# Terminology for sequential estimation with randomized stopping time

Consider a sequence of i.i.d Bernoulli RV's $\{X_i\}_{i \in \mathbb N}$ with parameter $p$. Based on this sequence we build a sequential estimator of $p$ (using inverse binomial sampling for example). This means there's a stopping time $N$ on the sequence, and the estimate $\hat p$ depends on the values of the sequence up to the stopping time, $X_1$, $X_2$, ... $X_N$.

If the estimate depends on some auxiliary random variables independent from $X_i$ (say you flip a coin and based on that you change the third decimal of $\hat p$), I usually say that the estimator $\hat p$ is randomized. (Is that the standard name?)

Now assume that the stopping time $N$ depends on auxiliary random variables too. I'd say the stopping time is randomized. How to call the sequential procedure in that case? If I say "randomized sequential procedure" it may give the impression that $\hat p$, and not $N$, is randomized. I'd like to say something like "randomized-stopping time, randomized-estimator sequential procedure". Is there any term for that, or what would you call it?