# Chapman–Robbins bound

In statistics, the Chapman–Robbins bound or Hammersley–Chapman–Robbins bound is a lower bound on the variance of estimators of a deterministic parameter. It is a generalization of the Cramér–Rao bound; compared to the Cramér–Rao bound, it is both tighter and applicable to a wider range of problems. However, it is usually more difficult to compute.

The bound was independently discovered by John Hammersley in 1950,[1] and by Douglas Chapman and Herbert Robbins in 1951.[2]

## Statement

Let θRn be an unknown, deterministic parameter, and let XRk be a random variable, interpreted as a measurement of θ. Suppose the probability density function of X is given by p(x; θ). It is assumed that p(x; θ) is well-defined and that p(x; θ) > 0 for all values of x and θ.

Suppose δ(X) is an unbiased estimate of an arbitrary scalar function g: RnR of θ, i.e.,

$E\{\delta(X)\} = g(\theta)\text{ for all }\theta.\,$

The Chapman–Robbins bound then states that

$\mathrm{Var}(\delta(X)) \ge \sup_\Delta \frac{\left[ g(\theta+\Delta) - g(\theta) \right]^2}{E_{\theta} \left[ \tfrac{p(X;\theta+\Delta)}{p(X;\theta)} - 1 \right]^2}.$

Note that the denominator in the lower bound above is exactly the $\chi^2$-divergence of $p(\cdot; \theta+\Delta)$ with respect to $p(\cdot; \theta)$.

## Relation to Cramér–Rao bound

The Chapman–Robbins bound converges to the Cramér–Rao bound when Δ → 0, assuming the regularity conditions of the Cramér–Rao bound hold. This implies that, when both bounds exist, the Chapman–Robbins version is always at least as tight as the Cramér–Rao bound; in many cases, it is substantially tighter.

The Chapman–Robbins bound also holds under much weaker regularity conditions. For example, no assumption is made regarding differentiability of the probability density function p(x; θ). When p(x; θ) is non-differentiable, the Fisher information is not defined, and hence the Cramér–Rao bound does not exist.