A note on a family of criteria for evaluating test statistics

ABSTRACT

In noting that the usual criteria for choosing an optimal test, Uniform Power and Local Power are at opposite ends of a spectrum of dominance criteria, a complete “Power Dominance” family of criteria for classifying and choosing optimal tests on the basis of their power characteristics is identified, wherein successive orders of dominance attach increasing weight to power close to the null hypothesis. Indices of the extent to which a preferred test has superior power characteristics over other members in its class, and an index of the proximity of a test to the envelope function of alternative tests are also provided. The ideas are exemplified using various optimal test statistics for Normal and Laplace population distributions.

Notes

1 A similar analysis is possible for two sided tests, but is omitted here for reasons of brevity.

2 Sometimes poorness indices P(x) are studied (Atkinson 1987) in which case U(x) = −P(x).

3 The intuition for how the result is arrived at follows in the discussion below.

4 The comparison procedures have been empirically implemented in several ways, see, for example, Anderson (1996, 2004), Barrett and Donald (2003), Davidson and Duclos (2000), Linton et al. (2005), Knight and Satchell (2008), and McFadden (1989).

5 Knight and Satchell (2008) empirically implemented this order of dominance comparison.

6 It follows from the Davidson and Duclos (2000) lemma that if Test 1 first-order power dominates locally, as the (LMP) does, it will dominate at some higher order over the whole range of μ₁.

7 Integration was approximated using the trapezoidal rule, which for small increments of θ (0.005), provides a more than adequate approximation.