On Quantitative Trait Locus mapping with an interference phenomenom

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a quantitative trait locus, i.e. a gene with quantitative effect on a trait) on the interval $[0,T]$ representing a chromosome. The observation is the trait and the composition of the genome at some locations called ''markers''. As in Rebai et al. (95), we focus on the interference phenomenom : a recombination event inhibes the formation of another nearby. We give the asymptotic distribution of the LRT process under the null hypothesis that there is no QTL on $[0,T]$ and under local alternatives with a QTL at $t^{\star}$ on $[0,T]$. We show that the LRT process is asymptotically the square of a ''linear interpolated and normalized process '' whereas the LRT process obtained recently by Azais et al., for a model without interference, was the square of a ''non linear interpolated and normalized process ''. The computation of the supremum of our LRT process becomes easy due to the interpolation. Besides, we proove that the MCQMC method to compute thresholds for QTL detection, proposed by Azais et al., is still suitable for our model with interference