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]
Let be the set of parameters for a family of probability distributions .
For any two , let be the -divergence from to . Then
Theorem — Given any scalar function on the parameter , and any two , we have .
A generalization to the multivariable case is [3]
Theorem — Given any multivariate function on the model , and any ,
By the variational representation of chi-squared divergence, [3]
Plug in , to obtain
Switch the denominator and the left side, and take supremum over to obtain the single-variate case. For the multivariate case, we define for any . Then plug in in the variational representation to obtain
Take supremum over , using the linear algebra fact that , we obtain the multivariate case.
The expression inside the supremum in the Chapman–Robbins bound converges to the Cramér–Rao bound when , 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.
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