Sannadkii 1961-kii James iyo Stein waxay daabaceen warqadda Qiyaasta oo leh Khasaaraha Quadratic . Qaado xogta sida caadiga ah loo qaybiyey oo leh micne aan la garanayn \(\mu\) iyo kala duwanaansho \(1\) . Haddii aad hadda ka doorato qiime random \(x\) xogtan oo ay tahay inaad qiyaasto celceliska \(\mu\) taas oo ku saleysan tan, dareen \(x\) waa qiyaas macquul ah \(\mu\) (maadaama qaybinta caadiga ahi jirto, si bakhtiyaa nasiib ah loo doortay \(x\) waxay u dhowdahay \(\mu\) ).
Hadda tijaabada waa la soo celiyay - markan waxaa la isticmaalayaa saddex xog oo madax-bannaan, oo caadi ahaan loo qaybiyay, mid walbana wuxuu leeyahay kala duwanaansho \(1\) iyo macne \(\mu_1\) , \(\mu_2\) , \(\mu_3\) . Ka dib marka la helo saddex qiime oo aan kala sooc lahayn \(x_1\) , \(x_2\) , iyo \(x_3\) , \(\mu_1=x_1\) \(\mu_2=x_2\) iyo \(\mu_3=x_3\) ayaa la qiyaasayaa (iyadoo la isticmaalayo isla habka).
Natiijada la yaabka leh ee James iyo Stein ayaa ah in ay jirto qiyaas ka wanaagsan \( \left( \mu_1, \mu_2, \mu_3 \right) \) (ie isku darka saddexda xog ee madaxbannaan) marka loo eego \( \left( x_1, x_2, x_3 \right) \) . "James Stein estimator" waa markaa::
$$ \begin{pmatrix}\mu_1\\\mu_2\\\mu_3\end{pmatrix} = \left( 1-\frac{1}{x_1^2+x_2^2+x_3^2} \right) \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} \neq \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} $$
Celceliska leexashada labajibbaaran ee qiyaasahan ayaa markaa had iyo jeer ka yar celceliska leexashada labajibbaaran \( E \left[ \left|| X - \mu \right||^2 \right] \)
Waa wax lala yaabo oo laga yaabo inay is-barbar-dhigaan in qiyaasaha James-Stein uu u beddelo qiyaasaha caadiga ah (wax yar oo sii yaraanaya) xagga asalka oo uu sidaas ku keeno natiijo wanaagsan inta badan kiisaska. Tani waxay khusaysaa cabbirrada \( \geq 3 \) , laakiin kuma jiraan kiiska laba-geesoodka ah.
Sharaxaad joomatari wanaagsan oo ku saabsan sababta tani u shaqeyso waxaa bixiya Brown & Zao . Ogsoonow in tani aysan macnaheedu ahayn inaad leedahay qiyaas ka wanaagsan mid kasta oo xog ah - waxaad haysataa qiyaas ka wanaagsan oo leh khatar yar oo la isku daray .