Generalized Likelihood Ratio test for correlated data

Question

Consider a sequence of random variables $\overline{z} = \{z(k-M+1), ..., z(k)\}$, with probability density depending upon a scalar parameter $\theta$. It is intended to decide between two hypotheses

$$ H_0: \theta = \theta_0 \hspace{3em} for \hspace{3em} k-M+1 \leq i \leq k \\ H_1: \theta = \theta_1 \hspace{3em} for \hspace{3em} k-M+1 \leq i \leq k \tag{1} $$

It is assumed that all the parameters are known except for $\theta_1$ which is replaced by its MLE estimate. Then, the GLR test is

$$L(\overline{z}) = \frac{p\left(\overline{z}, \; \hat{\theta_1}; \; H_1\right)} {p \left(\overline{z}, \; {\theta_0}; \; H_0 \right)} \tag{2}$$

where $\overline{z} = \{z(k-M+1), ..., z(k)\}$ is iid.

Then GLR test can be simplified to

$$L(\overline{z}) = \max\limits_{\theta_1} \left( \sum\limits_{i = k-M+1}^{k} ln \frac{p(z(i)|\theta_1)}{p(z(i)|\theta_0)} \right)\tag{3} $$

But what if $\overline{z} = \{z(k-M+1), ..., z(k)\}$ are correlated. I need to simplify $(2)$ for simplified Gaussian process.

Any help, referred references for correlated GLR test is highly appreciated.

Answer 1

A simple approach would be to transform your observations $\bar{z}$ into independent random variables is by using whitening transformation, i.e., if the covariance matrix $\Sigma$ is known, then find $W$ such that $W^TW = \Sigma^{-1}$. Next, define a new random variable $\bar{y}$ as $W \bar{z}$. The above transformation is only possible when $\Sigma$ is invertible.