### Abstract

The problem of testing the equality of two normal means when variances are not known is called the Behrens-Fisher Problem. This problem has three known exact solutions, due, respectively, to Chapman, to Prokof'yev and Shishkin, and to Dudewicz and Ahmed. Each procedure has level alpha and power beta when the means differ by a given amount delta, both set by the experimenter. No single-sample statistical procedures can make this guarantee. The most recent of the three procedures, that of Dudewicz and Ahmed, is asymptotically optimal. We review the procedures, and then compare them with respect to both asymptotic efficiency and also (using simulation) in finite samples. Of these exact procedures, based on finite-sample comparisons the Dudewicz-Ahmed procedure is recommended for practical use.

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

Pages (from-to) | 1584-1605 |

Number of pages | 22 |

Journal | Journal of Statistical Planning and Inference |

Volume | 137 |

Issue number | 5 |

DOIs | |

State | Published - 1 May 2007 |

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### Keywords

- Asymptotically optimal tests
- Behrens-Fisher Problem
- Chapman procedure
- Comparisons
- Dudewicz-Ahmed procedure
- Exact level tests
- Exact solutions
- Finite-sample efficiency
- Heteroscedasticity
- Prokof'yev-Shishkin procedure
- Recommendation for practical use
- Testing equality of means
- Tests of hypotheses
- Tests with specified power

### Cite this

*Journal of Statistical Planning and Inference*,

*137*(5), 1584-1605. https://doi.org/10.1016/j.jspi.2006.09.007

}

*Journal of Statistical Planning and Inference*, vol. 137, no. 5, pp. 1584-1605. https://doi.org/10.1016/j.jspi.2006.09.007

**Exact solutions to the Behrens-Fisher Problem : Asymptotically optimal and finite sample efficient choice among.** / Dudewicz, Edward J.; Ma, Yan; Mai, Enping (Shirley); Su, Haiyan.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Exact solutions to the Behrens-Fisher Problem

T2 - Asymptotically optimal and finite sample efficient choice among

AU - Dudewicz, Edward J.

AU - Ma, Yan

AU - Mai, Enping (Shirley)

AU - Su, Haiyan

PY - 2007/5/1

Y1 - 2007/5/1

N2 - The problem of testing the equality of two normal means when variances are not known is called the Behrens-Fisher Problem. This problem has three known exact solutions, due, respectively, to Chapman, to Prokof'yev and Shishkin, and to Dudewicz and Ahmed. Each procedure has level alpha and power beta when the means differ by a given amount delta, both set by the experimenter. No single-sample statistical procedures can make this guarantee. The most recent of the three procedures, that of Dudewicz and Ahmed, is asymptotically optimal. We review the procedures, and then compare them with respect to both asymptotic efficiency and also (using simulation) in finite samples. Of these exact procedures, based on finite-sample comparisons the Dudewicz-Ahmed procedure is recommended for practical use.

AB - The problem of testing the equality of two normal means when variances are not known is called the Behrens-Fisher Problem. This problem has three known exact solutions, due, respectively, to Chapman, to Prokof'yev and Shishkin, and to Dudewicz and Ahmed. Each procedure has level alpha and power beta when the means differ by a given amount delta, both set by the experimenter. No single-sample statistical procedures can make this guarantee. The most recent of the three procedures, that of Dudewicz and Ahmed, is asymptotically optimal. We review the procedures, and then compare them with respect to both asymptotic efficiency and also (using simulation) in finite samples. Of these exact procedures, based on finite-sample comparisons the Dudewicz-Ahmed procedure is recommended for practical use.

KW - Asymptotically optimal tests

KW - Behrens-Fisher Problem

KW - Chapman procedure

KW - Comparisons

KW - Dudewicz-Ahmed procedure

KW - Exact level tests

KW - Exact solutions

KW - Finite-sample efficiency

KW - Heteroscedasticity

KW - Prokof'yev-Shishkin procedure

KW - Recommendation for practical use

KW - Testing equality of means

KW - Tests of hypotheses

KW - Tests with specified power

UR - http://www.scopus.com/inward/record.url?scp=33846171958&partnerID=8YFLogxK

U2 - 10.1016/j.jspi.2006.09.007

DO - 10.1016/j.jspi.2006.09.007

M3 - Article

VL - 137

SP - 1584

EP - 1605

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 5

ER -