= Likelihood-ratio test for an exponential support endpoint
{title2=$R=(1-X_{(1)}/\overline X)^n$}
{title2=$\frac{n-1}{n}(-2\log R)\sim\chi_2^2\text{ under }H_0$}
Let $X_i=\theta+E_i$ for $n\geq2$ <independent and identically distributed> observations, with $E_i$ having an <exponential distribution> of unknown scale $\beta>0$. Test the <null hypothesis> $\theta=0$ within $\theta\geq0$. For nondegenerate data, the <likelihood function> is
$$
L(\theta,\beta)=\beta^{-n}\exp[-n(\overline X-\theta)/\beta]
\mathbf1_{\{0\leq\theta\leq X_{(1)}\}}.
$$
Maximizing over the scale gives $\widehat\beta(\theta)=\overline X-\theta$, and the resulting profile <likelihood> increases with $\theta$. Thus the unrestricted <maximum-likelihood estimates> are $\widehat\theta=X_{(1)}$ and $\widehat\beta=\overline X-X_{(1)}$, while the null estimate is $\widetilde\beta=\overline X$. The <likelihood ratio> is consequently
$$
\boxed{R=\left(1-\frac{X_{(1)}}{\overline X}\right)^n.}
$$
Under the null, put $A=nX_{(1)}$ and $D=\sum_i(X_i-X_{(1)})$. The <memoryless property> gives independent <gamma distributions> of shapes $1$ and $n-1$ and common scale $\beta$. For a direct verification, fix which observation is the minimum and write its value as $z$ and the other observations as $z+e_j$. Their joint density is proportional to $e^{-nz/\beta}\prod_j e^{-e_j/\beta}$, which factors into an exponential density for the minimum and independent exponential densities for the excesses. The minimum's index does not affect the resulting laws of $A,D$.
By <beta sampling by gamma ratios>, $D/(A+D)\sim\operatorname{Beta}(n-1,1)$. Therefore
$$
\Pr_0(R\leq k)=k^{(n-1)/n},\qquad0\leq k\leq1.
$$
An exact <test size> $\alpha$ is obtained by rejecting for $R\leq\alpha^{n/(n-1)}$. Equivalently,
$$
\boxed{\frac{n-1}{n}(-2\log R)\sim\chi_2^2.}
$$
Although the alternative adds one endpoint parameter, a regular one-parameter <Wilks theorem> calibration is incorrect: the support depends on the endpoint, and the null lies on its boundary. In particular, $-2\log R$ tends to a <chi-squared distribution> with two <statistical degrees of freedom>, not one.
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