### Abstract

Consider a one-way layout of the analysis of variance assuming independence, normality, and homogeneity of variance. Test the null hypothesis H_{0} that the means, μ_{i}, of each of k columns, i = 1 , . . . , k are equal versus the alternative that they follow an umbrella pattern. That is, the alternative is H_{1} - H_{0} where H_{1} : μ_{1} ≥ μ_{2} ≥ . . . ≥ μ_{m} ≤ μ_{m+1} ≤ . . . ≤ μ_{k}, and m is known. We derive class of tests which are unbiased and lie in a nontrivial complete class. We recommend specific tests within the class. A simulation of the power functions of some tests is contrasted with the simulated power function of the likelihood ratio test.

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

Pages (from-to) | 2807-2817 |

Number of pages | 11 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 25 |

Issue number | 11 |

DOIs | |

State | Published - Jan 1 1996 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Keywords

- Complete class
- Cone order monotonicity
- Dual cone
- Generators of cones
- Likelihood ratio test
- Order restricted inference
- Unbiased test

### Cite this

*Communications in Statistics - Theory and Methods*,

*25*(11), 2807-2817. https://doi.org/10.1080/03610929608831870

}

*Communications in Statistics - Theory and Methods*, vol. 25, no. 11, pp. 2807-2817. https://doi.org/10.1080/03610929608831870

**Tests for the umbrella alternative under normality.** / Cohen, Arthur; Sackrowitz, Harold.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Tests for the umbrella alternative under normality

AU - Cohen, Arthur

AU - Sackrowitz, Harold

PY - 1996/1/1

Y1 - 1996/1/1

N2 - Consider a one-way layout of the analysis of variance assuming independence, normality, and homogeneity of variance. Test the null hypothesis H0 that the means, μi, of each of k columns, i = 1 , . . . , k are equal versus the alternative that they follow an umbrella pattern. That is, the alternative is H1 - H0 where H1 : μ1 ≥ μ2 ≥ . . . ≥ μm ≤ μm+1 ≤ . . . ≤ μk, and m is known. We derive class of tests which are unbiased and lie in a nontrivial complete class. We recommend specific tests within the class. A simulation of the power functions of some tests is contrasted with the simulated power function of the likelihood ratio test.

AB - Consider a one-way layout of the analysis of variance assuming independence, normality, and homogeneity of variance. Test the null hypothesis H0 that the means, μi, of each of k columns, i = 1 , . . . , k are equal versus the alternative that they follow an umbrella pattern. That is, the alternative is H1 - H0 where H1 : μ1 ≥ μ2 ≥ . . . ≥ μm ≤ μm+1 ≤ . . . ≤ μk, and m is known. We derive class of tests which are unbiased and lie in a nontrivial complete class. We recommend specific tests within the class. A simulation of the power functions of some tests is contrasted with the simulated power function of the likelihood ratio test.

KW - Complete class

KW - Cone order monotonicity

KW - Dual cone

KW - Generators of cones

KW - Likelihood ratio test

KW - Order restricted inference

KW - Unbiased test

UR - http://www.scopus.com/inward/record.url?scp=0010557311&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010557311&partnerID=8YFLogxK

U2 - 10.1080/03610929608831870

DO - 10.1080/03610929608831870

M3 - Article

AN - SCOPUS:0010557311

VL - 25

SP - 2807

EP - 2817

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 11

ER -