### Abstract

For i = 1,..., p, let X_{i1},..., X_{ini} denote independent random samples from normal populations. The ith population has unknown mean μ_{i} and unknown variance σ_{i}
^{2}. We consider the sign testing problem of testing H_{0}: μ_{i} ≤ a_{i}, for some i=1,..., p, versus H_{1}: μ_{i} > a_{i}, for all i=1,..., p, where a_{1},..., a_{p} are fixed constants. Here, H_{1} might represent p different standards that a product must meet before it is considered acceptable. For 0 < α < 1/2, we first derive the size-α likelihood ratio test (LRT) for this problem, and then we describe an intersection-union test (IUT) that is uniformly more powerful than the likelihood ratio test if the sample sizes are not all equal. For a more general model than the normal, we describe two intersection-union tests that maximize the size of the rejection region formed by intersection. Applying these tests to the normal problem yields two tests that are uniformly more powerful than both the LRT and IUT described above. A small power comparison of these tests is given.

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

Pages (from-to) | 187-205 |

Number of pages | 19 |

Journal | Journal of Statistical Planning and Inference |

Volume | 107 |

Issue number | 1-2 |

DOIs | |

State | Published - Sep 1 2002 |

Externally published | Yes |

### Fingerprint

### Keywords

- Independence
- Intersection-union test
- Likelihood ratio test
- Min test
- Normal population
- t distribution

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Applied Mathematics
- Statistics and Probability

### Cite this

*Journal of Statistical Planning and Inference*,

*107*(1-2), 187-205. https://doi.org/10.1016/S0378-3758(02)00252-5

**More powerful tests for the sign testing problem.** / Saikali, Khalil G.; Berger, Roger L.

Research output: Contribution to journal › Article

*Journal of Statistical Planning and Inference*, vol. 107, no. 1-2, pp. 187-205. https://doi.org/10.1016/S0378-3758(02)00252-5

}

TY - JOUR

T1 - More powerful tests for the sign testing problem

AU - Saikali, Khalil G.

AU - Berger, Roger L.

PY - 2002/9/1

Y1 - 2002/9/1

N2 - For i = 1,..., p, let Xi1,..., Xini denote independent random samples from normal populations. The ith population has unknown mean μi and unknown variance σi 2. We consider the sign testing problem of testing H0: μi ≤ ai, for some i=1,..., p, versus H1: μi > ai, for all i=1,..., p, where a1,..., ap are fixed constants. Here, H1 might represent p different standards that a product must meet before it is considered acceptable. For 0 < α < 1/2, we first derive the size-α likelihood ratio test (LRT) for this problem, and then we describe an intersection-union test (IUT) that is uniformly more powerful than the likelihood ratio test if the sample sizes are not all equal. For a more general model than the normal, we describe two intersection-union tests that maximize the size of the rejection region formed by intersection. Applying these tests to the normal problem yields two tests that are uniformly more powerful than both the LRT and IUT described above. A small power comparison of these tests is given.

AB - For i = 1,..., p, let Xi1,..., Xini denote independent random samples from normal populations. The ith population has unknown mean μi and unknown variance σi 2. We consider the sign testing problem of testing H0: μi ≤ ai, for some i=1,..., p, versus H1: μi > ai, for all i=1,..., p, where a1,..., ap are fixed constants. Here, H1 might represent p different standards that a product must meet before it is considered acceptable. For 0 < α < 1/2, we first derive the size-α likelihood ratio test (LRT) for this problem, and then we describe an intersection-union test (IUT) that is uniformly more powerful than the likelihood ratio test if the sample sizes are not all equal. For a more general model than the normal, we describe two intersection-union tests that maximize the size of the rejection region formed by intersection. Applying these tests to the normal problem yields two tests that are uniformly more powerful than both the LRT and IUT described above. A small power comparison of these tests is given.

KW - Independence

KW - Intersection-union test

KW - Likelihood ratio test

KW - Min test

KW - Normal population

KW - t distribution

UR - http://www.scopus.com/inward/record.url?scp=0036742644&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036742644&partnerID=8YFLogxK

U2 - 10.1016/S0378-3758(02)00252-5

DO - 10.1016/S0378-3758(02)00252-5

M3 - Article

AN - SCOPUS:0036742644

VL - 107

SP - 187

EP - 205

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 1-2

ER -