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In a previous article (Grané and Fortiana, 2006), we studied a flexible class ofgoodness-of-fit tests associated with an orthogonal sequence, the Karhunen&-Loèvedecomposition of a stochastic process derived from the null hypothesis. Generallyspeaking, these tests outperform Kolmogorov&-Smirnov and Cramér&-von Mises, but we registered several exceptions. In this work we investigate the cause of theseanomalies and, more precisely, whether and when such poor behavior may beattributed to the orthogonal sequence itself, by replacing it with the Legendrepolynomials, a commonly used basis for smooth tests. We find an easily computableformula for the Bahadur asymptotic relative efficiency, a helpful quantity in choosingan adequate basis.