The Noisy Secretary Problem and Some Results on Extreme Concomitant Variables

The classical secretary problem for selecting the best item is studied when the actual values of the items are observed with noise. One of the main appeals of the secretary problem is that the optimal strategy is able to find the best observation with the nontrivial probability of about 0.37, even when the number of observations is arbitrarily large. The results are strikingly different when the quality of the secretaries are observed with noise. If there is no noise, then the only information that is needed is whether an observation is the best among those already observed. Since observations are assumed to be i.i.d. this is distribution free. In the case of noisy data, the results are no longer distrubtion free. Furthermore, one needs to know the rank of the noisy observation among those already seen. Finally, the probability of finding the best secretary often goes to 0 as the number of obsevations, n, goes to infinity. The results depend heavily on the behavior of pn, the probability that the observation that is best among the noisy observations is also best among the noiseless observations. Results involving optimal strategies if all that is available is noisy data are described and examples are given to elucidate the results.