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For fixed $n$, the likelihood ratio is strictly decreasing as $t^2$ increases. A likelihood-ratio test rejects for small $\lambda$, so its rejection rule is equivalently large $|t|$. Under the null hypothesis, the standard normal distribution and chi-squared distribution results give independent quantitiesTheir ratio therefore has Student's t-distribution with $n-1$ statistical degrees of freedom. The exact level-$\alpha$ rule isThis is precisely the two-sided Student's t-test; choosing the corresponding cutoff for $\lambda$ yields the same rejection region and the same p-value.

Write $a=x_{(1)}={\min }_i{x}_i$, $T={\sum }_i{x}_i$ and $\overline{x}=T/n$. For nondegenerate data from the shifted exponential distribution, the likelihood function isFor fixed feasible $\theta$, differentiating its log-likelihood with respect to $\beta$ givesThe derivative changes from positive to negative there. Substituting gives the profile likelihood $e^{-n}(\overline{x}-\theta {)}^{-n}$, which increases with $\theta$. Therefore the unrestricted maximum-likelihood estimates arewhile under the null hypothesis $\theta =0$ the estimate is $\tilde{\beta }=\overline{x}$. Their likelihood ratio isSince the likelihood-ratio test rejects for small $R$, the requested form is obtained withThe formulas assume $\overline{x}>a$, which holds almost surely for a continuous sample with $n\ge 2$; coincident observations give an unbounded unrestricted likelihood and can be treated by its supremum.

An exact calibration is available and also explains why Wilks theorem should not be applied here. Under the null, the memoryless property makes $A=n{X}_{(1)}$ and $D={\sum }_i(X_i-X_{(1)})$ independent gamma distributions with common scale $\beta$ and shapes $1$ and $n-1$. By beta sampling by gamma ratios,Consequently, for $0\le k\le 1$,This yields an exact test size $\alpha$, independent of the unknown scale. The support endpoint and boundary null make this a likelihood-ratio test for an exponential support endpoint, with a different calibration from a regular one-parameter test.

Put $A={\sum }_i{X}_i$, $B={\sum }_j{Y}_j$ and $N=n+m$. Independence gives the likelihood functionThe unrestricted maximum-likelihood estimators are $\hat{\theta }=A/n$ and $\hat{\mu }=B/m$. Under the null hypothesis, the common mean $\eta =\theta =\mu$ maximizes ${\eta }^{-N}{e}^{-(A+B)/\eta }$ at $\hat{\eta }=(A+B)/N$. These are maxima because each one-parameter score function has the sign of its observed sum minus its sample size times its proposed mean.

Define the generalized likelihood ratio to be the constrained maximum divided by the unrestricted maximum. The exponential factors cancel, givingSince $T=A/(A+B)$ and $1-T=B/(A+B)$,It has maximum one at $T=n/N$, when the two fitted means agree, and is small toward either endpoint. A likelihood-ratio test rejects for small $\Lambda$, or equivalently large $W=-2\log \Lambda$. This is the likelihood-ratio test for equality of exponential means.

A likelihood-ratio test rejects for small $\lambda$. The derivative of the natural logarithm of its variable factor isThus $t^n(1-t{)}^m$ increases up to $n/(n+m)$ and decreases thereafter, tending to zero at both endpoints. Every nontrivial lower level set of $\lambda$ therefore consists of two tails. For a prescribed test size $0<\alpha <1$, the exact critical region isWriting $F_{n,m}$ for the cumulative distribution function of the beta distribution with parameters $(n,m)$, the required two equations areThe first makes both cutoffs have the same likelihood ratio; the second calibrates the null hypothesis rejection probability. Equivalently, the second left-hand side isAs the common likelihood ratio threshold increases from zero to one, this probability increases continuously and strictly from zero to one; hence the equations determine unique cutoffs. The two tail probabilities need not each equal $\alpha /2$ when $n\ne m$. Exact calibration holds for every common rate because the given null distribution of $T$ is independent of that rate. Boundary equalities have zero probability.

Let $S={\sum }_i{X}_i$, which has a binomial distribution with parameters $(n,p)$. For $0<p_1<1/2$, the alternative-to-null likelihood ratio of the independent Bernoulli trials isSince $p_1/(1-p_1)<1$, this is strictly decreasing in $S$. A threshold on large $R$ is therefore a lower-tail likelihood-ratio test, and a threshold between $R(2)$ and $R(1)$ gives the stated critical region $S<2$. The Neyman-Pearson lemma also proves it is a most powerful test at its attained test size.

Using the binomial probability mass, its test size and statistical power are respectivelyThe associated power function is $\pi (p)=(1-p{)}^n+np(1-p{)}^{n-1}$. These formulas hold for positive integer $n$; when $n=1$ the test always rejects and has size one. If the boundary value $p_1=0$ is admitted, all alternative probability is at $S=0$; the same test still has statistical power one and can be represented using a threshold with ties in the likelihood ratio.

The number of free dimensions remaining after specified constraints often determines a reference sampling distribution. In a full-rank normal linear model with $n$ observations and $p$ mean coefficients, the residual space has dimension $n-p$. A regular nested likelihood-ratio test instead uses the difference in free statistical parameter dimensions. Boundary constraints may invalidate this regular calibration.