How to show that likelihood ratio test statistic for exponential distributions' rate parameter $\lambda$ has $\chi^2$ distribution with 1 df? Let $X_1, X_2, ..., X_n$ be iid. random variables with pdf. exponential. Let $H_0 = \theta = \theta_0 $ and $H_1 = \theta = \theta_1$. Then test statistic for likelihood ratio would be:
$$ LR = \frac{f(X_1, X_2, ..., X_n \mid\theta_0)}{f(X_1, X_2, ..., X_n \mid\theta_1)} = \frac{\theta_0^{-n}\exp \{-\sum_i  X_i/\theta_0 \}}{\theta_1^{-n}\exp \left\{-\sum_i  X_i/\theta_1 \right\}} = \left(\frac{\theta_0}{\theta_1}\right)^{-n}e^{\left(\frac{1}{\theta_1} - \frac{1}{\theta_0}\right) \sum_{i} X_{i}}$$
How to show that $LR \rightarrow \chi^2_1 \ $ ? Thanks in advance!
 A: Let $x \mapsto f(x \mid \theta)$ be a parametrised density function on a sample space $\mathrm{X}$ (so that $x$ are the possible observations) and with parameter $\theta$ assumed a priori to lie in some $\Omega_0 \subset \mathbf{R}^{d_0}.$ (Do not confuse this with a priori Bayesian distributions.) Suppose that $n$ independent samples are taken and write
$$
f_n(\mathbf{x}_n \mid \theta) = f_n(x_1, \ldots, x_n \mid \theta) = \prod_{k = 1}^n f(x_k \mid \theta)
$$
for the likelihood of the sample.
Suppose $\Omega_1$ is an open set such that $\Omega_0 \subset \Omega_1$ and we further assume that $\Omega_0$ is embedded in a $d_0$-dimensional subspace of $\mathbf{R}^{d_1}$ ($d_0 = 0$ not excluded, meaning $\Omega_0$ is a single point). Let
$$
\ell_n^*(\mathbf{x}_n \mid \mathrm{H}_i) = \sup_{\theta \in \Omega_i} \log f_n(\mathbf{x}_n \mid \theta)
$$
the log-likelihood optimised over the hypothesis $\mathrm{H}_i$ that $\theta \in \Omega_i.$
Wilk's theorem: Suppose that $f$ satisfy some measure-theoretic regularity conditions, and that for each $n,$ $\theta \mapsto f_n(\mathbf{x}_n \mid \theta)$ has maximimisers
$$
\theta_1^* = \theta_{1,n}^*(\mathbf{x}_n)
$$ in the interior of $\Omega_1,$ and
$$
\theta_0^* = \theta_{0,n}^*(\mathbf{x}_n)
$$
the interior of $\Omega_0$ relative to the subspace it is embbed into.
In other words, we assume that
$$
\ell_n^*(\mathbf{x_n} \mid \mathrm{H}_i) = \log f_n(\mathbf{x}_n \mid \theta_i^*).
$$
Then, as $n \to \infty$ ("as the number of sampled observations increases unlimitedly")
$$
-2\log \dfrac{f_n(\mathbf{x}_n \mid \theta_0^*)}{f_n(\mathbf{x}_n \mid \theta_1^*)} = 2 \big( \ell_n^*(\mathbf{x}_n \mid \mathrm{H}_1) - \ell_n^*(\mathbf{x}_n \mid \mathrm{H}_0) \big) \to \chi^2_{d_1 - d_0}
$$
Notes:

*

*Wilk's theorem uses the ratio of likelihoods principle. The likelihood of a sample is a metric that, well, estimates how likely is to see what we saw if the parameter we assume is $\theta.$ The ratio of likelihoods (likelihood ratio) $\dfrac{f_n(\mathbf{x}_n \mid \theta_0^*)}{f_n(\mathbf{x}_n \mid \theta_1^*)}$ is always $\leq 1,$ so the log-likelihood ratio is therefore $\geq 0.$ The likelihood ratio principle state that if the likelihood ratio is too small (equiv. the log-likelihood is too big) then the hypothesis $\mathrm{H}_0$ is incompatible with the data and as such there is evidence suggesting that $\mathrm{H}_1$ holds. How unlikely depends on the actual density $f,$ however, Wilk's theorem states that for large samples, we do not depend (thankfully) on $f$ but on the $\chi^2$-disitribution (so, Wilk's theorem is sort of a central limit theorem for the likelihood ratio principle).


*Wilk's theorem can be extended to the case where $\Omega_0$ is a single point (i.e. specifies the parameter exactly).


*I am not sure if Wilk's theorem has an extension (or what would that extension even mean) if you specify both parameters. The usual proof uses that $\Omega_0 \subset \Omega_1$.


*To use Wilk's theorem, you need to find the ratio of the likelihoods evaluated at the optimisers of both hypotheses.


*The degrees of freedom will then be the difference in dimensions of the two hypothesis, namely $d_1 - d_0.$ When you specify both hypothesis to be a single parameter, you will reach a zero, so certainly something is wrong with your formulation.


*Check the following link too https://stats.stackexchange.com/questions/101520/what-are-the-regularity-conditions-for-likelihood-ratio-test
