Figure 4: Power curves: Student’s t
Let S be the score test statistic, defined in the usual way. Under the
null hypothesis (p = ∞), S has a limiting χ2 distribution. Under Hn, as we
show in Appendix C.1,
S →d χ12(δ),
with non-centrality parameter
In conclusion, the score test and the IM tests have the same non-centrality
parameter in their limiting distribution. The power curves (as a function
of e) of 5%-level tests are given in Figure 4. The differences in power are
entirely due to differences in degrees of freedom: 1 for the score test, 2 for
the IM test with the ML estimator, and 3 for the IM test using any other
M-estimator. The difference in power between the IM tests is small.
4.3 Skewed normal alternative
Let Z — N(0, 1) and denote the distribution of ZI(Z ≤ 0) + (1 + γ) ZI(Z >
0) as FYk (Fernandez and Steel, 1998). Under the sequence of local alterna-
tives
Hn : Y
- Fn = Fesk√n,
15
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