On empirical processes for quantitative trait locus mapping under the presence of a selective genotyping and an interference phenomenon

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (i.e. a gene with quantitative effect on a trait) on the interval [ 0 , T ] representing a chromosome. The originality lies in the fact that we consider a selective genotyping (i.e. only the individuals with extreme phenotypes are genotyped) and an interference phenomenon (i.e. a recombination event inhibits the formation of another recombination event nearby). We show that, under the null hypothesis and contiguous alternatives, the LRT process is asymptotically the square of a “linear interpolated and normalized Gaussian process”. We have an easy formula in order to compute the supremum of the square of this linear interpolated process. We prove that we have to genotype symmetrically and that the threshold is exactly the same as in the situation without selective genotyping and without interference.