# A Proof of Tukey's Inequality

Suppose that $W_1,W_2,...,W_n$ are uncorrelated unbiased estimators of a parameter $\theta$.

Consider $W=\sum_{i=1}^na_iW_i$ such that $E(W)=\theta$ and $Var(W_i)=\sigma^2_i$, where the $a_i$'s are constants. So it is trivial to see that $\sum_{i=1}^na_i=1$. Call all such unbiased estimators $W$ "linear unbiased estimators" of $\theta$.

Let $W^{\star}$ be the linear unbiased estimator having minimum variance. It is easy to see, by a simple application of Cauchy-Schwarz Inequality, that $W^{\star}=\dfrac{\sum_{i=1}^nW_i/\sigma_i^2}{\sum_{i=1}^n(1/\sigma_i^2)}$ and $Var(W^{\star})=\dfrac{1}{\sum_{i=1}^n\dfrac{1}{\sigma_i^2}}$. The corresponding optimal weights are $a_i^{\star}=\dfrac{\dfrac{1}{\sigma_i^2}}{\sum_{i=1}^n\dfrac{1}{\sigma_i^2}}$.

Let $W'$ be any other linear unbiased estimator of $\theta$ such that $a_i\geq0$ for all $i\in\{1,2,...,n\}$.

Then Tukey's Inequality is: $\dfrac{Var(W')}{Var(W^{\star})}\leq\dfrac{1}{1-\lambda^2}$ where $\dfrac{1+\lambda}{1-\lambda}=\dfrac{b_{\max}}{b_{\min}}$, with $b_{\max}=\max_i(\dfrac{a_i}{a_i^{\star}})$ and similarly $b_{\min}=\min_i(\dfrac{a_i}{a_i^{\star}})$, $i\in\{1,2,...,n\}$.

How can this inequality be proved?