Abstract This article examines some problems of significance testing for one-sided hypotheses of the form H 0 : θ ≤ θ 0 versus H 1 : θ > θ 0, where θ is the parameter of interest. In the usual setting, let x be the observed data and let T(X) be a test statistic such that the family of distributions of T(X) is stochastically increasing in θ. Define Cx as {X : T(X) — T(x) ≥ 0}. The p value is p(x) = sup θ≤θ0 Pr(X ∈ Cx | θ). In the presence of a nuisance parameter η, there may not exist a nontrivial Cx with a p value independent of η. We consider tests based on generalized extreme regions of the form Cx (θ, η) = {X : T(X; x, θ, η) ≥ T(x; x, θ, η)}, and conditions on T(X; x, θ, η) are given such that the p value p(x) = sup θ≤θ0 Pr(X ∈ Cx (θ, η)) is free of the nuisance parameter η, where T is stochastically increasing in θ. We provide a solution to the problem of testing hypotheses about the differences in means of two independent exponential distributions, a problem for which the fixed-level testing approach...
[1]
W. A. Thompson.
Optimal significance procedures for simple hypotheses
,
1985
.
[2]
E. Lehmann.
Testing Statistical Hypotheses
,
1960
.
[3]
Jean D. Gibbons,et al.
P-values: Interpretation and Methodology
,
1975
.
[4]
Oscar Kempthorne,et al.
Probability, Statistics, and data analysis
,
1973
.
[5]
H. Jeffreys,et al.
The Theory of Probability
,
1896
.
[6]
Samaradasa Weerahandi,et al.
TESTING REGRESSION EQUALITY WITH UNEQUAL VARIANCES
,
1987
.
[7]
Jerald F. Lawless,et al.
Statistical Models and Methods for Lifetime Data.
,
1983
.
[8]
G. Casella,et al.
Reconciling Bayesian and Frequentist Evidence in the One-Sided Testing Problem
,
1987
.