# Finding an efficient estimator for $\beta$ in a sample of $n$ random variables having the $\text{Gamma}(\alpha,\beta)$-distribution.

Problem: Suppose that we have i.i.d. random variables $X_{1},\dots,X_{n} \sim \text{Gamma}(\alpha,\beta)$, where $\alpha > 0$ is known. Find an efficient estimator for $\beta$.

Recall that the probability density function of the $\text{Gamma}(\alpha,\beta)$-distribution is given by $$\forall x > 0: \quad f(x;\alpha,\beta) = \frac{1}{\Gamma(\alpha) \beta^{\alpha}} \cdot x^{\alpha - 1} e^{- x/\beta}.$$

I am a little lost, but I am guessing that I need to find the Cramér-Rao Lower Bound (CRLB), look for an unbiased estimator and then compare it to the CRLB. Any help would be greatly appreciated.

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There are two formulations for $\Gamma(\alpha,\beta)$, the $\beta$ in one of them being the inverse in the other. When working with gamma distributions, you should always specify which formulation you are working with. – Learner Feb 10 '13 at 7:11
Thanks for the advice. I added the the equation defined as I have been using it. – user45185 Feb 11 '13 at 5:11
Thanks a lot. I will take note of that for my previous posts – user45185 Feb 11 '13 at 7:50
The efficiency of an estimator depends precisely on the Cramér-Rao Lower Bound. You may have confused the concept of ‘efficient estimator’ with that of ‘minimum-variance unbiased estimator’. – Haskell Curry Feb 11 '13 at 9:38

The following is a solution, presented in the spirit of rigorous measure-theoretic statistics.

Fix $\alpha > 0$. We are considering the one-parameter family $(P_{\beta})_{\beta > 0}$ of probability distributions, where each $P_{\beta}$ is the $\text{Gamma}(\alpha,\beta)$-distribution. According to this webpage, an efficient estimator for $(P_{\beta})_{\beta > 0}$ using a random sample of size $n$ is the following Borel-measurable function $T: \mathbb{R}^{n} \to \mathbb{R}$: $$\forall (x_{1},\ldots,x_{n}) \in \mathbb{R}^{n}: \quad T(x_{1},\ldots,x_{n}) \stackrel{\text{def}}{=} \frac{1}{n \alpha} \sum_{k=1}^{n} x_{k}.$$ To prove that $T$ is an efficient estimator for $(P_{\beta})_{\beta > 0}$, we need to do the following:

• Prove that $T$ is an unbiased estimator for $(P_{\beta})_{\beta > 0}$: $$\forall \beta > 0: \quad X_{1},\ldots,X_{n} \sim P_{\beta} ~~ \Longrightarrow ~~ \text{E} \left[ T(X_{1},\ldots,X_{n}) \right] = \text{E} \left[ \frac{1}{n \alpha} \sum_{k=1}^{n} X_{k} \right] = \beta.$$

• Derive the Cramér-Rao Inequality for $T$ with respect to $(P_{\beta})_{\beta > 0}$: $$\forall \beta > 0: \quad X_{1},\ldots,X_{n} \sim P_{\beta} ~~ \Longrightarrow ~~ \text{Var} \left[ T(X_{1},\ldots,X_{n}) \right] \geq \frac{\beta^{2}}{n \alpha}.$$

• Prove that equality is actually attained by $T$ in the inequality above: $$\text{Var} \left[ T(X_{1},\ldots,X_{n}) \right] = \frac{\beta^{2}}{n \alpha}.$$

Conclusion: $T$ is an efficient estimator for $(P_{\beta})_{\beta > 0}$ using a random sample of size $n$.

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