Abstract
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.
| 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 |
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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
More powerful tests for the sign testing problem. / Saikali, Khalil G.; Berger, Roger L.
In: Journal of Statistical Planning and Inference, Vol. 107, No. 1-2, 01.09.2002, p. 187-205.Research output: Contribution to journal › Article
}
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 -