thats a two parameter exponential distribution. Reading this link might help
Find the MLE for Lamda and Theta
f(x|Lambda, Theta) = Lambda*exp(-Lambda(x-Theta))Indicator[x>=Theta]
I got Lambda-hat = (Theta + 1)/Xbar
and Theta-hat is nLambda=0, so lambda = 0 (Which seems like it must be wrong)
but assuming It was correct I just plugged in lambda = 0 into the MLE for lambda hat (not even sure if this is what you are supposed to do) and got Theta = -1.
I get the feeling im compltetly wrong can someone check this, I can't find where I have gone wrong.