Questions tagged [cramer-rao]

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Why does $T$ being an unbiased estimator for $g(\theta)$ imply that $g(\theta) = ET = \int T(\mathbf{y}) f_\theta(\mathbf{y}) \ d\mathbf{y}$?

I am currently studying the Cramer-Rao lower bound. My notes say the following: Theorem: Cramer-Rao lower bound Let $Y_1, \dots, Y_n$ have a joint distribution $f_\theta (\mathbf{y})$, where $f_\...
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How do these results show that $T(\mathbf{X})$ is an unbiased estimator of $E_\varphi[T(\mathbf{X})]$ that achieves the Cramer-Rao lower bound?

Let's say that $X_1, \dots, X_n$ has the joint distribution $f_\varphi(\mathbf{x})$ that belongs to the one-parameter exponential family $$f_\varphi(\mathbf{x}) = \exp{\left\{ c(\varphi) T(\mathbf{x}) ...
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How can one show that $\bar{X}$ is the best unbiased estimator for $\lambda$ without using the Cramèr-Rao lower bound?

Assume we have the random sample $X_1, \dots, X_n$ with mean $\mu$ and variance $\sigma^2 < \infty$. We have that $E[S^2] = \sigma^2$, where $S^2 = \sum_{i = 1}^n \dfrac{(X_i - \bar{X})^2}{n - 1}$ ...