Applicability of different formulas for Fisher's information

Question

Question: Let $X_1, ..., X_n \sim \text{Exp}(\lambda)$ where $E(X) = \lambda$. Find Fisher's information of $\lambda$.

From the description, $f_X(x) = \lambda^{-1}\exp(-\lambda^{-1}x)$

I first tried to calculate the Fisher's information using the formula $I(\lambda) = -E(\frac{d^2}{d\lambda^2}[l(\lambda;x)])$, where $l$ is the log-likelihood function.

$\frac{d^2}{d\lambda^2}[l(\lambda;x]) = \frac{d}{d\lambda}[\frac{-1}{\lambda} + \frac{x}{\lambda^2}] = \frac{1}{\lambda^2} - \frac{x}{\lambda^3}$

So $-E(\frac{d^2}{d\lambda^2}[l(\lambda;X)]) = -(\frac{1}{\lambda^2} - \frac{E(X)}{\lambda^3})= 0$

I don't believe this is right, so I tried again using the less differentiated formula, $I(\lambda) = E((\frac{d}{d\lambda}[l(\lambda;x)])^2)$

This time I got $E((\frac{-1}{\lambda} + \frac{x}{\lambda^2})^2) = E(\frac{X^2}{\lambda^4} - 2\frac{X}{\lambda^3} + \frac{1}{\lambda^2}) = \frac{1}{\lambda^2}$, which looks a lot more reasonable, and I believe this one is correct.

My question is whether the inapplicability of the first formula is due to its differentiability issue at 0. I would also like to verify that the first formula is only applicable when the log-likelihood function is twice differentiable everywhere in the parameter space.

Furthermore, on a somewhat separate note, an estimator $X_n$ of $\lambda$ is efficient iff $\frac{1}{I(\lambda)\text{Var}(X_n)} = 1$ How do I interpret this if $\frac{1}{I(\lambda)\text{Var}(X_n)} \neq 1$?

Thanks in advance!

Answer 1

1. You just have a mistake in the second derivative $$ \frac{\partial}{\partial \lambda } \left( - \frac{1}{\lambda} + \frac{X}{\lambda^2}\right) = \frac{1}{\lambda ^ 2} - \frac{2X}{\lambda^3}, $$ hence, $$ I_X = -E\left[\frac{1}{\lambda ^ 2} - \frac{2X}{\lambda^3}\right] = -\frac{1}{\lambda ^ 2} + \frac{2\lambda}{\lambda^3} = \frac{1}{\lambda^2}. $$

The two definitions are equivalent under appropriate (technical) regularity conditions. See, e.g., Appendix E here for the list of the conditions: Applicability of different formulas for Fisher's information.

2. This relates to the Cramer-Rao lower bound for estimator $T$ variance. If the ratio between the variance of the estimator $Var(T)$ and the bound is greater than $1$, the estimator $T$ is inefficient (in the sense of "risk" or variance). See here for further details: https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound