Question: There is a riddle which I heard a while ago and haven't came up with a satisfying approach yet.

Say a book is written by person $A$, two other people, $B$ and $C$ review the text and each of them find 40 and 30 errors respectively and among their founded errors, there are 15 common words. Estimate the true number of errors in the book with this given information.

I know there might be various approaches, so any ideas would be appreciated.

Answer: 
From the Wikipedia page for Wilks' theorem:

> 
> The theorem no longer applies when any one of the estimated parameters is at its upper or lower limit: Wilks’ theorem assumes that the ‘true’ but unknown values of the estimated parameters lie within the interior of the supported parameter space. The likelihood maximum may no longer have the assumed ellipsoidal shape if the maximum value for the population likelihood function occurs at some boundary-value of one of the parameters, i.e. on an edge of the parameter space. In that event, the likelihood test will still be valid and optimal as guaranteed by the Neyman-Pearson lemma,[2] but the significance (the p-value) can not be reliably estimated using the chi-squared distribution with the number of degrees of freedom prescribed by Wilks.
> 
> 
> 

In this case the estimated parameter $b$ is at its lower limit $x\_{\left(n\right)}$, as the likelihood is zero for $b < x\_{\left(n\right)}$.

Other proofs such as the asymptotic normality of the MLE also rely on the assumption that the true value of the parameter lies within the parameter space. So $\hat{b}$ is not asymptotically normal either.
### Wilks' theorem does not apply here

Before getting to Wilks' theorem, one problem here (as with many questions about uniform distributions on this site) is that you have not taken account of the support of the uniform distribution in your mathematics, and this leads you to an incorrect form for the likelihood-ratio statistic. You didn't specify your alternative hypothesis, but I'm going to assume it is $H\_\text{A}:b>1$ (i.e., a one-sided test). Given data $\mathbf{x} = (x\_1,...,x\_n)$ the likelihood function is:

$$L\_\mathbf{x}(b) = \frac{\mathbb{I}(x\_{(n)} \leqslant b)}{b^n} 
\quad \quad \quad \quad \quad \text{for all } b > 0,$$

so the maximised likelihood over any parameter subspace $\mathscr{B} \subseteq (0, \infty)$ is:

$$\begin{align}
\hat{L}\_\mathbf{x}(\mathscr{B})
\equiv \sup\_{b \in \mathscr{B}} L\_\mathbf{x}(b) 
&= \sup\_{b \in \mathscr{B}} \frac{\mathbb{I}(x\_{(n)} \leqslant b)}{b^n} \\[12pt]
&= \begin{cases}
(\inf \mathscr{B})^{-n} & \text{if } \mathscr{B} \cap [x\_{(n)}, \infty) \neq \varnothing, \\[6pt]
0 & \text{if } \mathscr{B} \cap [x\_{(n)}, \infty) = \varnothing. \\[6pt]
\end{cases} \end{align}$$

(Note that this maximised likelihood is *zero* if there are no values $b \in \mathscr{B}$ with $b \geqslant x\_{(n)}$.) Consequently, using the hypotheses $H\_0:b=1$ versus $H\_\text{A}:b>1$ you get:

$$\begin{align}
\hat{L}\_\mathbf{x}(H\_\text{A})
&= \hat{L}\_\mathbf{x}((1,\infty)) = 1, \\[12pt]
\hat{L}\_\mathbf{x}(H\_0)
&= \hat{L}\_\mathbf{x}([1]) = \mathbb{I}(x\_{(n)} \leqslant 1).
\end{align}$$

This gives the likelihood-ratio statistic:

$$\begin{align}
\Delta\_{LR}
\equiv 2 \log \Bigg( \frac{\hat{L}\_\mathbf{x}(H\_\text{A})}{\hat{L}\_\mathbf{x}(H\_0)} \Bigg) 
&= 2 [ \hat{\ell}\_\mathbf{x}(H\_\text{A}) - \hat{\ell}\_\mathbf{x}(H\_0) ] \\[12pt]
&= 2 [ 0 - \log \mathbb{I}(x\_{(n)} \leqslant 1)] \\[12pt]
&= \begin{cases}
0 & & \text{if } x\_{(n)} \leqslant 1, \\[6pt]
\infty & & \text{if } x\_{(n)} > 1. \\[6pt]
\end{cases}
\end{align}$$

This statistic has a distribution concentrated on two possible values, so it does not obey the approximate chi-squared distribution. The reason that [Wilks' theorem](https://en.wikipedia.org/wiki/Wilks%27_theorem) does not apply to this case is that the maximising parameter values for each maximised likelihood value occur on the boundary of the parameter range rather than the interior of the parameter range.