Estimation of binomial parameters when both , are unknown

We revisit the classic problem of estimation of the binomial parameters when both parameters n,p are unknown. We start with a series of results that illustrate the fundamental difficulties in the problem. Specifically, we establish lack of unbiased estimates for essentially any functions of just n or just p. We also quantify just how badly biased the sample maximum is as an estimator of n. Then, we motivate and present two new estimators of n. One is a new moment estimate and the other is a bias correction of the sample maximum. Both are easy to motivate, compute, and jackknife. The second estimate frequently beats most common estimates of n in the simulations, including the Carroll–Lombard estimate. This estimate is very promising. We end with a family of estimates for p; a specific one from the family is compared to the presently common estimate and the improvements in mean-squared error are often very significant. In all cases, the asymptotics are derived in one domain. Some other possible estimates such as a truncated MLE and empirical Bayes methods are briefly discussed.