Question

Suppose X1, x2, x, i Uniform(a, 10). And the actual sample realizations turn out to be 11, x2 -7, x3 5. From the given data, calculate the maximum likelihood estimator of a.

Answer 1

Here, $X_{i}\sim Uniform(a,10)$

So, $f(x_{i})=\frac{1}{10-a},a<x_{i}<10$

Likelihood function is given by:

$L=\prod_{i=1}^{n}f(x_{i})=\prod_{i=1}^{3}f(x_{i})=(\frac{1}{10-a})^{3}$

$log L=-3 log(10-a)$

$\frac{\partial logL}{\partial a}=\frac{3}{10-a}$

Equating this to 0, we cannot find the MLE of a.

Let X₍₁₎, X₍₂₎ and X₍₃₎ be the order statistics. Then, $a<X_{(1)}<X_{(2)}<X_{(3)}<10$

So, the value of likelihood function would be maximized for the least value among X_is.

S0, MLE of a = $\widehat{a}=X_{(1)}=1$