# Does efficiency imply unbiased and consistency?

If I can prove that for an estimator $\hat{k}( \theta)$ I can write: $$\frac{\partial l(X_1, \dots , X_n)}{\partial \theta} = a(n, \theta)(\hat{\theta} - \theta)$$

Am i sure that the estimator is unbiased? and consistent?

NB:

• $l$: is the log likelihood
• $X_1$ is generated from a regular model
• $\hat{\theta}$ is the estimator for $\theta$
• $a(\cdot,\cdot)$ is a function of $n$ and $\theta$ (without any particular meaning i guess)
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ok,i did. thanks for the hint. – Aslan986 Jun 28 '12 at 1:14
I'm not sure how your expression relates to efficiency... Could you please explain? Usually efficiency is only defined for unbiased estimators, so it doesn't make much sense to talk about biased efficient estimators (other than asymptotically). – MånsT Jun 28 '12 at 8:25
1. The equation appears to have no relationship to $\hat{k}(\theta)$ at all. Is it missing something? 2. Because one can define $a(n,\theta)=\frac{\partial l(X_1, \dots , X_n)}{\partial \theta} / (\hat{\theta}-\theta)$, the equation appears to add no information whatsoever. – whuber Jun 28 '12 at 13:16