Find the MLE density function of uniform [-\theta,\theta] [duplicate]

Question

For $X_1,\dots,X_n$, i.i.d $X_n \sim \mathrm{unif}[-\theta,\theta]$, the ML: $\hat\theta_{MLE}=\mathrm{max}\{-X_{(1)},X_{(n)}\}$. Find the density function. Hint: For $x_1,\dots,x_n$ : $\textrm{max}\{-x_{(1)},x_{(n)}\}=max\{|x_1|,\dots,|x_n|\}$.

Here is my work:

First i find the CDF for $\hat\theta_{MLE}$ and then differentiate with respect to $x$:

\begin{align} F_{\hat\theta_{MLE}}(x) &= P(\text{max}\{-X_{(1)},X_{(n)}\}\leq x) \\ &= P(\text{max}\{|X_{(1)}|, \dots ,|X_{(n)}|\}\leq x) \\ &= P(\text{max}\{|X_{(1)}|\}, \dots , \text{max}\{|X_{(n)}|\}\leq x) \\ &= \prod_{i=1}^{n} P(|X_i|\leq x) \\ &= \prod_{i=1}^{n} P(-x\leq X_i \leq x)\\ &= \prod_{i=1}^{n} F_X(x)-F_X(-x)\\ &= \left(F_X(x)-F_X(x)\right)^{n}\\ &= \left(\frac{x}{\theta}\right)^{n} \end{align}

\begin{align*} \frac{d}{dx}\left[\left(\frac{x}{\theta}\right)^{n}\right] &= n\frac{x^{n-1}}{\theta^n}\\ &= f_{\hat\theta_{MLE}}(\hat\theta) \ \text{,}\ 0\leq x \leq \theta \end{align*}

Is this correct?

Answer 1

A few extra (and less) steps: \begin{align}\require{cancel} F_{\hat\theta_{MLE}}(x) &= \mathbb P_\theta(\text{max}\{-X_{(1)},X_{(n)}\}\leq x)\\ &= \mathbb P_\theta(-X_{(1)}\le x,X_{(n)}\leq x) \tag{definition}\\ &= \mathbb P_\theta(X_{(1)}\ge -x,X_{(n)}\leq x) \tag{inversion}\\ &= \mathbb P_\theta(X_1\ge -x,\ldots,X_n\ge -x,\\ &\qquad \qquad X_1\leq x,\ldots,X_n\le x) \\ &= \mathbb P_\theta(|X_1|\le x,\ldots,|X_n|\le x)\\ &= \cancel{\mathbb P_\theta(\text{max}\{|X_{(1)}|, \dots ,|X_{(n)}|\}\leq x)}\tag{useless} \\ &= \cancel{ \mathbb P_\theta(\text{max}\{|X_{(1)}|\}, \dots , \text{max}\{|X_{(n)}|\}\leq x)}\tag{incorrect}\\ &= \prod_{i=1}^{n} \mathbb P_\theta(|X_i|\leq x) \\ &= \prod_{i=1}^{n} \mathbb P_\theta(-x\leq X_i \leq x)\\ &= \prod_{i=1}^{n} \{F_X(x)-F_X(-x)\}\\ &= \{F_X(x)-F_X(-x)\}^{n}\tag{typo}\\ &= \left(\frac{\min\{\theta,x\}}{\theta}\right)^{n}\tag{correction} \end{align}