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 tijaabadii waa lagu soo noqnoqday - markan oo leh saddex madax-bannaan, mar labaad si caadi ah loo qaybiyey xogta waxay mid kastaaba kala duwan tahay \(1\) iyo celceliska qiyamka \(\mu_1\) , \(\mu_2\) , \(\mu_3\) . Kadib markii la helo saddex qiime oo random \(x_1\) , \(x_2\) iyo \(x_3\) , hal qiyaas (iyadoo la isticmaalayo hab isku mid ah) \(\mu_1=x_1\) , \(\mu_2=x_2\) iyo \(\mu_3=x_3\) .
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 .