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**1 - 2**of**2**### A Lepage-type test for the GSHD

"... Numerical example We introduce a new location-scale rank test efficient for the generalised secant hyperbolic distribution (GSHD). The GSHD consists of symmetric unimodal distributions of various tails, and is interesting in applications where the lack of normality is explained by the tail behaviour ..."

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Numerical example We introduce a new location-scale rank test efficient for the generalised secant hyperbolic distribution (GSHD). The GSHD consists of symmetric unimodal distributions of various tails, and is interesting in applications where the lack of normality is explained by the tail behaviour of the data distributions. The new test is a family of Lepage-type tests, each of which combines the standardised location and scale rank statistics efficient under their alternative hypotheses for a specific distribution.

### was introduced in 2002 in [1]. The GSHD:

"... Is a location-scale family of symmetric unimodal distributions of various tails; Includes the Cauchy and the uniform distributions as its limiting heavy-tail and light-tail cases; and is interesting in applications where the lack of normality is explained by the tail behaviour of the data distributi ..."

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Is a location-scale family of symmetric unimodal distributions of various tails; Includes the Cauchy and the uniform distributions as its limiting heavy-tail and light-tail cases; and is interesting in applications where the lack of normality is explained by the tail behaviour of the data distributions. A member of the distribution is completely specified by the location, scale, and shape (tail) parameters. We introduce a new location-scale rank test efficient for the GSHD. The new test is a family of Lepage-type tests, each of which combines the standardised location and scale rank statistics efficient under their alternative hypotheses for a specific distribution. The two-sample linear rank procedures of location and scale alternatives efficient for the GSHD were introduced in 2006 in [2, 3]. Both the location and scale rank procedures are robust to distributional misspecifications. However, the scale procedures are extremely sensitive to the presence of outliers. Moreover, the location estimators are regular almost for the whole family, while the scale ratio rank estimator is only regular conditioned on no difference in location. Location-scale rank test We consider the two-sample location-scale problem with two independent samples of sizes m and n. Let m + n = N, let c(i) = 1 mn