BLUE in correlated case

Question

Let $X_1,...,X_n$ be real-valued observations such that $E(X_j)=\mu$, $V(X_j)=\sigma^2$ and the correlation coefficient between $X_i$ and $X_j$ is $r_{ij}$, $1 \le i < j \le n$. Assume that $r_{ij}$'s are known. Derive the Best Linear Unbiased Estimator (BLUE) $\hat{\mu}$ of $\mu$.

My approach so far

Suppose $T = \sum_{j=1}^{n} c_j X_j$ is the BLUE. Then $E(T)=\mu \implies \sum_{j=1}^{n} c_j =1$.

Also, we need to minimize $V(T) = \sum_{j=1}^{n} c_j ^2 V(X_j) + 2 \sum \sum_{i<j} c_i c_j \sigma^2 r_{ij}$ with respect to $c_1,...,c_n$.

I tried Lagrange Multipliers but the entire thing becomes too much tedious and I could not get a closed-form solution for the $c_j $'s. Is there any other approach to this?

Answer 1

Suppose $\boldsymbol X=(X_1,X_2,\ldots,X_n)'$, $\boldsymbol c=(c_1,c_2,\ldots,c_n)'$ and $\boldsymbol 1=(1,1,\ldots,1)'$. And let $\Sigma$ be the positive definite dispersion matrix of $\boldsymbol X$.

Then $T=\boldsymbol c'\boldsymbol X$, so the problem is to minimize $\operatorname{Var}(T)=\boldsymbol c'\Sigma \boldsymbol c$ with respect to $\boldsymbol c$ subject to $\boldsymbol c'\boldsymbol 1=1$.

Using a generalization of Cauchy-Schwarz inequality, one has

$$(\boldsymbol c'\Sigma \boldsymbol c)(\boldsymbol1'\Sigma^{-1}\boldsymbol 1)\ge (\boldsymbol c'\boldsymbol 1)^2=1$$

Here equality holds if and only if $\boldsymbol c\propto \Sigma^{-1}\boldsymbol 1$, from which the optimal choice of $\boldsymbol c$ is

$$\widehat{\boldsymbol c}=\frac{\Sigma^{-1}\boldsymbol 1}{\boldsymbol1'\Sigma^{-1}\boldsymbol 1}$$

BLUE of $\mu$ is therefore

$$\hat \mu=\widehat{\boldsymbol c}'\boldsymbol X=\frac{\boldsymbol 1'\Sigma^{-1}\boldsymbol X}{\boldsymbol1'\Sigma^{-1}\boldsymbol 1}$$