The random threshold and the bisection

A random threshold is a random variable that cannot be observed directly, but only by means of a response/no-response experiment. Bisection is the well known method to find the value of a constant threshold with minimum cost. When the threshold is random, an infinite number of bisection steps yields a random variable whose distribution is generally singular. If the random threshold lies outside the initial interval with probability zero, then the result or the bisection procedure can be considered as a biased estimator of the mean of the random threshold.