91Ó°ÊÓ

\({X_1},...,{X_n}\)is a random sample whose probability function is given as

\(\begin{array}{c}f\left( {x|\theta } \right) = {e^{\theta - x}}\,\,,\theta > 0\\ = 0\,\,\,{\rm{otherwise}}\end{array}\)

We need to calculate

(a) M.L.E. of θ does not exist

(b)pdf for which M.L.E. of θ will exist, and find this estimator

Let s be the sum of observations in the sample.

Then the likelihood function is given by

\(\begin{array}{c}f\left( {x|\theta } \right) = {e^{n\theta - s}}\,\,{\rm{for}}\,\min \left\{ {{X_1},...,{X_n}} \right\} > \theta \\ = 0\,\,\,\,{\rm{otherwise}}\end{array}\)

Now, for each value of X the likelihood function must satisfy the strict inequality . This implies there is no such parameter such that \(\theta = \min \left\{ {{X_1},...,{X_n}} \right\}\) .

Hence the M.L.E. of θ does not exist.

Let us suppose the pdf in part (a) has been changed to the following equivalent pdf in which strict inequality is removed.

\(\begin{array}{c}f\left( {x|\theta } \right) = {e^{\theta - x}}\,\,\,\,\,x \ge \theta \\ = 0\,\,\,\,\,x < \theta \end{array}\)

Then the likelihood function \(f\left( {x|\theta } \right)\) will be non-zero for \(\theta \le \min \left\{ {{X_1},...,{X_n}} \right\}\) and Hence the M.L.E. will be \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \theta } = \min \left( {{x_1},{x_2},...,{x_n}} \right)\)