Using large order statistics of runs for tracking a changing Bernoulli probability

We discuss the advantages of using estimators based on large order statistics of the runs of 0's and 1's in the estimation of the success probability associated with a sequence of independent Bernoulli trials, when this probability might be changing. Through theoretical arguments as well as Monte Carlo simulations, we show that appropriate linear combinations of these statistics offer the ability of following, relatively rapidly, the underlying probability when it is changing monotonically. In order to define our estimator, we introduce a coefficient that can be used in testing the null hypothesis that the underlying success probability has remained constant throughout the sequence of independent Bernoulli trials.