Apply Rao Blackwell theorem to estimate $\lambda$

expected valueprobabilitystatistical-inferencestatistics

I am very new to Rao Blackwell theorem.

It says that, if $\delta(X)$ is an unbiased estimator for a parameter $\theta$, then the estimator $T_0(X)=E\left(\delta|T\right)$, whete $T(X)$ is a sufficient statistic is also unbiased with variance no more than variance of $\delta(X)$.

So i was trying an example with Poisson distribution:

Its PMF is $$P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!} \:x>0,\lambda>0$$

Let $X_1,X_2,..X_n \sim poisson(\lambda) $

Let $\delta(X)=X_1$ and this is obviously an unbiased estimator for $\lambda$.

Also By factorization theorem $T(X)=\sum_{i=1}^{n}X_i$ is a sufficient statistic.

Now we have:
$$T_0=E\left(X_1|T=t\right)$$

Now how can i proceed to Rao blackwellization?

Best Answer

In this case we need to use little about the Poisson distribution due to a trick.

You would have:

  • $E[X_i|T]$ is independent of $i$ by symmetry
  • $\sum_i E[X_i|T]=E[T|T]=T$

This implies that:

$$T_0=E[X_1|T]=\frac{T}{N}$$

which is indeed an estimator of the same parameter with lower variance (the usual sample mean).

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