The Best Linear Unbiased Estimator

When there is no MVU estimator to be found, or that the MVU is too complicated to use in practice, or PDF of data may be unknown. We then find the next best thing, “good enough”.

The BLUE estimator is given by

$$\hat{\theta }=\sum _{n=0}^{N-1}{a}_{n}x[n]$$

Where we find the set of weights $\{a_n\}$ so that we minimize the estimator variance.

Finding the BLUE is a minimization problem

$$a_{min}=\text{arg}\underset{a}{\min }{a}^{T}Ca+\lambda (a^{T}s-1)$$

Giving us

Formula

$\hat{\theta }={\sum }_{n=0}^{N-1}{a}_{n}x[n]=\frac{s^T{C}^{-1}x}{s^T{C}^{-1}s}$

Or on vector form

Formula

$\hat{\theta }=Ax[n]=(H^T{C}^{-1}H{)}^{-1}{H}^T{C}^{-1}x$

Where $C=E\left[(x-E[x])(x-E[x]{)}^T\right]$, and $E[x[n]]=s_n\theta$. $s_n$ is referred to as the scaled mean, and is assumed known.

In the vector case, $C$ is the same, while $H$ is given by $E[x]=H\theta$

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