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 θ ∈ Rⁿ be an unknown, deterministic parameter, and let X ∈ R^k 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: Rⁿ → R of θ, i.e.,

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

The Chapman–Robbins bound then states that

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

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

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.

See also

• Cramér–Rao bound
• Estimation theory

References

1. Hammersley, J. M. (1950), "On estimating restricted parameters", Journal of the Royal Statistical Society, Series B, 12 (2): 192–240, JSTOR 2983981, MR 40631
2. Chapman, D. G.; Robbins, H. (1951), "Minimum variance estimation without regularity assumptions", Annals of Mathematical Statistics, 22 (4): 581–586, doi:10.1214/aoms/1177729548, JSTOR 2236927, MR 44084

Further reading

• Lehmann, E. L.; Casella, G. (1998), Theory of Point Estimation (2nd ed.), Springer, pp. 113–114, ISBN 0-387-98502-6