### Abstract

In this article, we propose a Stein-type two-sample procedure for comparing the means of k(> 1) experimental normal populations among themselves and with the reference to the mean of a controlled normal population, when the variances of all (k + 1) populations are unequal and unknown. Our selection formulation follows closely to that by Bechoffer and Turnbull (1978), who considered the comparison of k normal means with a specific nonrandom standard value, when the variances are either known or unknown and equal. Instead of comparing the k experimental populations to a nonrandom standard value, our comparison is made with reference to a random controlled normal population. Moreover, we broaden their assumption of equal unknown variances to unequal unknown variances. The proposed procedure satisfies two probability requirements: (1) the probability of selecting the control is at least prespecified (Formula presented.) when the largest experimental mean is significantly smaller than the mean of the control and (2) the probability of selecting the largest experimental mean is at least prespecified (Formula presented.) when the largest experimental mean is significantly largest than the second largest experimental mean and the mean of the control.

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

Pages (from-to) | 441-460 |

Number of pages | 20 |

Journal | Sequential Analysis |

Volume | 34 |

Issue number | 4 |

DOIs | |

State | Published - Oct 2 2015 |

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### Keywords

- Heteroscedasticity
- Least favorable configuration
- Normal distribution
- Ranking and selection

### ASJC Scopus subject areas

- Modeling and Simulation
- Statistics and Probability

### Cite this

*Sequential Analysis*,

*34*(4), 441-460. https://doi.org/10.1080/07474946.2015.1099935

**A Stein-Type Two-Sample Procedure for Comparing Normal Means.** / Buzaianu, Elena M.; Chen, Pinyuen.

Research output: Contribution to journal › Article

*Sequential Analysis*, vol. 34, no. 4, pp. 441-460. https://doi.org/10.1080/07474946.2015.1099935

}

TY - JOUR

T1 - A Stein-Type Two-Sample Procedure for Comparing Normal Means

AU - Buzaianu, Elena M.

AU - Chen, Pinyuen

PY - 2015/10/2

Y1 - 2015/10/2

N2 - In this article, we propose a Stein-type two-sample procedure for comparing the means of k(> 1) experimental normal populations among themselves and with the reference to the mean of a controlled normal population, when the variances of all (k + 1) populations are unequal and unknown. Our selection formulation follows closely to that by Bechoffer and Turnbull (1978), who considered the comparison of k normal means with a specific nonrandom standard value, when the variances are either known or unknown and equal. Instead of comparing the k experimental populations to a nonrandom standard value, our comparison is made with reference to a random controlled normal population. Moreover, we broaden their assumption of equal unknown variances to unequal unknown variances. The proposed procedure satisfies two probability requirements: (1) the probability of selecting the control is at least prespecified (Formula presented.) when the largest experimental mean is significantly smaller than the mean of the control and (2) the probability of selecting the largest experimental mean is at least prespecified (Formula presented.) when the largest experimental mean is significantly largest than the second largest experimental mean and the mean of the control.

AB - In this article, we propose a Stein-type two-sample procedure for comparing the means of k(> 1) experimental normal populations among themselves and with the reference to the mean of a controlled normal population, when the variances of all (k + 1) populations are unequal and unknown. Our selection formulation follows closely to that by Bechoffer and Turnbull (1978), who considered the comparison of k normal means with a specific nonrandom standard value, when the variances are either known or unknown and equal. Instead of comparing the k experimental populations to a nonrandom standard value, our comparison is made with reference to a random controlled normal population. Moreover, we broaden their assumption of equal unknown variances to unequal unknown variances. The proposed procedure satisfies two probability requirements: (1) the probability of selecting the control is at least prespecified (Formula presented.) when the largest experimental mean is significantly smaller than the mean of the control and (2) the probability of selecting the largest experimental mean is at least prespecified (Formula presented.) when the largest experimental mean is significantly largest than the second largest experimental mean and the mean of the control.

KW - Heteroscedasticity

KW - Least favorable configuration

KW - Normal distribution

KW - Ranking and selection

UR - http://www.scopus.com/inward/record.url?scp=84953232699&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84953232699&partnerID=8YFLogxK

U2 - 10.1080/07474946.2015.1099935

DO - 10.1080/07474946.2015.1099935

M3 - Article

VL - 34

SP - 441

EP - 460

JO - Sequential Analysis

JF - Sequential Analysis

SN - 0747-4946

IS - 4

ER -