Say I have the critical region for a test statistic $T$ and some constant $c$, as follows,

$$ n(T - 1)^2 \ge c $$

where $nT \sim \Gamma(n-1, 1)$ and the probability of rejection is $\alpha = P(n(T - 1)^2 \ge c)$.

So I have to prove that the critical region approximates to $2nT \le \chi_{1-\alpha/2}^2(2n-2)$ or $2nT \ge \chi_{\alpha/2}^2(2n-2)$

My try

By CLT, $\sqrt{n}(T -1 ) \overset{d}{\to} N(0,1)$ so $n(T -1)^2\overset{d}{\to} \chi^2(1) $ but I can't proceed.


You should not need to us CLT, since the distribution of the test statistic is already given to you.

Hint --

If $X \sim \Gamma(v/2, 2)$ (shape-scale parameterization), then $X \sim \chi^2_v$.

So, looking at the "Scaling" and "Related distributions" section on this link, assuming shape-scale parameterization, \begin{align*} X &\sim \Gamma(n-1, 1) \\ \Rightarrow 2X & \sim \Gamma(2n-2, 2)\\ \Rightarrow 2X &\sim \chi^2_{2n-2}\,. \end{align*}


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