Informal explanation: In the course of my research I've run into the following problem: I am observing a machine that outputs random numbers. Most (if not all) of these random numbers come from the Gaussian distribution $\mathcal{N}(0,\sigma_0^2)$. I know the value of the variance $\sigma_0^2$. However, some proportion of the numbers I observe may come from another distribution $A$. I know that the mean of $A$ is zero, and that $A$ is symmetric (not sure if symmetry helps us). I also know that its variance is $\sigma_a^2>\sigma_0^2$, but I do not know the exact value of $\sigma_a^2$ (unlike $\sigma_0^2$.) The presence of random numbers from $A$ indicates a problem with the machine I am watching. Note that even in that case, my machine will mostly output random values from $\mathcal{N}(0,\sigma_0^2)$.
I would like to construct a hypothesis test on the set $\mathbf{X}$ of $n$ observations I have collected from my machine to determine whether my machine is malfunctioning. Assume that $n$ is large. Null hypothesis is that machine operates normally (i.e. $x_i\sim \mathcal{N}(0,\sigma_0^2)$ where $x_i\in\mathbf{X}$). Alternate hypothesis is that machine is malfunctioning (that is, $x_i \sim \mathcal{N}(0,\sigma_0^2)$ with $x_i\in \mathbf{X}_N$, and $y_i\sim A$ with $y_i\in \mathbf{X}_A$, where $\mathbf{X}_N\cup \mathbf{X}_A=\mathbf{X}$ and $\mathbf{X}_N\cap \mathbf{X}_A=\emptyset$.)
My question is: what is the proportion $\pi$ of observations generated by $A$ in my set of observations $\mathbf{X}$ so that big given enough size of that set $n$, I achieve arbitrarily small type I and type II errors? I.e. how many out of $n$ observations in $\mathbf{X}$ have to come from $A$ for me to determine with negligible error that my machine is broken?
My intuition, based on the CLT, suggests that that if more than $\sqrt{n}$ observations come from $A$, then null hypothesis will be rejected with vanishingly small sum of errors for large $n$. Is my intuition correct? If so, how can I back it up? If not, why? Does it depend on the form of $A$ (what if I told you that $A$ is the sum of $\mathcal{N}(0,\sigma_0^2)$ and another, possibly Gaussian, distribution?)
Attempt at a formal statement of the problem and conjecture: For a zero-mean symmetric mixture distribution $(1-\pi)\mathcal{N}(0,\sigma_0^2)+\pi A(\sigma_a^2)$, I construct a test between two hypotheses:
$$\begin{align} H_0:&\pi=0\\ H_1:&\pi\neq 0 \end{align}$$
with $0\leq\pi\leq1$, $\sigma_a^2>\sigma_0^2$, with $\sigma_0^2$ known. Null hypothesis states that the distribution is a known "pure" Gaussian, while the alternate hypothesis is that it's a Gaussian "contaminated" with $A$.
For that test I need to characterize the value of $\pi$, in terms of $n$, that allows me to separate the two hypotheses. I conjecture that for $\pi>1/\sqrt{n}$ I can reject $H_0$ with arbitrarily small sum of probabilities of type I and type II errors given arbitrarily large (but finite) number of observations $n$. I am ok with $n$ depending on $\sigma_0^2/\sigma_a^2$ -- I assume that those are given to me and do not change as sampling progresses (I think given large enough $n$, whether I can separate the hypothesis depends only on $\pi$, but please correct me if I am wrong.)
I think someone in the statistics community has to have looked at this, and the answer is out there, as this seems like a problem that would come up fairly often. Unfortunately, I've started getting into mathematical statistics very recently: my background is computer science with some graduate-level course work on probability theory, signal theory, and information theory. I would appreciate any help you have to offer as I learn this.
Attempt at solution: I have tried the following track for solving this: suppose I collect $n$ observations in $\mathbf{X}$ and use chi-square test for testing the following hypothesis:
$$\begin{align} H_0:&\sigma^2=\sigma_0^2\\ H_1:&\sigma^2>\sigma_0^2\\ \end{align}$$
If $\sqrt{n}$ observations from $A$ separate the hypotheses for this test, then I've got my proof.
I can compute the statistic for the test (sample variance times $(n-1)/\sigma_0^2$). Given a very large $n$ that I have, it becomes normally-distributed by CLT. Now, let's assume that out of $n$ observations, $\sqrt{n}$ came from $A$ (rest are from $\mathcal{N}(0,\sigma_0^2)$. My test statistic then is a sum of two random variables: the sum of squares of $\sqrt{n}$ values coming from $A$ and the sum of squares of $n-\sqrt{n}$ values coming from $\mathcal{N}(0,\sigma_0^2)$. These two random variables are also (approximately) normally-distributed (very large $n$). Then I think I can look at the performance (i.e. errors) of the test, given the distribution of the test statistic -- I think I need to bound it asymptotically. Unfortunately, the Chernoff and Chebyshev bounds seem to be in the "wrong" direction. That's where I'm stuck.
Am I on the right path? Did I make mistake(s) in my assumptions? Any ideas on how to proceed from here?