Uniformly minimum-variance unbiased estimator
Section of Rao–Blackwell and UMVU, part of Statistical inference
Uniformly minimum-variance unbiased estimator
| Type | Section |
| Taught in | Rao–Blackwell and UMVU |
| Part | Statistical inference |
| Questions using it | 1 |
An estimator U of g(\theta) is a uniformly minimum-variance unbiased estimator, written UMVU, when it is unbiased and, for every other unbiased estimator V and every \theta,
\operatorname{Var}_\theta(U)\le\operatorname{Var}_\theta(V).
A minimum-variance-bound estimator is UMVU, because nothing unbiased can have smaller variance than the bound. The interesting case is a UMVU estimator that does not attain the bound.
Let T be a complete sufficient statistic, and let U(T) be unbiased for \theta with finite variance. Then U(T) is the UMVU estimator of \theta, and it is unique up to a set of probability zero.
Rao–Blackwell reduces the search to functions of T: any unbiased estimator that is not a function of T has a conditional expectation given T with the same mean and a smaller variance. Suppose q(T) is another unbiased function of T. Then \mathbb{E}_\theta\{U(T)-q(T)\}=0 for every \theta. Completeness forces U(T)-q(T)=0 almost surely. That is the Lehmann–Scheffé theorem in the form used here: one unbiased function of a complete sufficient statistic is the only one.
Completeness is essential for uniqueness. Without it, two different functions of a sufficient statistic can share a mean. A curved subfamily of an exponential family is the standard warning, recorded with the definition of completeness.
The normal sample variance shows a UMVU estimator that is not a minimum-variance-bound estimator of the information type for \sigma^2 in the one-parameter sense. For X_i independent \operatorname{N}(\mu,\sigma^2), the complete sufficient statistic for (\mu,\sigma^2) is (\overline X,\sum(X_i-\overline X)^2), and
s^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\overline X)^2
is unbiased for \sigma^2. It is the UMVU estimator. The divisor n has smaller mean squared error, but it is biased, so it is not in the competition this theorem describes.
Used in
- Taught in Rao–Blackwell and UMVU
- Also used in Information and bounds · Large-sample estimation · Exponential families and conjugacy
- Questions that use it (1) — An unbiased Poisson probability
- All 2 questions on Rao–Blackwell and UMVU
See also
Cramér–Rao lower bound · Fisher information · Score function · Rao–Blackwell theorem · Lehmann–Scheffé theorem