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# Point Estimation (cont.)

## Example: Uniform

Suppose $X_{i}\overset{iid}{\sim}U\left(0,\theta\right)$ where $\theta\gt 0$ is unknown.

The likelihood function equals $L\left(\left.\theta\right|x_{1}..x_{n}\right)=\Pi_{i=1}^{n}f\left(\left.x_{i}\right|\theta\right)=\Pi_{i=1}^{n}\frac{1}{\theta}1\left(0\leq x_{i}\leq\theta\right)$

Since $x_{i}$’s are draws from $U\left(0,\theta\right)$, the $0\leq x_{i}$ constraint will always be satisfied. However, since we are uncertain about the true value of $\theta$, the upper constraint may be binding.

This yields the following likelihood: $\Pi_{i=1}^{n}\frac{1}{\theta}1\left(0\leq x_{i}\leq\theta\right)=\Pi_{i=1}^{n}\frac{1}{\theta}1\left(x_{i}\leq\theta\right)=\frac{1}{\theta^{n}}1\left(x_{\left(n\right)}\leq\theta\right)$

Notice that $L\left(\left.\cdot\right|x_{1}..x_{n}\right)$ is not differentiable at $\theta=x_{\left(n\right)}$. We separate the problem:

• $L\left(\left.\cdot\right|x_{1}..x_{n}\right)=0$ if $\theta\lt x_{\left(n\right)}$; this reveals the impossibility that a value is generated above $\theta$.
• $L\left(\left.\cdot\right|x_{1}..x_{n}\right)=\frac{1}{\theta^{n}}$ if $\theta\geq x_{\left(n\right)}$; it is decreasing in $\theta$, so constraint is active, and $\widehat{\theta}_{ML}=x_{\left(n\right)}$.

Notice that the maximum likelihood estimator is different from the method of moments, $\widehat{\theta}_{ML}=x_{\left(n\right)}$ while $\widehat{\theta}_{MM}=2\overline{x}$.

Unlike the method of moments, we cannot obtain an estimator st $x_{i}\gt \widehat{\theta}_{ML}$.

However, as we will discuss later, there are some bad news.

The fact that we can never obtain $\widehat{\theta}_{ML}\gt \theta_{0}$, where $\theta_{0}$ is the true value of parameter $\theta$, means that the maximum likelihood estimator is likely to systematically underestimate the true parameter value.