Approximation of the critical value for $\alpha$ of $\Gamma(n-1,1)$

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.

1 Answer

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*}