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I have the following two hypotheses:

$\hspace{5cm}\mathcal{H}_0: y=w\\\hspace{5cm}\mathcal{H}_1: y=\sum_{i=1}^{N}h_ix_i+w$

Here $w\sim \mathcal{N}(0,1)$ represents Gaussian noise. $x_i \sim Bern(p), \forall i$ are i.i.d Bernoulli random variables.

Furthermore, $h_i \sim \mathcal{N}(0,\sigma^2), \forall i$ are i.i.d Gaussian random variables.

My aim: Derive a testing scheme to distinguish the 2 hypotheses based on the observation $y$?

My approach: I know $f(y|\mathcal{H}_0)=\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}$. Now, If I can compute $f(y|\mathcal{H}_1)$, then I could compute their ratios and perform the Likelihood Ratio Test.

But, I am unable to compute $f(y|\mathcal{H}_1)$ explicitly. I was able to derive an expression involving $N$ Gaussians and 1 impulse using the idea here. But this is useless to me as for large $N$ values the likelihood ratio is hard to compute.

Questions:

Can someone help me derive a good testing rule to distinguish between the 2 Hypotheses?

If my approach is the only way to go about this, can someone help me compute $f(y|\mathcal{H}_1)$ and perhaps simplify it for large $N$ values.

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  • $\begingroup$ Are $N$, $p$, and $\sigma$ known in advance? $\endgroup$
    – Matt F.
    Jun 17 at 14:17
  • $\begingroup$ @MattF. Yes, they are known in advance. $\endgroup$
    – wanderer
    Jun 19 at 14:49
  • $\begingroup$ And by “the observation $y$“ you mean a single observation or a set of observations? $\endgroup$
    – Matt F.
    Jun 19 at 20:18
  • $\begingroup$ @MattF. Here, I mean a single observation. I was hoping that if I could do this for single observation case, then I can extend it to a multiple observation case. In short, either is fine for me. $\endgroup$
    – wanderer
    Jun 20 at 3:54
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The variance under $H_1$ exceeds the variance under $H_0$. Extreme values for $y$ are less likely under $H_0$ than under $H_1$.

Reject $H_0$ in favor of the alternative $H_1$ if you observe an extreme value $y$.

The distribution of $y$ under $H_0$: $\quad P_{H_0}(y) \sim N(0,1)$.

Pick a false negative error rate $\alpha$ you can live with. Reject $H_0$ when $|y|$ exceeds a threshold $\theta$,
$$P_{H_0}(y < -\theta)+P_{H_0}(y > +\theta) = P( |y| > \theta) = \alpha.$$

Reject $H_0$ if $|y| > \theta = \Phi^{-1}(\frac{\alpha}{2})$, where $\Phi^{-1}()$ is the inverse cumulative normal function.

For example, specifying $\alpha = 0.01$, reject $H_0$ if $|y| > 2.58$.

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