I know bootstrapping is an asymptotic result (your sample needs to be large enough to look like the actual empirical distribution), but what do you do if when you construct the likelihood of the bootstrap sample, one of the parameters never appears, so it cannot be estimated? In my specific model, I am estimating a strength parameter for each player in a sequence of games, but due to small sample sizes, when I resample with replacement, it is very likely that at least one player will not appear in the new dataset at all, so we can’t provide any estimate of his parameter. I need some thoughts on a valid way to address this.

Note: the actual model is Bayesian and works fine due to the prior information, but I am asked to entertain a frequentist approach and need some way to get at least something that looks like a confidence interval.