Discussion
Suppose that \(x_{1} = -2\), \(x_{2} = 1\), \(x_{3} = 3\), \(x_{4} = -4\) be the observed values of a random sample from the c.d.f. with density,
\[ \begin{aligned} f(x, \theta) &= \frac{e^{-x}}{e^{\theta} - e^{-\theta}}, \\ &-\theta \lt x \lt \theta, \; \theta \gt 0. \end{aligned} \]
Then, the maximum likelihood estimator of \(\theta\) is :
Answer: C
The maximum likelihood estimate of \(\theta\) is 4.
The likelihood is
\[ L(\theta) = e^{-\sum x_{i}}\,(e^{\theta} - e^{-\theta})^{-4} \]
provided every observation lies in \((-\theta, \theta)\), that is \(\theta \gt \max |x_{i}| = 4\); otherwise it is 0. The denominator \(e^{\theta} - e^{-\theta} = 2\sinh\theta\) increases with \(\theta\), so L decreases in \(\theta\) and is largest at the smallest admissible value. The MLE is \(\max |x_{i}| = |-4| = 4\).
3 is the largest observation, which ignores the negative value \(-4\) that also has to lie inside \((-\theta, \theta)\). 0.5 and 1.5 are not admissible at all, since they exclude several observations.
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