An explicit representation of the limit of the LRT for interval mapping of quantitative trait loci

Consider the interval mapping method for quantitative trait loci (QTL) detection of an experimental organism in a backcross population. In a recent paper, Chen and Chen [2005. On some statistical aspects of the interval mapping for QTL detection. Statist. Sinica 15, 909-925] show that the limiting null distribution of the likelihood ratio test (LRT) statistic for testing existence of a QTL flanked by two markers is that of the supremum of a χ2-process. In this note, an explicit representation of the supremum in terms of the recombination fraction between two markers of the χ2-process is presented. Based on the explicit representation, a simulation algorithm for determining a limiting threshold of the LRT is given.