Abstract
Some general results about the GLR tests, for testing simple hypothesis versus two-sided hypothesis, in the family with support dependent on the parameter, are obtained. In addition, we show that such GLR tests are equivalent to the UMP tests in the same problems. Moreover, we derive the general form of the UMP tests for testing an interval hypothesis versus two-sided alternative.
Highlights
Introduction and BackgroundThe problem of finding uniformly most powerful (UMP) test for testing the simple hypothesis H0 : θ = θ0 against two-sided alternative H1 : θ= θ0 has been an interesting problem in testing statistical hypotheses
Birnbaum [3] and Pratt [12] offered the solutions for UMP tests for composite hypotheses in the uniform distributions with both end points of the support dependent on the parameter
Migliorati [10] demonstrated that a UMP test exists for testing two-sided hypothesis in the family with support dependent on the parameter
Summary
The UMP two-sided test in the family with right and left extreme points of the support dependent on the parameter is obtained by Sayyareh et al [13]. We obtain the GLR tests for testing hypothesis H0 : θ = θ0 versus two-sided alternative H1 : θ= θ0, in the family of distributions of the following forms fθ(x) = a(θ)b(x) ; c < x < θ, a(θ) > 0,. This paper is organized as follows: In Section 2, we obtain the general form of the GLR test for testing simple hypothesis versus two-sided alternative in the family with the right or left extreme point of the support dependent on the parameter. Remark 2 The GLR tests obtained in Theorems 1 and 2 are coincided with the UMP tests in the same problems (see Theorems 4.3 and 4.4 in [13])
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