Question

Consider a gamma distribution with pdf \[ f(x;\beta)= \begin{cases} \dfrac{1}{24\beta^{5}}x^{4}e^{-x/\beta}, & x>0,\\ 0, & \text{elsewhere}, \end{cases} \] with $\beta>0$. For $\beta=2$, find the Cramér–Rao lower bound (rounded to one decimal place) for the variance of any unbiased estimator of $\beta^{2}$ based on a random sample of size 8.

Hint:

For one–parameter families $f(x;\beta)$, $\operatorname{Var}(\hat g)\ge\dfrac{(g'(\beta))^{2}}{nI_{1}(\beta)}$. Compute $I_{1}$ from $(\partial\log f/\partial\beta)^{2}$ and take expectations under the model.

Answer 1

Correct Answer: 1.6

The given pdf is a Gamma distribution with shape $k=5$ and scale $\beta$. For one observation, \[ \log f(x;\beta)= -\log 24 -5\log\beta +4\log x -\frac{x}{\beta}. \] Differentiate with respect to $\beta$: \[ \frac{\partial}{\partial\beta}\log f(x;\beta)= -\frac{5}{\beta}+\frac{x}{\beta^{2}}. \] Hence the Fisher information for one observation is \[ I_{1}(\beta)=\mathbb{E}\!\left[\!\left(\frac{\partial\log f}{\partial\beta}\right)^{2}\!\right] =\frac{5}{\beta^{2}}. \] For $n=8$ independent observations, $I(\beta)=\dfrac{8\times5}{\beta^{2}}=\dfrac{40}{\beta^{2}}.$
For the function $g(\beta)=\beta^{2}$, the Cramér–Rao lower bound is \[ \operatorname{Var}(\hat g)\ge \frac{[g'(\beta)]^{2}}{I(\beta)} =\frac{(2\beta)^{2}}{40/\beta^{2}} =\frac{4\beta^{4}}{40} =\frac{\beta^{4}}{10}. \] At $\beta=2$, the bound is $\displaystyle \frac{16}{10}=1.6.$ Thus the CRLB is $\boxed{1.6}$.