\n\u3000\u7d71\u8a08\u5206\u6790\u306e\u5fa9\u7fd2<\/p>\n
\\(n\\)\u500b\u306e\u30c7\u30fc\u30bf\u306e\u7d44 \\((x_1,y_1),...,(x_n,y_n)\\) \u306b\u5bfe\u3057\u3066\u3001\\(y_i=\\alpha+\\beta x_i+\\varepsilon_i\\) \u3068\u3044\u3046\u5358\u56de\u5e30\u30e2\u30c7\u30eb\u3092\u8003\u3048\u3066\u3001\\(\\alpha, \\beta\\) \u306e\u6700\u5c0f\u4e8c\u4e57\u63a8\u5b9a\u91cf(OLSE: Ordinary Least Squares Estimation)\u3092\u3001\\(\\hat\\alpha, \\hat\\beta\\) \u3068\u304a\u3053\u3046\u3002\u305f\u3060\u3057\u3001\\(\\hat\\beta=(x,y\\mathrm{\u306e\u5171\u5206\u6563})\/(x\\mathrm{\u306e\u5206\u6563})\\) \u3067\u3042\u308a\u3001\\(\\hat\\alpha = \\bar y-\\hat\\beta \\bar x\\) \u306e\u95a2\u4fc2\u304c\u3042\u308b\u3002 \u3053\u306e\u3068\u304d\u3001\u6b8b\u5dee \\(e_i\\) \u3068\u306f\u5b9f\u969b\u306e\u5024 \\(y_i\\) \u3068\u4e88\u6e2c\u5024 \\(\\hat y_i=\\hat\\alpha+\\hat\\beta x_i\\) \u306e\u5dee \\(y_i-\\hat y_i\\) \u3067\u3042\u3063\u3066\u3001\u4ee5\u4e0b\u306e2\u3064\u306e\u516c\u5f0f\u3092\u6e80\u305f\u3059\u3002<\/p>\n
\u3000\u305d\u3082\u305d\u3082OLSE\u3068\u3044\u3046\u306e\u306f\u3001\u6b8b\u5dee\u306e\u4e8c\u4e57\u548c \\(\\displaystyle f(a,b) = \\sum_i (y_i-(a+bx_i))^2\\) \u3092\u6700\u5c0f\u5316\u3059\u308b\u3088\u3046\u306a \\(a, b\\)\u306e\u3053\u3068\u3092\u3044\u3046\u306e\u3067\u3042\u3063\u305f\u3002\u3053\u308c\u306f\u3059\u306a\u308f\u3061\u3001 $$\\left . \\frac{\\partial f}{\\partial a}\\right |_{(a,b)=(\\hat\\alpha,\\hat\\beta)}=0$$ $$\\left . \\frac{\\partial f}{\\partial b}\\right |_{(a,b)=(\\hat\\alpha,\\hat\\beta)}=0$$\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u304b\u3089\u3001\u5b9f\u969b\u306b\u504f\u5fae\u5206\u3057\u3066\u307f\u308b\u3068$$\\left . \\frac{\\partial f}{\\partial a}\\right |_{(a,b)=(\\hat\\alpha,\\hat\\beta)}=-2\\sum_i(y_i-(\\hat\\alpha+\\hat\\beta x_i))=-2\\sum_i e_i$$ $$\\left . \\frac{\\partial f}{\\partial b}\\right |_{(a,b)=(\\hat\\alpha,\\hat\\beta)}=-2\\sum_i(y_i-(\\hat\\alpha+\\hat\\beta x_i))x_i=-2\\sum_i e_ix_i$$
\n\u3068\u306a\u3063\u3066\u76f4\u3061\u306b\u5f97\u3089\u308c\u308b\u3002<\/p>\n
\u3000\u4ee5\u4e0a\u3088\u308a\u3001\u6b8b\u5dee \\(e_i\\) \u306f2\u3064\u306e\u516c\u5f0f\u3092\u6e80\u305f\u3059\u3001\u8a00\u3044\u63db\u3048\u308c\u3070\u30012\u3064\u306e\u5236\u7d04\u304c\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002\u5c11\u3005\u7d71\u8a08\u3092\u304b\u3058\u3063\u305f\u4eba\u306a\u3089\u3070\u3001\u81ea\u7531\u5ea6\u304c \\(n-2\\) \u3068\u306a\u308b\u3053\u3068\u306f\u5bb9\u6613\u306b\u60f3\u50cf\u304c\u3067\u304d\u3088\u3046\u3002<\/p>\n
\u3000\u3055\u3066\u3001\u56de\u5e30\u5206\u6790\u3092\u3059\u308b\u4e0a\u3067\u91cd\u8981\u306b\u306a\u308b\u306e\u306f\u8aa4\u5dee\u9805 \\(\\varepsilon_i\\) \u306e\u5206\u6563 \\(\\sigma^2\\)\u3067\u3042\u308b\u304c\u3001\u3053\u308c\u306f\u5f53\u7136\u3001\u672a\u77e5\u3067\u3042\u308b\u304b\u3089\u3001\u63a8\u5b9a\u3092\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002\u5358\u56de\u5e30\u5206\u6790\u306b\u304a\u3044\u3066\u8aa4\u5dee\u9805\u306b\u5bfe\u5fdc\u3059\u308b\u9805\u306f\u6b8b\u5dee\u9805\u3067\u3042\u308b\u304b\u3089\u3001\u6b8b\u5dee\u306e\u5206\u6563\u3092\u3082\u3063\u3066\u8aa4\u5dee\u9805\u306e\u5206\u6563\u3092\u63a8\u5b9a\u3059\u308b\u306e\u304c\u81ea\u7136\u3067\u3042\u308b\u3002
\n\u3000\u81ea\u7531\u5ea6\u304c \\(n-2\\) \u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u5ff5\u982d\u306b\u7f6e\u3051\u3070\u3001(\u4e0d\u504f\u5206\u6563\u306e\u5206\u6bcd\u304c \\(n-1\\) \u3067\u3042\u3063\u305f\u3088\u3046\u306b)\u6b8b\u5dee\u306e\u5206\u6563\u304c $$s^2=\\frac{1}{n-2}\\sum_i e_i^2$$\u3067\u3042\u308b\u3053\u3068\u306f\u660e\u3089\u304b\u3067\u3042\u308b\u304c\u3001\u3053\u308c\u304c\u672c\u5f53\u306b \\(\\sigma^2\\) \u306e\u4e0d\u504f\u63a8\u5b9a\u91cf\u3067\u3042\u308b\u3053\u3068\u3001\u6570\u5f0f\u306b\u3059\u308c\u3070\u3001\\(E(s^2)=\\sigma^2\\)\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u3002<\/p>\n
\u3000\u7c21\u5358\u306e\u305f\u3081\u306b\u4ee5\u4e0b\u306e\u5909\u6570\u3092\u7528\u610f\u3057\u3066\u304a\u3053\u3046\u3002$$A=\\sum_i (x_i-\\bar x)^2$$$$w_i=\\frac{x_i-\\bar x}{A}$$\u3000\u307e\u305f\u3001\u6b21\u306e\u516c\u5f0f\u3092\u793a\u3057\u3066\u304a\u3053\u3046\u3002$$\\sum_i(x_i-\\bar x)x_i=\\sum_i(x_i-\\bar x)^2$$$$\\sum_i(x_i-\\bar x)y_i=\\sum_i(x_i-\\bar x)(y_i-\\bar y)$$\u3000\u3053\u308c\u306f\u8003\u3048\u3066\u307f\u308c\u3070\u3069\u3046\u3068\u3044\u3046\u3053\u3068\u306f\u306a\u304f\u3066\u3001\\(\\displaystyle \\sum_i(x_i-\\bar x)=0\\) \u3092\u5229\u7528\u3057\u3066\u5de6\u8fba\u304b\u3089\u30bc\u30ed\u3092\u5f15\u3044\u3066\u3044\u308b\u3060\u3051\u306e\u8a71\u3067\u3042\u308b\u3002
\n\u3000\u307e\u305f\u3001\u3053\u308c\u3089\u3092\u5229\u7528\u3057\u3066\\(\\hat\\beta\\) \u306b\u3064\u3044\u3066\u66f8\u304d\u4e0b\u3057\u3066\u307f\u308b\u3068\u3001$$\\begin{eqnarray}\\hat\\beta&=&\\frac{\\sum_i (x_i-\\bar x)(y_i-\\bar y)}{A}\\\\
\n&=&\\frac{\\sum_i (x_i-\\bar x)y_i}{A}\\\\
\n&=&\\frac{\\sum_i (x_i-\\bar x)(\\alpha+\\beta x_i+\\varepsilon_i)}{A}\\end{eqnarray}$$\u3000\u3053\u3053\u3067\u5206\u5b50\u306b\u3064\u3044\u3066\u8003\u3048\u3066\u307f\u308b\u3068\u3001$$\\begin{eqnarray}\\sum_i (x_i-\\bar x)(\\alpha+\\beta x_i+\\varepsilon_i)&=&\\alpha\\sum_i(x_i-\\bar x)+\\beta\\sum_i(x_i-\\bar x)x_i+\\sum_i(x_i-\\bar x)\\varepsilon_i\\\\
\n&=&\\beta\\cdot A+\\sum_i(x_i-\\bar x)\\varepsilon_i\\end{eqnarray}$$\u3060\u304b\u3089\u3001\u7d50\u5c40$$\\hat\\beta=\\beta+\\sum_i w_i\\varepsilon_i$$\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n
\u3000\u307e\u305a\u306f\u6b8b\u5dee\u306e\u4e8c\u4e57\u548c\u304b\u3089\u30b9\u30bf\u30fc\u30c8\u3057\u3088\u3046\u3002$$\\begin{eqnarray}\\sum_i e_i^2&=&\\sum_i e_i\\cdot e_i\\\\
\n&=&\\sum_i e_i(y_i-\\hat y_i)\\\\
\n&=&\\sum_i e_i(\\alpha+\\beta x_i+\\varepsilon_i-(\\hat\\alpha+\\hat\\beta x_i))\\\\
\n&=&(\\alpha-\\hat\\alpha)\\sum_i e_i + (\\beta-\\hat\\beta)\\sum_i e_ix_i +\\sum_i e_i\\varepsilon_i\\\\
\n&=&\\sum_i e_i\\varepsilon_i\\quad\\cdots(1)\\end{eqnarray}$$\u6700\u5f8c\u306e\u5909\u5f62\u306f\u516c\u5f0f(i), (ii)\u3092\u4f7f\u3063\u305f\u3002
\n\u3000\u6b21\u306b\u3001\u3082\u3046\u4e00\u3064\u306e \\(e_i\\) \u3092\u5225\u306e\u5f62\u3067\u66f8\u304d\u8868\u3057\u3066\u307f\u308b\u3002$$\\begin{eqnarray}e_i&=&y_i-\\hat y_i=y_i-(\\hat\\alpha+\\hat\\beta x_i)\\\\
\n&=&y_i-(\\bar y-\\hat\\beta \\bar x+\\hat\\beta x_i)\\\\
\n&=&(y_i-\\bar y)-\\hat\\beta(x_i-\\bar x)\\end{eqnarray}$$\u3000\u3053\u3053\u3067\u3001\\(\\bar y=\\alpha+\\beta \\bar x+\\bar \\varepsilon\\) (\u305f\u3060\u3057 \\(\\displaystyle \\bar \\varepsilon=\\frac 1n \\sum_i \\varepsilon_i\\))\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001
\n$$\\begin{eqnarray}e_i&=&\\beta(x_i-\\bar x)+\\varepsilon_i-\\bar\\varepsilon -\\hat\\beta(x_i-\\bar x)\\\\
\n&=&(\\beta-\\hat\\beta)(x_i-\\bar x)+\\varepsilon_i-\\bar\\varepsilon\\end{eqnarray}$$\u3000\u3055\u3089\u306b\u3001\\(\\displaystyle \\hat\\beta-\\beta=\\sum_i w_i\\varepsilon_i\\)\u3060\u3063\u305f\u3053\u3068\u304b\u3089\u3001$$e_i=-(x_i-\\bar x)\\sum_j w_j\\varepsilon_j+\\varepsilon_i-\\bar\\varepsilon$$\u3068\u306a\u308b\u3002\u3053\u308c\u3092(1)\u306b\u623b\u3057\u3066\u3084\u308b\u3068\u3001$$\\begin{eqnarray}\\sum_i e_i^2&=&\\sum_i e_i\\varepsilon_i\\\\
\n&=&-\\sum_{i,j}(x_i-\\bar x)\\varepsilon_iw_j\\varepsilon_j+\\sum_i(\\varepsilon_i-\\bar\\varepsilon)\\varepsilon_i\\\\
\n&=&-\\sum_{i,j}(x_i-\\bar x)\\varepsilon_iw_j\\varepsilon_j+\\sum_i\\varepsilon_i^2-\\frac1n \\sum_{i,j}\\varepsilon_i\\varepsilon_j\\\\
\n&=&-\\sum_{i,j}\\left((x_i-\\bar x)w_j+\\frac1n\\right)\\varepsilon_i\\varepsilon_j+\\sum_i\\varepsilon_i^2\\end{eqnarray}$$
\n\u3000\u3053\u3053\u3067\u4e21\u8fba\u306e\u671f\u5f85\u5024\u3092\u53d6\u308b\u306e\u3060\u304c\u3001\u56de\u5e30\u5206\u6790\u306e\u6a19\u6e96\u7684\u4eee\u5b9a\u306b\u3088\u308a\u3001\\(i\\neq j\\) \u306a\u3089\u3070 \\(E(\\varepsilon_i\\varepsilon_j)=0\\) \u304c\u6210\u308a\u7acb\u3063\u3066\u3044\u308b\u306e\u3067\u3001\\(\\displaystyle E\\left(\\sum_{i,j}\\varepsilon_i\\varepsilon_j\\right)=E\\left(\\sum_i \\varepsilon_i^2\\right)\\) \u3068\u306a\u308b\u3002\u3086\u3048\u306b
\n$$\\begin{eqnarray}E\\left(\\sum_i e_i^2\\right)&=&E\\left(-\\sum_i\\left((x_i-\\bar x)w_i-\\frac1n\\right)\\varepsilon_i^2+\\sum_i \\varepsilon_i^2\\right)\\\\
\n&=&-\\sum_i\\left((x_i-\\bar x)w_i+\\frac1n\\right)E(\\varepsilon_i^2)+\\sum_i E(\\varepsilon_i^2)\\end{eqnarray}$$
\n\u3000\u3053\u3053\u3067 \\(E(\\varepsilon_i^2)\\) \u3068\u306f\u8aa4\u5dee\u9805\u306e\u5206\u6563 \\(V(\\varepsilon_i)=\\sigma^2\\) \u306b\u307b\u304b\u306a\u3089\u306a\u3044\u3002\u3086\u3048\u306b$$E\\left(\\sum_i e_i^2\\right)=-\\sigma^2\\sum_i\\left((x_i-\\bar x)w_i+\\frac1n\\right)+n\\sigma^2$$\u3067\u3042\u308b\u3002\u3055\u3089\u306b\u3001$$\\begin{eqnarray}\\sum_i (x_i-\\bar x)w_i&=&\\sum_i (x_i-\\bar x)\\frac{x_i-\\bar x}{A}\\\\
\n&=&\\frac{\\sum_i(x_i-\\bar x)^2}{A}\\\\
\n&=&A\/A\\\\
\n&=&1\\end{eqnarray}$$\u306a\u306e\u3067\u3001\u7d50\u5c40$$E\\left(\\sum_i e_i^2\\right)=-\\sigma^2(1+1)+n\\sigma^2=(n-2)\\sigma^2$$
\n\u3088\u3063\u3066\u3001$$E\\left(\\frac{1}{n-2}\\sum_i e_i^2\\right)=\\sigma^2$$\u304c\u793a\u3055\u308c\u3001\u6b8b\u5dee\u5206\u6563\u306f\u8aa4\u5dee\u9805\u306e\u5206\u6563\u306e\u4e0d\u504f\u63a8\u5b9a\u91cf\u3067\u3042\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f\u3002<\/p>\n
\u3000\u91cd\u56de\u5e30\u306e\u3068\u304d\u3082\u540c\u3058\u611f\u3058\u3067\u884c\u3051\u305d\u3046\u306a\u96f0\u56f2\u6c17\u304c\u3042\u308b\u304c\u3001\u308f\u3056\u308f\u3056\u8a3c\u660e\u3059\u308b\u5fc5\u8981\u304c\u306a\u3044\u306e\u3067\u3053\u308c\u3067\u3088\u3057\u3068\u3059\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u3000\u7d71\u8a08\u5206\u6790\u306e\u5fa9\u7fd2 \\(n\\)\u500b\u306e\u30c7\u30fc\u30bf\u306e\u7d44 \\((x_1,y_1),...,(x_n,y_n)\\) \u306b\u5bfe\u3057\u3066\u3001\\(y_i=\\alpha+\\beta x_i+\\varepsilon_i\\) \u3068\u3044\u3046\u5358\u56de\u5e30\u30e2\u30c7\u30eb\u3092\u8003\u3048\u3066\u3001\\(<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[87],"tags":[86],"_links":{"self":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/1051"}],"collection":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/comments?post=1051"}],"version-history":[{"count":27,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/1051\/revisions"}],"predecessor-version":[{"id":1293,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/1051\/revisions\/1293"}],"wp:attachment":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/media?parent=1051"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/categories?post=1051"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/tags?post=1051"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}