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.,

The Chapman–Robbins bound then states that

Note that the denominator in the lower bound above is exactly the -divergence of with respect to .

Relation to Cramér–Rao bound

The expression inside the supremum in 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

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 0040631
  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 0044084

Further reading

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