Urgent Help - Statistics Question

Question

A random sample of n observations, Y₁, Y₂,..., Y_n is selected from a population where Y_i, follows an exponential distribution with mean β.

1. Determine the method of moments estimators for the unknown parameter β.
2. Determine the MLE for the unknown parameter β.

Answer 1

As we talk of the distribution having mean ß, clearly, our method of moments estimator is simply the sample mean. We could calculate the mean by calculating I[0, ∞) x/ß e^-x/ß dx, but that would simply return ß. The MOM estimator of ß is ∑x_i /n = x-bar

Now, to the MLE estimator, maximize the log-likelihood. The log likelihood equals - ln ß - x₁/ß - lnß - x₂/ß - .... - ln ß - x_n/ß = -n ln ß - 1/ß∑x_i

We take the derivative and set it equal to 0.

-n/ß + 1/ß²∑x_i = 0

1/ß²∑x_i = n/ß

Multiply both sides by ß²/n

∑x_i /n = ß

The sample mean is also the MLE estimate of ß