As is well known, the **Cramér-Rao bound** (or **information inequality**) sets a lower bound on the variance of estimators of a parameter.

Consider the case when the parameter is a **scalar**, the estimator is **unbiased**, and sample size is **fixed**. From the Wikipedia, suppose $\theta$ is an unknown (deterministic) parameter which is to be estimated from a vector of (random) measurements $x$, distributed according to some probability density function $f(x;\theta)$. Then the variance of any unbiased estimator $\hat{\theta}$ of $\theta$ is bounded by the reciprocal of the **Fisher information**.

My focus is on the case when the measurements $x$ are $n$ **i.i.d Bernoulli** variables with parameter $p$, and $p$ which is to be estimated from $x$. In this case, the bound becomes
$$
\mathrm{Var}(\hat p) \geq \frac{p(1-p)}{n}.
$$

There's also a **sequential** version of the Cramér-Rao bound, in the which sample size $n$ is a random **stopping time** with respect to a sequence of observations $x$. See for example Wolfowitz, 1957. The bound holds with sample size replaced by its expected value $\mathrm E[n]$, under some regularity conditions. So, in the Bernoulli case,
$$
\mathrm{Var}(\hat p) \geq \frac{p(1-p)}{\mathrm E[n]}.
$$

Consider now that, in the sequential Bernoulli case, $\hat p$ is a function not only of the observations $x$, but also of $y$, where $y$ is a set of **auxiliary random variables** independent of $x$. The estimator is then **randomized**, in the sense that given $x$ the value of $\hat p$ is still random, by virtue of $y$. The variance of $\hat p$ is defined with respect to both $x$ and $y$.

My question: **Does the Cramér-Rao bound hold for randomized estimators?** (i.e. when the estimator depends on $y$ in addition to $x$) I'm interested in the Bernoulli case, unbiased estimator; ideally in a sequential estimation setting, but a fixed-size result would also be interesting.

If this hasn't been studied before, or no result is known, could you point me in an appropriate direction to establish a randomized-estimator version of the Cramer-Rao bound?