Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation.[1] The theorem states that any estimator which is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann and Henry Scheffé, given their two early papers.[2][3]

If T is a complete sufficient statistic for θ and E(g(T)) = τ(θ) then g(T) is the uniformly minimum-variance unbiased estimator (UMVUE) of τ(θ).

Statement

Let $\displaystyle{ \vec{X}= X_1, X_2, \dots, X_n }$ be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) $\displaystyle{ f(x:\theta) }$ where $\displaystyle{ \theta \in \Omega }$ is a parameter in the parameter space. Suppose $\displaystyle{ Y = u(\vec{X}) }$ is a sufficient statistic for θ, and let $\displaystyle{ \{ f_Y(y:\theta): \theta \in \Omega\} }$ be a complete family. If $\displaystyle{ \varphi:\operatorname{E}[\varphi(Y)] = \theta }$ then $\displaystyle{ \varphi(Y) }$ is the unique MVUE of θ.

Proof

By the Rao–Blackwell theorem, if $\displaystyle{ Z }$ is an unbiased estimator of θ then $\displaystyle{ \varphi(Y):= \operatorname{E}[Z\mid Y] }$ defines an unbiased estimator of θ with the property that its variance is not greater than that of $\displaystyle{ Z }$.

Now we show that this function is unique. Suppose $\displaystyle{ W }$ is another candidate MVUE estimator of θ. Then again $\displaystyle{ \psi(Y):= \operatorname{E}[W\mid Y] }$ defines an unbiased estimator of θ with the property that its variance is not greater than that of $\displaystyle{ W }$. Then

$\displaystyle{ \operatorname{E}[\varphi(Y) - \psi(Y)] = 0, \theta \in \Omega. }$

Since $\displaystyle{ \{ f_Y(y:\theta): \theta \in \Omega\} }$ is a complete family

$\displaystyle{ \operatorname{E}[\varphi(Y) - \psi(Y)] = 0 \implies \varphi(y) - \psi(y) = 0, \theta \in \Omega }$

and therefore the function $\displaystyle{ \varphi }$ is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that $\displaystyle{ \varphi(Y) }$ is the MVUE.

Example for when using a non-complete minimal sufficient statistic

An example of an improvable Rao–Blackwell improvement, when using a minimal sufficient statistic that is not complete, was provided by Galili and Meilijson in 2016.[4] Let $\displaystyle{ X_1, \ldots, X_n }$ be a random sample from a scale-uniform distribution $\displaystyle{ X \sim U ( (1-k) \theta, (1+k) \theta), }$ with unknown mean $\displaystyle{ \operatorname{E}[X]=\theta }$ and known design parameter $\displaystyle{ k \in (0,1) }$. In the search for "best" possible unbiased estimators for $\displaystyle{ \theta }$, it is natural to consider $\displaystyle{ X_1 }$ as an initial (crude) unbiased estimator for $\displaystyle{ \theta }$ and then try to improve it. Since $\displaystyle{ X_1 }$ is not a function of $\displaystyle{ T = \left( X_{(1)}, X_{(n)} \right) }$, the minimal sufficient statistic for $\displaystyle{ \theta }$ (where $\displaystyle{ X_{(1)} = \min_i X_i }$ and $\displaystyle{ X_{(n)} = \max_i X_i }$), it may be improved using the Rao–Blackwell theorem as follows:

$\displaystyle{ \hat{\theta}_{RB} =\operatorname{E}_\theta[X_1\mid X_{(1)}, X_{( n)}] = \frac{X_{(1)}+X_{(n)}} 2. }$

However, the following unbiased estimator can be shown to have lower variance:

$\displaystyle{ \hat{\theta}_{LV} = \frac 1 {k^2\frac{n-1}{n+1}+1} \cdot \frac{(1-k)X_{(1)} + (1+k) X_{(n)}} 2. }$

And in fact, it could be even further improved when using the following estimator:

$\displaystyle{ \hat{\theta}_\text{BAYES}=\frac{n+1} n \left[1- \frac{\frac{X_{(1)} (1+k)}{X_{(n)} (1-k)}-1}{ \left (\frac{X_{(1)} (1+k)}{X_{(n)} (1-k)}\right )^{n+1} -1} \right] \frac{X_{(n)}}{1+k} }$

The model is a scale model. Optimal equivariant estimators can then be derived for loss functions that are invariant.[5]