maximum likelihood estimate Uniform

Question

Question

A random sample is taken from a U(-θ,2θ) distribution, where θ is a positive constant. This is the sample x1=-1.77, x2=-1.16, x3=-0.39, x4=0.24, x5=1.28, x6=2.25.

Find the maximum likelihood estimate voor θ.

What I've done is to use probability density function of uniform distribution and then multiply them and then use the log. But in this case i totally differ from the correct answers. Someone who can help me how to solve this correctly?

Answer 1

Your density is the following

$$f_X(x|\theta)=\frac{1}{3\theta}\mathbb{1}_{(-\theta;2\theta)}(x)$$

This means that

$$\begin{cases} -\theta < x_{(1)} \\ 2\theta >x_{(n)} \end{cases} $$

or

$$\begin{cases} \theta > -x_{(1)} \\ \theta >\frac{x_{(n)}}{2} \end{cases} $$

that is also $\theta > max[-x_{(1)};\frac{x_{(n)}}{2}]$

The likelihood is

$$L(\theta)=\frac{1}{(3\theta)^n}$$

The likelihood is strictly decreasing $\forall{\theta}$

so the ML estimator is the maximum between

1. $-min(X)=1.77$

2. $\frac{max(X)}{2}=\frac{2.25}{2}=1.125$

so the answer is 1. : 1.77