Constant time estimation of ranking statistics by analytic combinatorics
Résumé
We consider i.i.d. increments (or jumps) X_i that are integers in [-c,...,+d], the partial sums S_j, and the discrete walks (j,S_j) with 1 <= j <= n. Late conditioning by a return of the walk to zero at time n provides discrete bridges that we note B_j with 1<= j <= n. We give in this extended abstract the asymptotic law in the central domain of the height max_{1<= j <= n}B_j of the bridges as n tends to infinity. As expected, this law converges to the Rayleigh law which is the law of the maximum of a standard Brownian bridge. In the case where c=1 (only one negative jump), we provide a full expansion of the asymptotic limit which improves upon the rate of convergence O(log(n)/sqrt(n)) given by Borisov (78) for lattice jumps; this applies in particular to the case where X_i is in {-1,+d}, in which case the expansion is expressible as a function of n, d and of the height of the bridge. Applying this expansion for X_i in {-1,d/c} gives an excellent approximation of the case X_i in {-d,+c} and provides in constant time an indicator used in ranking statistics; this indicator can be used for medical diagnosis and bioinformatics analysis (see Keller et al. (2007)) who compute it in time O(n min(c,d)) by use of dynamical programming).
Domaines
Mathématique discrète [cs.DM]Origine | Fichiers produits par l'(les) auteur(s) |
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