# Thread: Cramér–Rao Lower Bound for a Function of a Parameter

1. ## Cramér–Rao Lower Bound for a Function of a Parameter

I'm trying to compute the Cramér–Rao Lower Bound for a function of the parameter $\lambda$ of a Poisson distribution.

The function is $g (\lambda) = e^{-\lambda}$.
Let $\hat{g}$ be an estimate of $g(\lambda)$

I'm using the formula:

$Var (\hat{g}) \ge \dfrac{(g'(\lambda))^2}{nI(\lambda)}$

Now from estimating $\lambda$ previously, I know that $nI(\lambda) = \dfrac {n}{\lambda}$

And
$g'(\lambda) = -e^{-\lambda}$
$(g'(\lambda))^2 = e^{-2\lambda}$

So
$Var (\hat{g}) \ge \dfrac{e^{-2\lambda}}{\dfrac {n}{\lambda}}$

$Var (\hat{g}) \ge \dfrac{\lambda e^{-2\lambda}}{n}$

Is this correct because I tried it another way and got?
$Var (\hat{g}) \ge \dfrac{e^{-\lambda}(1-e^{-\lambda})}{n}$

2. Originally Posted by alan4cult
I'm trying to compute the Cramér–Rao Lower Bound for a function of the parameter $\lambda$ of a Poisson distribution.

The function is $g (\lambda) = e^{-\lambda}$.
Let $\hat{g}$ be an estimate of $g(\lambda)$

I'm using the formula:

$Var (\hat{g}) \ge \dfrac{(g'(\lambda))^2}{nI(\lambda)}$

Now from estimating $\lambda$ previously, I know that $nI(\lambda) = \dfrac {n}{\lambda}$

And
$g'(\lambda) = -e^{-\lambda}$
$(g'(\lambda))^2 = e^{-2\lambda}$

So
$Var (\hat{g}) \ge \dfrac{e^{-2\lambda}}{\dfrac {n}{\lambda}}$

$Var (\hat{g}) \ge \dfrac{\lambda e^{-2\lambda}}{n}$

Is this correct because I tried it another way and got?
$Var (\hat{g}) \ge \dfrac{e^{-\lambda}(1-e^{-\lambda})}{n}$
Well I'm not going to work this out but you first form looks wrong as it has the same dimensions as $\lambda$ (lets say $\text{[T]}^{-1}$) but it should be a pure number. The second form is a pure number which is what we should expect.

CB

3. Hi thanks for your reply.

Can you see what's wrong with my end result by looking at the calculations that led me to it? Did I differentiate incorrectly?

The derivative of g with respect to $\lambda$ is $-e^{-2\lambda}$

And then I've just used that in the formula?
Do you think I'm using the formula incorrectly/out of context?

About the dimensions thing you're talking about, what does that mean? I haven't encountered dimensions before?

4. Anyone got any more ideas?