Question

A sample of 50 losses revealed that 10 were above $500 and 40 were below $500.

Using only this information, determine the maximum likelihood estimate of θ if we assume that an exponential distribution with mean θ is an appropriate model for the data.

Answer 1

If the loss, say L, has an exponential distribution with mean , then

Thus, the indicator variable of the event {L>500} is Bernoulli with probability

.

Thus, the number of times loss is more than 500 is Binom(50,p) and has probability mass function

We know that the MLE of p is

.

Now,

But MLE has the property that for any function f, the MLE of f(p) is f applied to the MLE of p.

Thus MLE of