Fréchet-Darmois-Cramér-Rao Inequality

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Theorem

Let $S$ be a sample of $n$ observations from a probability distribution with frequency function $\map f {x, \theta}$.

Let it be assumed that certain regularity conditions apply.

Let it also be assumed that the extremes do not depend on $\theta$.


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In particular: what those conditions are
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The lower bound of the variance of an estimator $T$ of a parameter $\theta$ is given by $\dfrac 1 I$ where:

$I = -n \map E {\dfrac {\partial^2 \ln f} {\partial \theta^2} }$


Proof


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Also known as

The Fréchet-Darmois-Cramér-Rao Inequality is usually known as:

the Cramér-Rao Inequality
the Cramér-Rao Bound
the Cramér-Rao Lower Bound.


Source of Name

This entry was named for Maurice René Fréchet, Georges Darmois, Carl Harald Cramér and Calyampudi Radhakrishna Rao.


Historical Note

The Fréchet-Darmois-Cramér-Rao Inequality was discovered in $1945$ by Calyampudi Radhakrishna Rao, and independently by Carl Harald Cramér in $1946$.


Sources

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