When testing hypotheses in two-sample problems, the Lepage test has often
been used to jointly test the location and scale parameters, and has been discussed by
many authors over the years. The Lepage test was a combination of the Wilcoxon
statistic and the Ansari–Bradley statistic. Various Lepage-type tests were proposed
with discussions of an asymptotic relative efficiency (Duran et al., Biometrika 63:173–
176, 1976; Goria, Stat Neerl 36:3–13, 1982), a robustness and a power comparison
(Neuhäuser, Commun Stat Theory Methods 29:67–78, 2000; Büning, J Appl Stat
29:907–924, 2002) and an adaptive test (Büning and Thadewald, J Stat Comput Sim
65:287–310, 2000). We derive an expression for the moment generating function of a
linear combination of two linear rank statistics. As a suggested Lepage-type test, we use
a combination of the generalized Wilcoxon statistic and the generalized Mood statistic.
Deriving the exact critical value of the statistic can be difficult when the sample sizes
are increasing. In this situation, an approximation method to the distribution function
of the test statistic can be useful with a higher order moment. We use a moment-
based approximation with an adjusted gamma polynomial to evaluate the upper tail
probability of a Lepage-type test for a finite sample size. We determine the asymptotic
efficiencies of the Lepage and Lepage-type tests for various distributions.

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• A moment generating function of a combination of linear rank tests and its asymptotic efficiency