  The problem of estimating the probability p=P(g(X<0) is considered when X
represents a multivariate stochastic input of a monotone function g. First, a
heuristic method to bound p is formally described, involving a specialized
design of numerical experiments. Then a statistical estimation of p is
considered based on a sequential stochastic exploration of the input space. A
maximum likelihood estimator of p based on successive dependent Bernoulli data
is defined and its theoretical convergence properties are studied. Under
intuitive or mild conditions, the estimation is faster and more robust than the
traditional Monte Carlo approach, therefore adapted to time-consuming computer
codes g. The main result of the paper is related to the variance of the
estimator. It appears as a new baseline measure of efficiency under monotone
constraints, which could play a similar role to the usual Monte Carlo estimator
variance in unconstrained frameworks. Furthermore the bias of the estimator is
shown to be corrigible via bootstrap heuristics. The behavior of the method is
illustrated by numerical tests led on a class of toy examples and a more
realistic hydraulic case-study.
  Keywords : monotone function, deterministic computer codes, Monte Carlo
acceleration, failure probability
