I ka makahiki 1961 ua hoʻopuka ʻo James lāua ʻo Stein i ka pepa Estimation with Quadratic Loss . Lawe i ka ʻikepili i puunaue maʻamau me ka mean ʻike ʻole \(\mu\) a me ka ʻokoʻa \(1\) . Inā koho ʻoe i kēia manawa i kahi waiwai maʻamau \(x\) mai kēia ʻikepili a pono ʻoe e koho i ka mean \(\mu\) ma ke kumu o kēia, ʻo \(x\) he kuhi kūpono no \(\mu\) (no ka loaʻa ʻana o ka puunaue maʻamau, ua kokoke paha ka \(x\) i koho ʻole ʻia i \(\mu\) ).
I kēia manawa ua hana hou ʻia ka hoʻokolohua - i kēia manawa me ʻekolu mau ʻikepili kūʻokoʻa, puʻunaue hou ʻia i kēlā me kēia me ka ʻokoʻa \(1\) a me nā kumu waiwai \(\mu_1\) , \(\mu_2\) , \(\mu_3\) . Ma hope o ka loaʻa ʻana o ʻekolu mau waiwai maʻamau \(x_1\) , \(x_2\) a me \(x_3\) , hoʻokahi koho (e hoʻohana ana i ke kaʻina hana like) \(\mu_1=x_1\) , \(\mu_2=x_2\) a me \(\mu_3=x_3\) .
ʻO ka hopena kamahaʻo o James lāua ʻo Stein ʻo ia ka ʻoi aku ka maikaʻi o ka manaʻo no \( \left( \mu_1, \mu_2, \mu_3 \right) \) (ʻo ia hoʻi ka hui ʻana o nā pūʻulu ʻikepili kūʻokoʻa ʻekolu) ma mua o \( \left( x_1, x_2, x_3 \right) \) . ʻO ka "James Stein estimator" a laila:
$$ \begin{pmatrix}\mu_1\\\mu_2\\\mu_3\end{pmatrix} = \left( 1-\frac{1}{x_1^2+x_2^2+x_3^2} \right) \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} \neq \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} $$
A laila, ʻoi aku ka liʻiliʻi o ka huinahā huinahā like o kēia mea hoʻohālikelike ma mua o ka huina huinahā like \( E \left[ \left|| X - \mu \right||^2 \right] \) o ka mea koho maʻamau.
He mea kupanaha a hoʻohālikelike paha ka hoʻololi ʻana o ka mea hoʻohālikelike ʻo James-Stein i ka mea hoʻohālikelike maʻamau (ma kahi kumu hoʻemi) i ke kumu a no laila e hāʻawi i kahi hopena maikaʻi aʻe i ka hapa nui o nā hihia. Pili kēia i nā ana \( \geq 3 \) , akā ʻaʻole i ka hihia ʻelua.
ʻO kahi wehewehe geometric maikaʻi no ke kumu i hāʻawi ʻia ai kēia hana e Brown & Zao . E hoʻomanaʻo , ʻaʻole ia he manaʻo ʻoi aku ka maikaʻi o kāu koho ʻana no kēlā me kēia papa helu helu - ʻoi aku ka maikaʻi o kāu koho me ka liʻiliʻi i hui pū ʻia.