Indida kaStein1122

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 .

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