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Consider a distribution with parameter $\lambda$ that has density

$$f_\lambda(x)=\frac{x^4}{24\lambda^5}e^{\frac{-x}{\lambda}},x>0$$

Let $X_1,...,X_n$ be $n$ independent random variables drawn from this distribution.

I test the hypotheses $H_0:\lambda=1$ vs $H_0:\lambda≠1$ by using the test

$$\psi=\textbf{1}\big(|\bar{X}_n-5|>C_{a,n}\big)$$

I want to find the smallest threshold $C_{a,n}$ so that the test $\psi$ has asymptotic level $\alpha$. I'm choosing this test so that $C_{a,n}$ does not depend on the estimate of $\lambda$.

How do I compute $C_{a,n}$?

I already computed the MLE which also depends on the sample mean $\bar{X}_n$ but I have no idea how to apply it. I'm used to calculating a test statistic and then finding the critical value or $p$-value in order to reject or fail to reject the null at a given asymptotic level $\alpha$. Here, I don't even know the exact distribution although it looks like it belongs to the canonical exponential family. I also don't know how to calculate the value of $\bar{X}_n$ without any observations it's technically unachievable.

I would really appreciate if someone could help me out.

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    $\begingroup$ By inspection the distribution is in the form of a gamma (the kernel is $x$ to a power times $e^{-x/c}$). It's really useful to be able to spot the 'standard'/common exponential family distributions (gamma, normal, poisson, binomial, inverse-Gaussian, ...) as well as related distributions to those that commonly arise in models and theoretical calculations (chi-squared, beta, F, t, lognormal, logistic, uniform etc). You might find it useful to start with a table of common distributions and their properties (Casella and Berger has a good list if I recall right). Wikipedia's also a handy resource $\endgroup$
    – Glen_b
    Commented Dec 18, 2022 at 23:14
  • $\begingroup$ Yes @Glen_b wikipedia is definitely bookmarked. Any suggestion on how I should tackle this one? If you need any clarification on something that I might have missed like for the other question let me know :) $\endgroup$
    – pecer10012
    Commented Dec 18, 2022 at 23:17
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    $\begingroup$ This is clearly a self-study I thought a hint was in order -- my previous comment comment was the hint. Given the identification as a gamma, it's possible towork out the distribution of $\sum_i X_i$ and hence of $\bar{X}$ -- though you don't need it if all you're concerned with is the asymptotics -- the mean and variance and enough information to verify that a standardized $\bar{X}$ converges to a standard normal would do. $\endgroup$
    – Glen_b
    Commented Dec 18, 2022 at 23:37
  • $\begingroup$ Do you know any theorems relating to distributions of standardized means? $\endgroup$
    – Glen_b
    Commented Dec 19, 2022 at 0:12
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    $\begingroup$ Alternatively, use the Lindeberg-Levy CLT lower down on that Wikipedia page. If you're trying to do problems that involve asymptotics, you need some theorems that deal with asymptotic behavior or you have nowhere to go. Many simple ones will use the law of large numbers, the central limit theorem and/or Slutsky's theorem. They each have wikipedia pages. It might also be worth checking the article relating to convergence en.wikipedia.org/wiki/Convergence_of_random_variables . If you don't have a suitable text go here: projects.iq.harvard.edu/stat110/home and click "Book". Free! $\endgroup$
    – Glen_b
    Commented Dec 19, 2022 at 10:53

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