# ʻO ke kuʻikahi o Stein1122

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.

Hope