Let $X_1, X_2, \dots, X_n$ be a sample of i.i.d. random variables, with density $$f_\theta=\frac{2}{3\theta}\left(1-\frac{x}{3\theta}\right) $$ for $0 < x < 3\theta$. And $f_\theta=0$ if $ x < 0$ or $ x>3\theta$
Let $\hat{\theta}=\overline{X}$ be an estimate for $\theta$
I showed that $\hat\theta$ is an unbiased estimator for $\theta$ and it's a consistent estimator.
My question is:
Why doesn't the Cramér-Rao lower bound apply to unbiased estimates of for this distribution?
Are you aware of the three regularity conditions that must be satisfied for the CR lower bound to apply? It looks like it violates the condition that the bounds of the distribution function must not depend upon the quantity being estimated. $\theta$ determines the bounds of the distribution. See the Wikipedia article, regularity condition 1: http://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound
The Cramer-Rao Lower Bound (CRLB) is valid only for densities that are sufficiently regular. In particular, the support of the density f(x; θ) cannot depend upon the parameter θ. This is because
f(x; θ) must be such that the order of integration of f(x; θ) with respect to x and differentiation of
f(x; θ) with respect to θ can be interchanged. For the example you provided, the support of f(x; θ) depends upon the parameter (0 < x < 3θ). Therefore, the CRLB does not apply.
The regularity conditions come from the proof of the CRLB. If they do not hold, the proof is invalid.
