Statistics on continuous IBD data: Exact distribution evaluation for a pair of full(half)-sibs and a pair of a (great-) grandchild with a (great-) grandparent

A technology is provided for numerical evaluation, with any given accuracy, of the cumulative probabilities of some statistics on continuous genome data for pairs of closely related individuals. In the case of a pair of full-sibs, the following statistics are considered: (i) the proportion of genome with 2 (at least 1) haplotypes shared identical-by-descent (IBD) on a chromosomal segment, (ii) the number of distinct pieces (subsegments) of a chromosomal segment, on each of which exactly 2 (at least 1) haplotypes are shared IBD. The natural counterparts of these statistics for the other relationships are also considered. Relevant Maple codes are provided for a rapid evaluation of the cumulative probabilities of such statistics. The genomic continuum model, with Haldane's model for the crossover process, is assumed.A technology, together with relevant software codes for its automated implementation, are provided for exact evaluation of the distributions of relevant statistics associated with continuous genome data on closely related individuals.Pairs of related individuals, such as full-sibs, are widely used in linkage analysis. Most of linkage tests are based on statistics associated with identity by descent (IBD) data. Evaluation of p-values requires relevant information on the distributions of such statistics. The current biotechnology provides data on very densely packed loci, and therefore, it may provide almost continuous IBD data for pairs of closely related individuals. The distribution theory for statistics on continuous IBD data has not been developed yet. Bickeboller and Thompson [1,2] provide approximations, based on the Poisson clumping heuristic, to the distribution of the proportion of genome shared IBD by half-sibs, while Stefanov [3] provides a methodology for exact evaluation of the cumulative probabilities for the proportion of genome shared IBD by two individuals in grandparent-type relationship. Browning [4,5] suggests a Monte-Carlo approach for s