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children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "が予測するために使用する変数(説明変数)になります。そして、与えられたデータから", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mo, { children: "⋯" }), _jsx(_components.mtext, { children: " " }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "k" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta_0,\\beta_1,\\cdots,\\beta_k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "(偏回帰係数という)を計算し、未知のデータを当てはめることで", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "y" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "y" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" })] }) })] }) }), "を予測できるようになります。"] }), "\n", _jsx(_components.p, { children: "では、とりあえずPythonでの線形回帰分析の実行例を見てみましょう。以下のようなBoston House Priceというデータをロードします。" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " pandas ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " pd\n", _jsx(_components.span, { className: "hljs-keyword", children: "from" }), " sklearn.datasets\n", _jsx(_components.span, { className: "hljs-keyword", children: "import" }), " load_boston\n", _jsx(_components.span, { className: "hljs-keyword", children: "import" }), " statsmodels.api ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " sm\n", _jsx(_components.span, { className: "hljs-comment", children: "#「boston house price」というデータをロード" }), "\nboston = load_boston()\ndf = pd.DataFrame(boston.data, columns=boston.feature_names)\ndf[", _jsx(_components.span, { className: "hljs-string", children: "'PRICE'" }), "] = boston.target\n", _jsx(_components.span, { className: "hljs-comment", children: "#boston house priceの上位5件を表示" }), "\ndf[:", _jsx(_components.span, { className: "hljs-number", children: "5" }), "]\n"] }) }), "\n", _jsx(_components.p, { children: "結果" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog2/boston_data.png", alt: "boston_data", title: "boston_data" }) }), "\n", _jsx(_components.p, { children: "次に線形回帰分析を使って、PRICEを予測してみます。" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-comment", children: "# 変数「CRIM」から「LSTAT」を説明変数として、「PRICE」を予測する" }), "\nX = df.drop(", _jsx(_components.span, { className: "hljs-string", children: "\"PRICE\"" }), ", ", _jsx(_components.span, { className: "hljs-number", children: "1" }), ")\ny = boston.target\n", _jsx(_components.span, { className: "hljs-comment", children: "# add_constantで定数項を挿入" }), "\nmodel = sm.OLS(y, sm.add_constant(X))\nresult = model.fit()\nresult.summary()\n"] }) }), "\n", _jsx(_components.p, { children: "結果(一部抜粋)" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog2/linear_reg2.png", alt: "線形回帰2", title: "線形回帰2" }) }), "\n", _jsx(_components.p, { children: "コードを書けば結果が出てきます。ちなみに、「coef」が偏回帰係数、「std_err」が標本誤差、「t」がt値、「P>|t|」がp値、[0.025,0.075]が信頼区間です。なんとなく決定係数が高ければ良いし、回帰係数とかは知っているし、p値は低い方が良いのは知っているが、具体的に何と言われると難しいって人は多いのではないのでしょうか(昔の私はそうでした)。" }), "\n", _jsx(_components.h2, { children: "統計学の考え方" }), "\n", _jsx(_components.h3, { children: "記述統計と推測統計" }), "\n", _jsxs(_components.p, { children: ["線形回帰分析においてまず念頭においておきたいのは", _jsx(_components.strong, { children: "推測統計" }), "という概念です。実際、統計学は大きく分けて", _jsx(_components.strong, { children: "記述統計" }), "と", _jsx(_components.strong, { children: "推測統計" }), "に分かれます。"] }), "\n", _jsx(_components.p, { children: "記述統計とは、今あるデータに対して標本平均や標本分散等の要約統計量を算出する統計です。データ分析を行う上でまずやらなければならないことですね。" }), "\n", _jsx(_components.p, { children: "推測統計とは、今あるデータから母集団のパラメータ(=母数)を推測する統計です。推測統計を考える基本的な考え方として、母集団と標本の概念が必要になってきます。" }), "\n", _jsx(_components.h3, { children: "母集団と標本" }), "\n", _jsxs(_components.p, { children: ["さて、推測統計の目的を考えてみましょう。推測統計は、今あるデータについての情報を知りたいのではなく、その先の得られていないデータを含めた全体のデータの統計量を知りたい、というのが目的です。この全体のデータを", _jsx(_components.strong, { children: "母集団" }), "と言います。そして、今得られたデータは母集団からランダムに抽出されて得られたデータだと仮定します。この抽出されたデータを", _jsx(_components.strong, { children: "標本" }), "と言います。言い換えれば、ある1標本から母数を求めようとしているのが推測統計なわけですね。"] }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog2/population_sample.png", alt: "母集団と標本", title: "母集団と標本" }) }), "\n", _jsxs(_components.p, { children: ["線形回帰分析の例だと、与えられたデータから求められた回帰式はあくまである", _jsx(_components.strong, { children: "1標本の回帰式" }), "でしかありません。推測統計の概念からいえば、今求めたいのは", _jsx(_components.strong, { children: "母集団が従う回帰式" }), "になります。当然出てくる疑問として、標本から求めた回帰式は母集団が従う回帰式とどれくらい違うの?とか、標本からの回帰式は母集団の回帰式に近づくの?とかあると思います。これを理論に基づいて証明していったのが線形回帰分析であり、例の結果として出てきたものになります。"] }), "\n", _jsx(_components.h2, { children: "線形回帰分析の理論" }), "\n", _jsxs(_components.p, { children: ["ここからが本題です。推測統計がある程度わかったところで、線形回帰分析の理論を軽く見てみましょう。なお、標本から求められた偏回帰係数を", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mo, { children: "^" })] }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\hat{\\beta}_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1523em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9579em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" })] }), _jsxs(_components.span, { style: { top: "-3.2634em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "、母集団が從う回帰式の偏回帰係数を", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\beta_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "とします。"] }), "\n", _jsx(_components.h3, { children: "偏回帰係数は不偏推定量である" }), "\n", _jsxs(_components.p, { children: ["標本偏回帰係数から母集団の偏回帰係数を推測しようとしたとき、当然のことながら値がずれるので、母集団の偏回帰係数を推定するのにある程度良い性質が必要となります。不偏推定量とは良い性質の一つで、", _jsx(_components.strong, { children: "標本から算出された統計量を平均すれば母集団のパラメータになる" }), "という性質を持つ統計量のことを言います。"] }), "\n", _jsx(_components.p, { children: "線形回帰分析においては、ある条件を満たせば、偏回帰係数は不偏推定量になります。(詳しく知りたい方はガウス・マルコフの定理を調べてみてください。)" }), "\n", _jsx(_components.p, { children: "つまり、いろんな標本で回帰分析を行えばその偏回帰係数の平均が母集団の偏回帰係数になるということができます。式で書けば以下のようになります。" }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "E" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsxs(_components.msub, { children: [_jsxs(_components.mover, { accent: "true", children: [_jsx(_components.mi, { children: "β" }), _jsx(_components.mo, { children: "^" })] }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " \\beta_i = E[\\hat{\\beta}_i]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2079em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "E" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord accent", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.9579em" }, children: [_jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" })] }), _jsxs(_components.span, { style: { top: "-3.2634em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "accent-body", style: { left: "-0.1667em" }, children: _jsx(_components.span, { className: "mord", children: "^" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1944em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "イメージとしてはこんな感じです。" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog2/huhen.png", alt: "不偏推定量", title: "不偏推定量" }) }), "\n", _jsx(_components.h3, { children: "偏回帰係数は一致推定量である" }), "\n", _jsxs(_components.p, { children: ["考え方は不偏推定量似ています。一致推定量とは、", _jsx(_components.strong, { children: "サンプルサイズを増やしていけば、標本から算出された統計量が母集団のパラメータに近づく" }), "という性質を持つ統計量のことを言います。(正確にいうと、近づくとは確率収束の意味です。ちなみに、標本平均が一致推定量であることと大数の弱法則が成り立つことは同値です。大数の法則や確率収束については", _jsx(_components.a, { href: "http://mathshingo.chillout.jp/blog1.html", children: "前回のブログ記事" }), "を参考にしてください。)"] }), "\n", _jsx(_components.p, { children: "線形回帰の偏回帰係数は、一致推定量でもあります。サンプルサイズを大きくしていけば今回のデータで出した偏回帰係数の分散が小さくなり、母集団の偏回帰係数に近づきます。数式で書くと以下のようになります。" }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∀" }), _jsx(_components.mi, { children: "ε" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mspace, { width: "2.8453em" }), _jsxs(_components.munder, { children: 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children: "ε" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " \\forall \\varepsilon>0 ,\\hspace{1cm}\\lim_{n\\rightarrow \\infty}P(|\\hat{\\beta}_i-\\beta_i|\\ge\\varepsilon)=0 " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7335em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord", children: "∀" }), _jsx(_components.span, { className: "mord mathnormal", children: "ε" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ">" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), 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_jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05278em" }, children: "β" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0528em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mord", children: "∣" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ε" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "イメージとしてはこんな感じです。" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog2/itti.png", alt: "一致推定量", title: "一致推定量" }) }), "\n", _jsx(_components.p, { children: "上記から、標本の選ばれ方やサンプルサイズによりますが標本偏回帰係数は母集団の偏回帰係数に様々な意味で近いことがわかります。従って、標本から得られたパラメータをもとに分析結果を解釈していくことが可能になるのです。" }), "\n", _jsx(_components.p, { children: "仮説検定とp値についても書こうと思っていたのですが、予想以上に長くなってしまったため、次回に回します。" }), "\n", _jsx(_components.h2, { children: "いったんまとめ" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "統計学には記述統計と推測統計がある。" }), "\n", _jsx(_components.li, { children: "線形回帰分析で求められるパラメータは標本からのものであるが、実際に知りたいのは母集団のパラメータである。" }), "\n", _jsx(_components.li, { children: "偏回帰係数は不偏性、一致性をもち、性質のいい推定量である。" }), "\n"] })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = Object.assign({}, _provideComponents(), props.components); return MDXLayout ? _jsx(MDXLayout, Object.assign({}, props, { children: _jsx(_createMdxContent, props) })) : _createMdxContent(props); } return { default: MDXContent }; a:["$","div",null,{"className":"container mx-auto px-4 py-8 max-w-7xl","children":["$","div",null,{"className":"flex flex-col md:flex-row gap-8","children":[["$","article",null,{"className":"w-full md:w-[70%] bg-white shadow-lg rounded-lg p-8","children":[["$","header",null,{"className":"mb-8 border-b 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