10.6.∗Adaptive Procedures 623left. The second selector statistic isQ 2 =U. 05 −L. 05
U. 5 −L. 5. (10.6.3)
Large values ofQ 2 indicate that the distribution is heavy-tailed, while small values
indicate that the distribution is light-tailed. Rules are needed for score selection,
and here we make use of the benchmarks proposed in an article by Hogg et al.
(1975). These rules are tabulated below, along with their benchmarks:
Benchmark Distribution Indicated Score Selected
Q 2 > 7 Heavy-tailed symmetric φ 2
Q 1 >2andQ 2 < 7 Right-skewed φ 4
Q 1 ≤2andQ 2 ≤ 2 Light-tailed symmetric φ 3
Elsewhere Moderate heavy-tailed φ 1Hogg et al. (1975) performed a Monte Carlo power study of this adaptive proce-
dure over a number of distributions with different kurtosis and skewness coefficients.
In the study, both the adaptive procedure and the Wilcoxon test maintain theirα
level over the distributions, but the Studenttdoes not. Moreover, the Wilcoxon test
has better power than thet-test, as the distribution deviates much from the normal
(kurtosis = 3 and skewness = 0), but the adaptive procedure is much better than
the Wilcoxon for the short-tailed distributions, the very heavy-tailed distributions,
and the highly skewed distributions that are considered in the study.
Remark 10.6.1(Computation for the Adaptive Procedure).An R implementation
of Hogg’s adaptive procedure as discussed in Example 10.6.1 can be found in the R
packagenpsmdeveloped by Kloke and McKean (2014); see their Section 3.6. The
R function ishogg.test. For illustration, consider the normal data discussed in
Example 10.5.3. Here are the code and results:
load("examp1053.rda"); hogg.test(y,x)
Scores Selected: Wilcoxon; p.value 0.11984
Hence, for this data, Hogg’s procedure selected Wilcoxon scores. As another ex-
ample, consider the waterwheel data given in Example 10.4.1. In this case the
computation results in:
load("waterwheel.rda"); hogg.test(grp2,grp1)
Scores Selected: bent; p.value 0.63494
The selected score is the bent score which is the score functionφ 4 (u) in Hogg’s
procedure. As the boxplot for the combined samples indicates the data are right-
skewed, an indication that the score selection is appropriate.
The adaptive distribution-free procedure that we have discussed is for testing.
Suppose we have a location model and were interested in estimating the shift in
locations Δ. For example, if the trueF is a normal cdf, then a good choice for
the estimator of Δ would be the estimator based on the normal scores procedure
discussed in Example 10.5.1. The estimators, though, are not distribution free and,
hence, the above reasoning does not hold. Also, the combined sample observations