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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gauss–Markov theorem ： ウィキペディア英語版 Gauss–Markov theorem In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator. Here "best" means giving the lowest variance of the estimate, as compared to other unbiased, linear estimators. The errors do not need to be normal, nor do they need to be independent and identically distributed (only uncorrelated with mean zero and homoscedastic with finite variance). The requirement that the estimator be unbiased cannot be dropped, since biased estimators exist with lower variance. See, for example, the James–Stein estimator (which also drops linearity) or ridge regression. == Statement == Suppose we have in matrix notation, :$\backslash underline\; =\; X\; \backslash underline\; +\; \backslash underline,\backslash quad\; (\backslash underline,\backslash underline\; \backslash in\; \backslash mathbb^n,\; \backslash beta\; \backslash in\; \backslash mathbb^K\; \backslash text\; X\backslash in\backslash mathbb^)$ expanding to, :$y\_i=\backslash sum\_^\backslash beta\_j\; X\_+\backslash varepsilon\_i\; \backslash quad\; \backslash forall\; i=1,2,\backslash ldots,n$ where $\backslash beta\_j$ are non-random but unobservable parameters, $X\_$ are non-random and observable (called the "explanatory variables"), $\backslash varepsilon\_i$ are random, and so $y\_i$ are random. The random variables $\backslash varepsilon\_i$ are called the "disturbance", "noise" or simply "error" (will be contrasted with "residual" later in the article; see errors and residuals in statistics). Note that to include a constant in the model above, one can choose to introduce the constant as a variable $\backslash beta\_$ with a newly introduced last column of X being unity i.e., $X\_\; =\; 1$ for all $i$. The Gauss–Markov assumptions are *$E(\backslash varepsilon\_i)=0,$ * (i.e., all disturbances have the same variance; that is "homoscedasticity"), and *$(\backslash varepsilon\_i,\backslash varepsilon\_j)\; =\; 0,\; \backslash forall\; i\; \backslash neq\; j$ for $i\backslash neq\; j$ that is, the error terms are uncorrelated. A linear estimator of $\backslash beta\_j$ is a linear combination :$\backslash widehat\backslash beta\_j\; =\; c\_y\_1+\backslash cdots+c\_y\_n$ in which the coefficients $c\_$ are not allowed to depend on the underlying coefficients $\backslash beta\_j$, since those are not observable, but are allowed to depend on the values $X\_$, since these data are observable. (The dependence of the coefficients on each $X\_$ is typically nonlinear; the estimator is linear in each $y\_i$ and hence in each random $\backslash varepsilon$, which is why this is "linear" regression.) The estimator is said to be unbiased if and only if :$E(\backslash widehat\backslash beta\_j)=\backslash beta\_j\backslash ,$ regardless of the values of $X\_$. Now, let $\backslash sum\_^K\backslash lambda\_j\backslash beta\_j$ be some linear combination of the coefficients. Then the mean squared error of the corresponding estimation is :$E\; \backslash left(\backslash left(\backslash sum\_^K\backslash lambda\_j(\backslash widehat\backslash beta\_j-\backslash beta\_j)\backslash right)^2\backslash right);$ i.e., it is the expectation of the square of the weighted sum (across parameters) of the differences between the estimators and the corresponding parameters to be estimated. (Since we are considering the case in which all the parameter estimates are unbiased, this mean squared error is the same as the variance of the linear combination.) The best linear unbiased estimator (BLUE) of the vector $\backslash beta$ of parameters $\backslash beta\_j$ is one with the smallest mean squared error for every vector $\backslash lambda$ of linear combination parameters. This is equivalent to the condition that :$V(\backslash tilde\backslash beta)-\; V(\backslash widehat\backslash beta)$ is a positive semi-definite matrix for every other linear unbiased estimator $\backslash tilde\backslash beta$. The ordinary least squares estimator (OLS) is the function :$\backslash widehat\backslash beta=(X\text{'}X)^X\text{'}y$ of $y$ and $X$ (where $X\text{'}$ denotes the transpose of $X$) that minimizes the sum of squares of residuals (misprediction amounts): :$\backslash sum\_^n\backslash left(y\_i-\backslash widehat\_i\backslash right)^2=\backslash sum\_^n\backslash left(y\_i-\backslash sum\_^K\backslash widehat\backslash beta\_j\; X\_\backslash right)^2.$ The theorem now states that the OLS estimator is a BLUE. The main idea of the proof is that the least-squares estimator is uncorrelated with every linear unbiased estimator of zero, i.e., with every linear combination $a\_1y\_1+\backslash cdots+a\_ny\_n$ whose coefficients do not depend upon the unobservable $\backslash beta$ but whose expected value is always zero. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gauss–Markov theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.