Statistics MLE question?

Question

Anyone know how to find the Maximum Likelihood Estimate (MLE) theta of function:
f(x|theta) = e^(theta - x), when x>theta,
0 otherwise

Answer 1

The context for an MLE estimate is when you are given a set of data values { x_i }. First, without explanation, the answer is that the MLE is: theta = min_i ( x_i ).

EXPLANATION: The likelihood is given by: \product_i [ exp(theta-x_i) ] when x_i >= theta and zero otherwise. Thus the likelihood will be zero if for any i x_i < theta. Thus we must have x_i >= theta for all i. As long as that constraint is satisfied, we want f(x|theta) to be as large as possible for all the { x_i }. Given the nature of exp(-x) (including the fact that it's monotonically decreasing with increasing x), all f(x_i|theta) will have the largest value possible is we set theta = min_i ( x_i ). This is easy to show/see graphically.

NOTE: I changed your "x>theta" to "x>=theta". Otherwise it would be a little messier because you would have to define it in terms of limits: theta = min_i ( x_i ) + epsilon, where epsilon -> 0.

Answer 2

http://www.itl.nist.gov/div898/handbook/apr/section4/apr412.htm
http://mathworld.wolfram.com/MaximumLikelihood.html
http://cnx.org/content/m11446/latest/