# Cramér-Rao bound for randomized estimator

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?

Let $\hat \theta$ be an estimator of the parameter $\theta$. The estimate is obtained as a (deterministic) function of $n$ observations $x_1, \ldots, x_n$ and $m$ auxiliary random variables $y_1, \ldots, y_m$. $\hat \theta$ is a randomized estimator, in the sense that the estimate depends not only on the observations $x_1, \ldots, x_n$ but also on the random variables $y_1, \ldots, y_m$.
Consider first the case $m=1$. From the law of total variance, $$\mathrm{Var}[\hat \theta] = \mathrm E_{y_1}(\mathrm{Var}[\hat \theta \,|\, y_1]) + \mathrm{Var}_{y_1}(\mathrm E[\hat \theta \,|\, y_1]).$$ The term $\mathrm{Var}[\hat \theta \,|\, y_1]$ is the variance of a deterministic estimator obtained from conditioning on $y_1$; and therefore satisfies the Cramér-Rao bound with respect to the observations $x_1, \ldots, x_n$. Thus $\mathrm E_{y_1}(\mathrm{Var}[\hat \theta \,|\, y_1])$ also satisfies the Cramér-Rao bound. The second summand on the right-hand side is clearly non-negative. Consequently, $\mathrm{Var}[\hat \theta]$ satisfies the Cramér-Rao bound.
For $m>1$ the reasoning is similar. The formula for the (total) variance now contains more terms. The first is the expected value of the conditional variance, $\mathrm E_{y_1,\ldots,y_m}(\mathrm{Var}[\hat \theta \,|\, y_1,\ldots,y_m])$, for which the standard Cramér-Rao bound applies; and the remaining terms are again non-negative.
• There is a simpler way. Marginalize out the $y$ vector. You then have a generative model. $p \rightarrow x \rightarrow \hat \theta$. The Cramer-Rao, which be though of as a data processing inequality, applies and gives you your result. (also, IMO it's best to apply CRao to all estimators, even the biased ones, but with the corrected formula) Aug 10, 2015 at 8:41