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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.

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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.

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  • $\begingroup$ Thanks Maxtron for the answer. However, I need kinda to derive it directly. I mean to derive equivalent of equation 3 for simple case of correlated Gaussian process (only change in mean value). $\endgroup$ – Mahdi Ghe Jan 22 at 13:40

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