Likelihood ratio test process for quantitative trait locus detection

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'. We give the asymptotic distribution of this LRT process under the null hypothesis that there is no QTL on [ 0, T] and under local alternatives with a QTL at t* on [0, T]. We show that the LRT is asymptotically the square of some Gaussian process. We give a description of this process as an 'nonlinear interpolated and normalized process'. We propose a simple method to calculate the maximum of the LRT process using only statistics on markers and their ratio. This gives a new method to calculate thresholds for the QTL detection.