Ngo-1961 uJames noStein bashicilela iphepha elithi Estimation with Quadratic Loss . Thatha idatha esabalaliswe ngokuvamile enencazelo engaziwa \(\mu\) kanye nokwehluka \(1\) . Uma manje ukhetha inani elingahleliwe \(x\) kule datha futhi kufanele ulinganisele incazelo \(\mu\) ngesisekelo salokhu, ngokunembile \(x\) isilinganiso esinengqondo se- \(\mu\) (njengoba ukusabalalisa okuvamile kukhona, okukhethwe ngokungahleliwe \(x\) cishe kuseduze \(\mu\) ).
Manje ukuhlolwa kuyaphindwa - kulokhu okuthathu okuzimele, futhi idatha esabalalisiwe ngokuvamile isetha ngayinye ngokuhluka \(1\) kanye namanani \(\mu_1\) \(\mu_1\) , \ \(\mu_2\) , \(\mu_3\) . Ngemva kokuthola amanani amathathu angahleliwe \(x_1\) , \(x_2\) kanye \(x_3\) , umuntu uyalinganisela (esebenzisa inqubo efanayo) \(\mu_1=x_1\) , \(\mu_2=x_2\) kanye \(\mu_3=x_3\) .
Umphumela omangalisayo ka-James noStein ukuthi kukhona isilinganiso esingcono sokuthi \( \left( \mu_1, \mu_2, \mu_3 \right) \) (okungukuthi inhlanganisela yamasethi amathathu edatha azimele) kune- \( \left( x_1, x_2, x_3 \right) \) . I-"James Stein estimator" yilapho:
$$ \begin{pmatrix}\mu_1\\\mu_2\\\mu_3\end{pmatrix} = \left( 1-\frac{1}{x_1^2+x_2^2+x_3^2} \right) \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} \neq \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} $$
Ukuchezuka kwesikwele okumaphakathi kwalesi silinganisi-ke kuhlala kuncane kunokuchezuka kwesikwele esimaphakathi \( E \left[ \left|| X - \mu \right||^2 \right] \) sesilinganiso esivamile.
Kuyamangaza futhi mhlawumbe kuyindida ukuthi isilinganiseli sika-James-Stein sishintsha isilinganiso esivamile (ngesici esinciphayo) siye emsuka futhi ngaleyo ndlela sinikeze umphumela ongcono ezimweni eziningi. Lokhu kusebenza kubukhulu \( \geq 3 \) , kodwa hhayi kusimo sezinhlangothi ezimbili.
Incazelo enhle yejiyomethri yokuthi kungani lokhu kusebenza inikezwa uBrown & Zao . Qaphela ukuthi lokhu akusho ukuthi unesilinganiso esingcono sayo yonke idathasethi eyodwa - unesilinganiso esingcono esinobungozi obuncane obuhlanganisiwe .