Ngowe-1961 uJames noStein bapapasha iphepha elithi Estimation with Quadratic Loss . Thatha idata esasazwe ngokuqhelekileyo ngentsingiselo engaziwayo $$\mu$$ kunye nokwahlukana $$1$$ . Ukuba ngoku ukhetha ixabiso elingacwangciswanga $$x$$ kule datha kwaye kufuneka uqikelele intsingiselo $$\mu$$ ngokwesiseko soku, ngentuitively $$x$$ luqikelelo olufanelekileyo lwe $$\mu$$ (ekubeni unikezelo oluqhelekileyo lukhona, okukhethiweyo ngokungenamkhethe $$x$$ mhlawumbi kufutshane $$\mu$$ ).

Ngoku uvavanyo luyaphindwa - ngeli xesha kunye ezintathu ezizimeleyo, kwakhona ngokuqhelekileyo kusasazwa idatha iseti nganye ngokwahluka $$1$$ kunye namaxabiso $$\mu_1$$ , $$\mu_2$$ , $$\mu_3$$ . Emva kokufumana amaxabiso amathathu angaqhelekanga $$x_1$$ , $$x_2$$ kunye $$x_3$$ , omnye uyaqikelela (esebenzisa inkqubo efanayo) $$\mu_1=x_1$$ , $$\mu_2=x_2$$ kunye $$\mu_3=x_3$$ .

Isiphumo esimangalisayo sikaJames noStein kukuba kukho uqikelelo olungcono lwe- $$\left( \mu_1, \mu_2, \mu_3 \right)$$ (o.k.t. indibaniselwano yeeseti ezintathu ezizimeleyo zedatha) kune $$\left( x_1, x_2, x_3 \right)$$ . "Umqikelelo kaJames Stein" ungoko:

$$\begin{pmatrix}\mu_1\\\mu_2\\\mu_3\end{pmatrix} = \left( 1-\frac{1}{x_1^2+x_2^2+x_3^2} \right) \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} \neq \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}$$

Intsingiselo yokutenxa isikwere esisikweri salo mqikelelo isoloko incinci kunentsingiselo yenxaso yesikwere $$E \left[ \left|| X - \mu \right||^2 \right]$$ yoqikelelo oluqhelekileyo.

Kuyamangalisa kwaye mhlawumbi kuyaxaka ukuba umqikelelo kaJames-Stein atshintshe uqikelelo oluqhelekileyo (ngento ecuthekayo) ukuya kwimvelaphi kwaye ngaloo ndlela enika iziphumo ezingcono kuninzi lwamatyala. Oku kusebenza kwimilinganiselo $$\geq 3$$ , kodwa kungekhona kwimeko ye-dimensional.

Inkcazo enhle yejometri yokuba kutheni le nto isebenza inikezelwa nguBrown & Zao . Qaphela ukuba oku akuthethi ukuba unoqikelelo olungcono lweseti yedatha nganye - unoqikelelo olungcono olunomngcipheko omncinci odityanisiweyo .

Emva