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Abstract

Let (Yi ,Xi ) , i =1,..., n , be a random sample from some p+1 variate distribution where Xi is a vector having length p. Many methods for testing the hypothesis that Y is independent of X are relatively insensitive to a broad class of departures from independence. Power improvements focus on the median of Y or some other quantile and test the hypothesis that the regression surface is a horizontal plane versus some unknown form. A wild bootstrap method (Stute et al. 1998) can be used based on quantiles, but with small or moderate sample sizes, control over the probability of a Type I error can be unsatisfactory when sampling from asymmetric distributions. He and Zhu (2003) is readily adapted to testing the hypothesis that the conditional & gamma; quantile of Y does not depend on X where critical values are determined via simulations. A modification is suggested that avoids the need for simulations to obtain critical values, and perform wells in terms of Type I errors even when sampling from asymmetric distributions.

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