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3

This is the definition of unbiasedness.

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The unbiasedness part holds trivially from the fact that you have stipulated that the unknown quantity you are estimating is the expected value of the estimator. Denoting this by the parameter $\theta \equiv \mathbb{E}_\varphi(T(\mathbf{X}))$ and noting that your estimator is $\hat{\theta} \equiv T(\mathbf{X})$, you then clearly have: \text{Bias}(\hat{\...

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The Lehmann-Scheffé Theorem says that any unbiased estimator that is a function of a complete sufficient statistic is the minimum variance unbiased estimator. Here $\sum_i X_i$ is a complete sufficient statistic for $\lambda$ and $\bar X$ is a function of it, and is unbiased for $\lambda$, so it is best unbiased. You might then ask if this really avoids ...

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