A note on testing two-dimensional normal mean

Nomakuchi, Kentaro; Sakata, Toshio

doi:10.1007/bf02491485

Cited by 16 publications

(15 citation statements)

References 12 publications

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“…4 Trivially, these tests have power functions uniformly above that of the LR test. The construction of these exact tests resembles those mentioned by Lehmann (1952), p. 542, and later by Nomakuchi and Sakata (1987), p. 492. See also Berger (1989) for related constructions.…”

Section: Asymptotic Problemmentioning

confidence: 69%

Improved tests for mediation

2022

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Testing for a mediation effect is important in many disciplines, but is made difficult -even asymptotically -by the influence of nuisance parameters. Classical tests such as likelihood ratio (LR) and Wald tests have very poor size and power properties in some parts of the parameter space, and many attempts have been made to produce improved tests, with limited success. In this paper we show that augmenting the critical region of the LR test can produce a test with much improved behaviour everywhere. In fact, we first show that there exists a test of this type that is (asymptotically) exact for certain test sizes α, including the common choices α = .01, .05, .10. This is evidently an important result, but we also observe that the critical region of this exact test has some undesirable properties. Thus, we then go on to show that there is a very simple class of augmented LR critical regions which provides tests that, while not exact, are very nearly so, and which avoid the issues inherent in the exact test. We suggest an optimal member of this class, and provide the tables needed to implement it. Although motivated by a simple two-equation linear model, the results apply to any model structure that reduces to the same testing problem asymptotically. A short application of the method to an entrepreneurial attitudes study is included for illustration.

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Section: Asymptotic Problemmentioning

confidence: 69%

Improved tests for mediation

2022

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“…A result of Lehmann (1952) shows that in some problems of the type we are considering, no unbiased, nonrandomized tests exists. Nomakuchi and Sakata (1987) also discuss this. But, in fact, tests do exist that have the same size as the LRT and are uniformly more powerful.…”

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confidence: 83%

Uniformly More Powerful Tests for Hypotheses concerning Linear Inequalities and Normal Means

Berger

1989

Journal of the American Statistical Association

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In this paper we consider some hypothesis testing problems regarding normal means. In these problems, the hypotheses are defined by linear inequalities on the means. We show that in certain problems the likelihood ratio rest (LRT) is not very powerful. We describe a test that has the same size, ex, as the LRT and is uniformly more powerful. The test is easily implem~nted since its critical values are standard normal percentiles. The increase in power with the new test can be substantial. For example the new test's power is l/2ex times bigger (10 times bigger for ex = .05) than the LRT's power fo"r some parameter points in a simple example.Specifically let :z = (XlI...,X p )' (p~2) be a multivariate normal random vector with unknown mean e = (PlI''''Pp)' and known, nonsingular covariance matrix~. We consider testing the null hypothesis parameter points. Let the size of the test, a, be less than .5. Define the integer J by the inequality J-I < l/2a i J. Define the constants co,...,c] by Co = CD; Cj = Zja, j=I,... ,J-I; and c] = O. (Again, z ja is the upper 100(ja) percentile of a standard normal distribution.) In many cases, a size-a test that is uniformly more power than the LRT is the test that rejects H o if J ]! £ u R. j=l J statistic. The set R 1 is the rejection region of the LRT. So this test is = c. l' = . J J "a:a. Ua. n

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“…We also have g(Xl,02) <_ 0 for xl < a by reversed version of Lemma 2. Nomakuchi and Sakata (1987) showed that the ~u discussed in the introduction is an unbiased test under the assumption of the Schur-concavity of the joint density function. In the following theorem the Schur-concavity is not assumed; therefore, the test and distribution axe generalized to the exponential family.…”

Section: The Lemmasmentioning

confidence: 99%

“…Inada (1978), Sasabuchi (1980Sasabuchi ( , 1988aSasabuchi ( , 1988b). The admissibility of this test was shown by Cohen et al (1983) and also Nomakuchi and Sakata (1987). This might sound as if there exist no uniformly more powerful tests than ~LR.…”

Section: Introductionmentioning

confidence: 99%

“…In fact it is well known that likelihood ratio tests are optimum in many testing problems. Interestingly, however, Nomakuchi and Sakata (1987) showed that ll ~DU(Xl' x2) --~ 0 if z(i/m) < Xl, x2 < z((i --1)/m), i = 1,..., m, otherwise, is a level 1/m unbiased test and uniformly most powerful than the likelihood ratio test. This test is essentially proposed by Lehmann (1952).…”

Section: Introductionmentioning

confidence: 99%

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