Information inequality (mathematical statistics)

Information inequality (mathematical statistics) is an inequality for an unbiased estimate with a locally minimal dispersion, which sets the lower bound for the variance of this estimate. It plays an important role in the theory of asymptotically effective estimates ^[1] .

Wording

We denote $X$ ${\ displaystyle X}$ - observational data, $\theta$ ${\ displaystyle \ theta}$ - parameter estimated on their basis, $p(X,\theta )$ ${\ displaystyle p (X, \ theta)}$ - conditional probability density of distribution, Fisher’s information as $I(\theta )=E\left[{\frac {\partial }{\partial \theta }}\ln p(X,\theta )\right]^{2}$ ${\ displaystyle I (\ theta) = E \ left [{\ frac {\ partial} {\ partial \ theta}} \ ln p (X, \ theta) \ right] ^ {2}}$ . Let be $I(\theta )>0$ ${\ displaystyle I (\ theta)> 0}$ , $\delta$ ${\ displaystyle \ delta}$ - any statistics with $E_{\theta }(\delta ^{2})<\infty$ ${\ displaystyle E _ {\ theta} (\ delta ^ {2}) <\ infty}$ for which the derivative with respect to $\theta$ ${\ displaystyle \ theta}$ from mathematical expectation $E_{\theta }(\delta )=\int \delta p_{\theta }d\mu$ ${\ displaystyle E _ {\ theta} (\ delta) = \ int \ delta p _ {\ theta} d \ mu}$ exists and can be obtained by differentiation under the sign of the integral. Then the information inequality is true ^[2] : $D_{\theta }(\delta )\geqslant {\frac {\left[{\frac {\partial }{\partial \theta }}E_{\theta }(\delta )\right]^{2}}{I(\theta )}}$ ${\ displaystyle D _ {\ theta} (\ delta) \ geqslant {\ frac {\ left [{\ frac {\ partial} {\ partial \ theta}} E _ {\ theta} (\ delta) \ right] ^ {2 }} {I (\ theta)}}}$ .

Notes

↑ Lehman, 1991 , p. 110.
↑ Lehman, 1991 , p. 116.

Literature

Lehman E. Theory of point estimation. - M .: Nauka, 1991 .-- 448 p. - ISBN 5-02-013941-6 .

[_40f2ff8c97f85b06-1] Lehman, 1991 , p. 110.

[_40f2ff8c97f85b00-2] Lehman, 1991 , p. 116.

Information inequality (mathematical statistics)

Wording

Notes

Literature

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