Likelihood ratio test always equal to $1$?

Consider a random sample of size $n$ from a discrete uniform distribution with pmf $f(x|N) = 1/N$, where $x=1,\dots,N$. Determine the likelihood ratio test for testing $H_0: N \leq N_0$, $H_1: N > N_0$.

I found the likelihood function is $1/N^n$. The maximum will be at $N=1$. So I believe the likelihood test will be $\Lambda = 1$. Is this correct - if so, what does this mean? It seems that we always will fail to reject $H_0$, which isn't really much of a test is it?

• Normally the likelihood ratio test would test the null hypothesis the the distribution is discrete uniform with N known and fixed versus an alternative that the distribution is some specific nonuniform discrete distribution. x would be the random variable and you would have a sequence of n independent xs all with the same discrete distribution on 1, 2, 3,...,N. Under the null hypothesis the likelihood would be $1/N^n$, But what sort of test are you doing ? Are you taking N to be random too? Commented May 6, 2018 at 4:04
• If you see $x_1=3$ (say), can $N$ actually be $1$? See this related question on MLE, which discusses likelihood for this case. Commented May 6, 2018 at 4:55

The likelihood ratio test (LRT) statistic does not always equal $$1$$.

Likelihood function given the sample $$x=(x_1,\ldots,x_n)\in\{1,2,\ldots,N\}$$ is $$L(N\mid x)=\frac{1}{N^n}I_{\{x_{(n)},x_{(n)}+1,\ldots\}}(N)\quad,\,N\in\{1,2,\ldots\}$$

Here $$x_{(n)}=\max\{x_1,x_2,\ldots,x_n\}$$ is the maximum order statistic as usual.

So the LRT statistic is

\begin{align} \Lambda(x)&=\frac{\sup_{N\le N_0}L(N\mid x)}{\sup_N L(N\mid x)} \\&=\frac{L(\tilde N\mid x)}{L(\hat N\mid x)}\,\,, \end{align}

where $$\hat N=x_{(n)}$$ is the (unrestricted) MLE of $$N$$ and $$\tilde N=\min(\hat N,N_0)$$ is the restricted MLE of $$N$$ when $$N\le N_0$$. Therefore the ratio is

$$\Lambda(x)=\begin{cases}1&,\text{ if }x_{(n)}\le N_0 \\ \frac{L(N_0\mid x)}{L(x_{(n)}\mid x)}&,\text{ if }x_{(n)}>N_0\end{cases}$$

That is,

$$\Lambda(x)=\begin{cases}1&,\text{ if }x_{(n)}\le N_0 \\ 0&,\text{ if }x_{(n)}>N_0\end{cases}$$

$$H_0$$ is trivially accepted when $$\Lambda=1$$. We reject $$H_0$$ for small values of $$\Lambda$$.

So a level $$\alpha$$ likelihood ratio test rejects $$H_0$$ if $$X_{(n)}>N_0$$.

The above test has size $$0$$. If you want the test to attain exact size $$\alpha$$, use the randomized test $$\phi(x)=\begin{cases}1&,\text{ if }x_{(n)}>N_0 \\ \alpha &, \text{ if }x_{(n)}\le N_0\end{cases}$$