If X1, X2, ..., Xn constitute a random sample of size n from an exponential population, show that Xbar is a sufficient estimator of the parameter theta.

Exponential distribution: g(x; theta) = (1/theta)(e^(-x/theta))

Hi,

In sufficient statistics, a statistic (estimator) T(X) is sufficient estimator for theta if the conditional probability distribution of X given T(X)=t is not a function of theta.

Using the factorization criterion:

A statistic T(X) for theta is a sufficient statistic if g can be expressed as a product of (or factored into) two functions:



For a exponential distribution , the mean of a sample (Xbar) is always a sufficient estimator for

Hope this helps. Let me know if you still have troubles with it.

Regards!