# Cramér-Rao inequality

Dina statistik, Ka-teusarua-an Cramér-Rao, ngaran keur ngahargaan ka Harald Cramér jeung Calyampudi Radhakrishna Rao, ngagambarkeun wates luhur dina présisi éstimator statistis, dumasar kana informasi Fisher.

Katangtuanna nyaéta informasi Fisher bulak balik , ${\displaystyle {\mathcal {I}}(\theta )}$, paraméter ${\displaystyle \theta }$, mangrupa wates handap variance paraméter éstimator unbiased (dilambangkeun ${\displaystyle {\widehat {\theta }}}$).

${\displaystyle \mathrm {var} \left({\widehat {\theta }}\right)\geq {\frac {1}{{\mathcal {I}}(\theta )}}={\frac {1}{\mathrm {E} \left[\left[{\frac {\partial }{\partial \theta }}\log f(X;\theta )\right]^{2}\right]}}}$

Dina sababaraha kasus, taya unbiased éstimator kapanggih dina wates handapna.

Cramér-Rao inequality disebut ogé Cramér-Rao bounds (CRB) atawa Cramér-Rao lower bounds (CRLB) sabab dicokot tina wates handap variance ${\displaystyle {\widehat {\theta }}}$.

## Bukti

Anggap variabel random X, mibanda probability density function f(x,θ). Di dieu T = t(X) nyaéta statistic dipaké salaku estimator keur θ. Lamun V mangrupa score, nyaéta

${\displaystyle V={\frac {\partial }{\partial \theta }}\log f(X;\theta ).}$

mangka expectation V, ditulikeun E(V), sarua jeung. Lamun urang nganggap covariance cov(V, T) V sarta T urang mibanda cov(V, T) = E(VT) sabab ekspektasi V sarua jeung zero. Ngalegaan tina rumus ieu urang mibanda

${\displaystyle {\rm {cov}}(V,T)=E\left(T\cdot {\frac {\partial }{\partial \theta }}\log f(X;\theta )\right).}$

Ieu bisa dilegaan ku ngagunakeun identitas

${\displaystyle {\frac {\partial }{\partial \theta }}\log Q={\frac {1}{Q}}{\frac {dQ}{d\theta }}}$

sarta harti ekspektasi nu dibérékeun, sanggeus nunda f(x; θ),

${\displaystyle \int t(x)\left\{{\frac {\partial }{\partial \theta }}f(x;\theta )\right\}\,dx.}$

Ayeuna lamun turunan ditukerkeun ku integral, mangka ieu ngan sakadar turunan (wrt θ) tina ekspektasi t(X), atawa

${\displaystyle {\frac {\partial }{\partial \theta }}E(T).}$

Sabab T mangrupa unbiased, ekspektasi-na θ; we are left with 1.

Cauchy-Schwarz inequality nembongkeun yen

${\displaystyle {\rm {var\ }}T\times {\rm {var\ }}V\geq {\rm {cov}}(V,T)=1,}$

mangka dina kasus ieu

${\displaystyle {\rm {var\ }}T\geq {\frac {1}{{\rm {var\ }}V}}={\frac {1}{I(\theta )}}}$

di mana I(θ) mangrupa Fisher information. Ieu mangrupa kateusaruaan Cramér-Rao; aya di wates dina varian tina unbiased éstimators.

Efisiensi T dihartikeun ku

${\displaystyle e(T)={\frac {1/I(\theta )}{{\rm {var\ }}T}}}$

atawa varian minimum nu mungkin keur unbiased éstimator dibagi ku varian nu sabenerna. Mangka wates handap Cramér-Rao dibérékeun ku e(T) ≤ 1.