Maximum Likelihood Estimator of the exponential function parameter based on Order Statistics

Question

Let $X_1, \ldots, X_n$ be a random sample from the exponential distribution $\exp(\lambda)$. Let $$M_n=\max\{X_1, \ldots, X_n\}$$ with probability density function $$g_{M_n}(x)=n\lambda e^{-\lambda x}(1-e^{-\lambda x})^{(n-1)}, \qquad x>0$$ Q1. If $M_n$ is the only information that you have from the sample, find a maximum likelihood estimator (mle) $\hat{\lambda}_n$ of $\lambda$.
Q2. Using $(1+x)^n>1+nx$ prove that $\hat{\lambda}_n$ is consistent, i.e. that $P(| \hat{\lambda}_n-\lambda|>\epsilon)\longrightarrow0$, for $n\rightarrow \infty$

Thanks.

Answer 1

As you already know, considering the function $$G_n(x)=\log\left(n+\frac{n-1}{x-1}\right),$$ defined on $(1,+\infty)$, $\hat\lambda_n$ solves the identity $$\hat\lambda_n M_n=G_n(\hat\lambda_n M_n).$$ As you noted, this identity has no analytical solution. However, the function $G_n$ decreases on $(1,+\infty)$ from $G_n(1)=+\infty$ to $G_n(+\infty)=\log(n)$ and, perhaps amazingly, this is all one needs to solve the exercise.

To wit, first note that $G_n\gt\log(n)$ on $(1,+\infty)$ hence the unique solution of the identity above is such that $\hat\lambda_n M_n\gt\log(n)$. In particular, $\hat\lambda_n M_n\to\infty$ hence $$G_n(\hat\lambda_n M_n)=\log(n+o(n))=\log(n)+o(1),$$ which implies $$\hat\lambda_n M_n/\log n\to1\ \text{almost surely.}\tag{$\ast$}$$ On the other hand, classic estimates of $M_n$ are as follows. For every nonnegative $x$, $$P(M_n\leqslant x)=(1-\mathrm e^{-\lambda x})^n,$$ hence, for every $\varepsilon$ in $(0,1)$, $$P(M_n\leqslant(1+\varepsilon)\log n/\lambda)=(1-n^{-1-\varepsilon})^n\to1,$$ and$$P(M_n\leqslant(1-\varepsilon)\log n/\lambda)=(1-n^{-1+\varepsilon})^n\to0,$$ that is, $$M_n/\log n\to1/\lambda\ \text{in probability.}\tag{$\dagger$}$$ Putting $(\ast)$ and $(\dagger)$ together, one sees that $\hat\lambda_n\to\lambda$ in probability, as desired.