The differentiation approach to derive the maximum likelihood estimator (mle) is not appropriate in all the cases. Let X1, X2, ..., Xn be a random sample of size n from the population of X. Consider the probability function of X:
. . . . .\(\displaystyle f(x;\, \theta)\, =\, \begin{cases} e^{-(x - \theta)}, & \mbox{if }\, \theta\, \leq\, x\, <\, \infty\, \mbox{ for }\, -\infty\, <\, \theta\, <\, \infty \\ {} & {} \\ 0, & \mbox{otherwise} \end{cases}\)
Derive the mean square error (mse) of mme.
. . . . .\(\displaystyle f(x;\, \theta)\, =\, \begin{cases} e^{-(x - \theta)}, & \mbox{if }\, \theta\, \leq\, x\, <\, \infty\, \mbox{ for }\, -\infty\, <\, \theta\, <\, \infty \\ {} & {} \\ 0, & \mbox{otherwise} \end{cases}\)
Derive the mean square error (mse) of mme.
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