation procedures, and stimulated the development of efficient nonparametric estimation theory. We use this familiar setting to illustrate a novel use of empirical likelihood method for estimation and testing. Empirical likelihood is a general nonparametric inference method, see Owen [Owen, A. (2001). Empirical Likelihood. London: Chapman and Hall]. However, for symmetric location problem (and some other problems) empirical likelihood has difficulties. Owen (2001) call them "challenges for the empirical likelihood". We propose and study a way to use the empirical likelihood with such problems by modifying the parameter space. We illustrate this approach by applying it to the symmetric location problem. We show that the usual asymptotic theory of empirical likelihood still holds and the asymptotic efficiency of the so obtained empirical NPMLE of location is studied.

AB - The problem of estimating the center of a symmetric distribution is well studied and many nonparametric procedures are available. It often serves as the test problem for many nonparametric estimation procedures, and stimulated the development of efficient nonparametric estimation theory. We use this familiar setting to illustrate a novel use of empirical likelihood method for estimation and testing. Empirical likelihood is a general nonparametric inference method, see Owen [Owen, A. (2001). Empirical Likelihood. London: Chapman and Hall]. However, for symmetric location problem (and some other problems) empirical likelihood has difficulties. Owen (2001) call them "challenges for the empirical likelihood". We propose and study a way to use the empirical likelihood with such problems by modifying the parameter space. We illustrate this approach by applying it to the symmetric location problem. We show that the usual asymptotic theory of empirical likelihood still holds and the asymptotic efficiency of the so obtained empirical NPMLE of location is studied.

KW - Asymptotic chi-square distribution

KW - Many constraints of symmetry

KW - Nonparametric information bound

UR - http://www.scopus.com/inward/record.url?scp=7544235800&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=7544235800&partnerID=8YFLogxK

U2 - 10.1081/STA-200026602

DO - 10.1081/STA-200026602

M3 - Article

AN - SCOPUS:7544235800

VL - 33

SP - 2233

EP - 2243

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 9 SPEC.ISS.

ER -