p-Values, Bayes Factors, and Sufficiency

Figures & data

Figure 1. The GLR and Wilks’s p-value. The gray region shows the possible values of p₀ under the ET embedding model given in (3(3) $$f(x;\theta )=\frac{f_0\left(x\right)·e^{\theta ·t\left(x\right)}}{M_T\left(\theta \right)},\theta \ge 0,$$(3) ), and the solid line the values of Wilks’s p-value, based on the asymptotic distribution of − 2log G₀₁(X) under the null distribution.

Figure 2. The Generalized Likelihood Ratio G₀₁ and possible Bayes factors B₀₁ (gray region), as functions of the p-value p₀, for the null distribution X ∼ N(0, 1) and the test statistic t(x) = x, which is a UMP one-tailed test for location.

Figure 3. Similar to Figure 2, except based on the “two-tailed” test statistic t(x) = x², which is a UMP one-tailed test for dispersion.