So we've got a sample data coming from exponential distribution with parameter $\lambda$, and we take an estimator $\lambda_n = \frac{n}{X_1+X_2+\cdots+X_n}$.
I need to show that this is a biased and consistent estimator. Normally what you'd do is say that the sample mean follows gamma distribution and find the expectation and variance, but we've not covered that topic yet, so using gamma distribution is not allowed. How else could we compute the expectation of the estimator?
$\frac{n}{E(X_1+X_2+\cdots+X_n)}$ gets me stuck as it's in the denominator
Best Answer
A special case of Jensen's inequality is that for a positive-valued random variable $\overline X,$ you have $$ \frac 1 {\operatorname E\left(\overline X\right)} \le \operatorname E\left( \frac {\,\,1\,\,} {\overline X}\right). $$ The inequality is strict if $\operatorname{var}\left( \overline X \right)>0.$
If $\overline X = (X_1+\cdots+X_n)/n,$ you therefore have $\operatorname E\left( 1\left/\overline X\right.\right) > 1 / \operatorname E\left( \overline X\right) = \lambda.$