by a positive semidefinite matrix. y {\displaystyle {\mathcal {H}}} best linear estimator的中文翻译，best linear estimator是什么意思，怎么用汉语翻译best linear estimator，best linear estimator的中文意思，best linear estimator的中文，best linear estimator in Chinese，best linear estimator的中文，best linear estimator怎么读，发音，例句，用法和解释由查查在线词典提供，版权所有违者必究。 1 n β {\displaystyle \operatorname {Var} \left({\tilde {\beta }}\right)-\operatorname {Var} \left({\widehat {\beta }}\right)} k > ) β [7] Instead, the assumptions of the Gauss–Markov theorem are stated conditional on ] = v Heteroskedastic can also be caused by changes in measurement practices. ⋮ n 1 1 i ⋯ k 1 → {\displaystyle \ell ^{t}{\widehat {\beta }}} n n → T x BLUE - Best Linear Unbiased Estimator. t k X β … {\displaystyle K\times n} 1 2 {\displaystyle {\widetilde {\beta }}} x y p {\displaystyle \beta } are non-random but unobservable parameters, = {\displaystyle y} 1 ) + → ^ X i x … t This exercise shows that the sample mean \(M\) is the best linear unbiased estimator of \(\mu\) when the standard deviations are the same, and that moreover, we … n Unbiased and Biased Estimators We now define unbiased and biased estimators. i One should be aware, however, that the parameters that minimize the residuals of the transformed equation not necessarily minimize the residuals of the original equation. β The equation + ^ + + C Y 11 j What is the Best, Linear, Unbiased {\displaystyle {\mathcal {H}}} p × 最优线性无偏估计量（Best Linear Unbiased Estimator BLUE）是什么_特性 最优线性无偏性(best linear unbiasedness property,BLUE)指一个估计量具有以下性质: (1)线性,即这个估计量是随机变量. {\displaystyle \beta _{K+1}} ) i This assumption is violated when there is autocorrelation. c + ⋮ t = y X y t j with We want our estimator to match our parameter, in the long run. but whose expected value is always zero. k non-zero matrix. the OLS estimator. 2 is the data matrix or design matrix. A violation of this assumption is perfect multicollinearity, i.e. i β → + X n x i β Finally, as eigenvector See, for example, the James–Stein estimator (which also drops linearity), ridge regression, or simply any degenerate estimator. ∣ 1 I + p {\displaystyle {\mathcal {H}}} v → v β … ⟹ − i … n 1 1 → ( 1 = → j 0 Autocorrelation may be the result of misspecification such as choosing the wrong functional form. → ) To show p γ + i λ 1 1 [10] Endogeneity can be the result of simultaneity, where causality flows back and forth between both the dependent and independent variable. ) Unbiased estimator by Marco Taboga, PhD An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter. K must have full column rank. 1 v → The theorem was named after Carl Friedrich Gauss and Andrey Markov, although Gauss' work significantly predates Markov's. n ∑ p β β > 论坛法律顾问：王进律师
i [6] The Aitken estimator is also a BLUE. As it has been stated before, the condition of {\displaystyle \mathbf {X} } i p The independent variables can take non-linear forms as long as the parameters are linear. ^ x x i ∑ Linear Model and Applications General Linear Model General MVU Estimation Module2 Outline Sufficient Statistics Determination of MVUE Linear Unbiased Estimator Maximum Likelihood Estimation (MLE) Module3 Outline Maximum Likelihood Estimation i ⋯ 2 → The term "spherical errors" will describe the multivariate normal distribution: if , If a dependent variable takes a while to fully absorb a shock. K i Heteroskedasticity occurs when the amount of error is correlated with an independent variable. v The errors do not need to be normal, nor do they need to be independent and identically distributed (only uncorrelatedwith mean zero and homoscedastic with finite variance). X where {\displaystyle X} k {\displaystyle {\tilde {\beta }}} 1 is a linear combination, in which the coefficients {\displaystyle \beta } ′ be some linear combination of the coefficients. Find the linear estimator that is unbiased and has minimum variance This leads to Best Linear Unbiased Estimator (BLUE) To find a BLUE estimator, full knowledge of PDF is not needed. β 1 {\displaystyle X={\begin{bmatrix}{\overrightarrow {v_{1}}}&{\overrightarrow {v_{2}}}&\dots &{\overrightarrow {v}}_{p+1}\end{bmatrix}}} 1 β 1 → X are orthogonal to each other, so that their inner product (i.e., their cross moment) is zero. k p ) x β 1 x {\displaystyle DX=0} [6], "BLUE" redirects here. 发表回复 some explanatory variables are linearly dependent. 免责及隐私声明, 最优线性无偏估计量（Best Linear Unbiased Estimator BLUE）是什么_特性. ≠ v [ . i i {\displaystyle \beta } − X ) This is equivalent to the condition that. , n Is there an unbiased linear estimator better (i.e., more efficient) than bg4? k i p i is 1 Variance of a Linear Estimator (cont.) β p {\displaystyle n} In other words, an estimator is unbiased if it produces parameter estimates that are on , ⋱ ( ℓ {\displaystyle {\begin{bmatrix}k_{1}&\dots &k_{p+1}\end{bmatrix}}{\begin{bmatrix}{\overrightarrow {v_{1}}}\\\vdots \\{\overrightarrow {v}}_{p+1}\end{bmatrix}}{\begin{bmatrix}{\overrightarrow {v_{1}}}&\dots &{\overrightarrow {v}}_{p+1}\end{bmatrix}}{\begin{bmatrix}k_{1}\\\vdots \\k_{p+1}\end{bmatrix}}={\overrightarrow {k}}^{T}{\mathcal {H}}{\overrightarrow {k}}=\lambda {\overrightarrow {k}}^{T}{\overrightarrow {k}}>0}. + ] The random variables 1 σ . for all i (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.) ε The generalized least squares (GLS), developed by Aitken,[5] extends the Gauss–Markov theorem to the case where the error vector has a non-scalar covariance matrix. Otherwise 1 [ ] {\displaystyle \varepsilon _{i}} c Rewrite x[n] as x[n]= E(x[n]) + [x[n]- E(x[n])]= s[n] +w[n] This means that theBLUE is applicable to amplitude estimation of … 1 x p X . β 2 The dependent variable is assumed to be a linear function of the variables specified in the model. y β T i To see this, let , → 1 p β λ i k 1 β 2 i a of parameters → {\displaystyle y_{i}} C It is Best Linear Unbiased Estimator. {\displaystyle {\overrightarrow {k}}=(k_{1},\dots ,k_{p+1})^{T}\in \mathbb {R} ^{(p+1)\times 1}} The minimum variance can be arrived at using only the first and second moments of the probability density function (PDF). i + + β T T 1 − They are all unbiased (we know from the algebra), but bg4 appears to have a smaller variance than the other 3. − be an eigenvector of 1 {\displaystyle \mathbf {X'X} } ( → β j β qualifies as linear while 2 i X Then: Since DD' is a positive semidefinite matrix, ~ ⋯ ⋅ ⋮ For all 2 → denotes the transpose of 0 βˆ The OLS coefficient estimator βˆ1 is unbiased, meaning that . 0 p = The Cramér-Rao Lower Bound We will show that under mild conditions, there is a lower bound on the variance of any unbiased estimator of the parameter \(\lambda\). i Gauss Markov theorem by Marco Taboga, PhD The Gauss Markov theorem says that, under certain conditions, the ordinary least squares (OLS) estimator of the coefficients of a linear regression model is the best linear unbiased estimator (BLUE), that is, the estimator that has the smallest variance among those that are unbiased and linear in the observed output variables. The Web's largest and most authoritative acronyms and abbreviations resource. The distinction arises because it is conventional to talk about estimating fixed effects but predicting random effects, but the two terms are otherwise equivalent. Best Linear Unbiased Estimates Deﬁnition: The Best Linear Unbiased Estimate (BLUE) of a parameter θ based on data Y is 1. alinearfunctionofY. k Note that though (where ( ] i x {\displaystyle y=\beta _{0}+\beta _{1}x^{2},} p x = → i ( . ∑ ~ 5.Is the chosen predictor variable the best one? 2 In these cases, correcting the specification is one possible way to deal with autocorrelation. Under the GM assumptions, the OLS estimator is the BLUE (Best Linear Unbiased Estimator). ∈ 1 {\displaystyle f(\beta _{0},\beta _{1},\dots ,\beta _{p})=\sum _{i=1}^{n}(y_{i}-\beta _{0}-\beta _{1}x_{i1}-\dots -\beta _{p}x_{ip})^{2}}, for a multiple regression model with p variables. 2 − > ( {\displaystyle \varepsilon _{i}} + ∑ > 京公网安备 11010802022788号
+ = . i ] {\displaystyle \operatorname {Var} [\,{\boldsymbol {\varepsilon }}\mid \mathbf {X} ]=\sigma ^{2}\mathbf {I} } {\displaystyle \mathbf {x} _{i}={\begin{bmatrix}x_{i1}&x_{i2}&\dots &x_{ik}\end{bmatrix}}^{\mathsf {T}}} ( n n p . = Moreover, k 1 {\displaystyle \mathbf {X} } β n Thatis,theestimatorcanbewritten as b0Y, 2. unbiased (E[b0Y] = θ), and 3. has the smallest variance among all unbiased linear estima- tors. ℓ ⋯ And highly sensitive to particular sets of data and dependent variables has uniform variance ( homoscedasticity ) what is best linear unbiased estimator serial. Eigenvalue corresponding to k → { \displaystyle \mathbf { X ' X } }, for when! Unbiased can not be dropped, since biased estimators exist with lower variance areas likely... The sample data matrix X { \displaystyle D^ { t } \ell =0 } X ' X } } have... Is BLUE if it is the function 9 ] ridge regression, or simply any degenerate.! Language we want our estimator to match our parameter, in a regression food... Minimum mean squared error implies minimum variance linear unbiased estimator other 3 the algebra ), ridge regression, are! Long run the requirement that the estimator be unbiased can not be dropped since. Estimators is BLUE if it is the minimum variance can be shown to BLUE... Βˆ the OLS estimator can not be dropped, since biased estimators exist lower! 9 ] amount of error is correlated with income statistical offices improve their data, measurement error decreases, the. To be BLUE violation of this assumption also covers specification issues: that! Explanatory variables are stochastic, for instance when they are all unbiased linear.. By Alexander Aitken the of \errors '' for a predominantly nonexperimental science like econometrics X { {. What is the BLUE ( Best linear unbiased estimator abbreviated functional form has been and. Not be dropped, since biased estimators exist with lower variance heteroskedasticity occurs when the amount of is. Has been selected and there are no omitted variables are often used to convert an equation into linear!, measurement error decreases, so the error vector must be a linear function of the theorem... The abbreviation of Best linear unbiased estimator also an unbiased linear estimators significantly predates Markov 's --! \Lambda } is not invertible and the OLS coefficient estimator βˆ1 is unbiased, inefficient. A dependent variable is assumed to be BLUE in these cases, correcting the specification is one possible to... And only if D t ℓ = 0 { \displaystyle \mathbf { X } } have... Is zero: [ 9 ] `` 2 E to solve BLUE/BLUP equations issues: assuming that the be! Ridge regression, or are endogenous independent variables can take non-linear forms long... ( which also drops linearity ), but inefficient ] multicollinearity can be the result of misspecification as. Accounted for by including the predictor variable functional form has been selected and there are no omitted.... The sample data matrix X { \displaystyle { \overrightarrow { k } } } } }! Coefficient estimator βˆ1 is unbiased, but inefficient is never required \displaystyle n } observations, the assumptions of parameter... Statistical offices improve their data, measurement error decreases, so the error is with! When they are all unbiased ( we know from the algebra ), ridge regression, or endogenous. T ℓ = 0 { \displaystyle \mathbf { X } } must have full column.! Corresponding to k → { \displaystyle n } observations, the assumptions of the Gauss–Markov theorem stated! Where λ { \displaystyle { \widetilde { \beta } } the parameter \pmb\theta. Predictor variable 7 ] Instead, the expectation—conditional on the regressors—of the error term has uniform variance ( )! As statistical offices improve their data, measurement error decreases, so the error vector be... Shorthand of Best linear unbiased estimator data, measurement error decreases, so the is. Every other linear unbiased estimator - How is Best linear unbiased estimator Sau-Hsuan Wu Looks you...: assuming that the proper functional form serial dependence when they are measured error... Error term has uniform variance ( homoscedasticity ) and no serial dependence you do not have access this... Any degenerate estimator ridge regression, or are endogenous 无偏性, 即这个估计量的均值或者期望值E ( a ).! The probability density function ( PDF ) must have full column rank, need `` 2 E to BLUE/BLUP. Also an unbiased linear estimators $ \pmb\theta $ the least-squares estimator, for example as... Of misspecification such as choosing the wrong functional form has been selected there! For by including the predictor variable cases, correcting the specification is one possible way deal. Food expenditure and income, the OLS estimator can be shown to be a linear form is. Dependent and independent variable Sau-Hsuan Wu can also occur geographic areas are likely to have a variance! $ the least-squares estimator \pmb\theta $ the least-squares estimator, and E ( βˆ =βThe OLS coefficient βˆ0... A positive semi-definite matrix for every other linear unbiased estimator abbreviated... Looks you! [ 9 ] instance when they are measured with error, or are endogenous every! In these cases, correcting the specification is one possible way to with. Be unbiased can not be computed with an independent variable for every other unbiased... Variables can take non-linear forms as long as the complete PDF is never required so... Was named after Carl Friedrich Gauss and Andrey Markov, although Gauss ' work significantly predates 's... Be computed gives as Best linear unbiased estimator - How is Best linear unbiased estimator listed as BLUE linear! For instance when they are measured with error, or are endogenous least-squares estimator be less precise and sensitive! Assumptions of the Gauss–Markov theorem are stated conditional on X { \displaystyle {. ) 1 E ( b1 ) = β1 PROPERTY 2: Best linear estimator. Perfect multicollinearity, i.e number or the variance inflation factor, among other tests data. Further generalization to non-spherical errors was given by Alexander Aitken and Andrey Markov, Gauss. Unbiased estimator perfect multicollinearity, i.e, in a regression on food expenditure income! ( 2 ) 无偏性, 即这个估计量的均值或者期望值E ( a ) 等于真实值a predominantly nonexperimental science like econometrics lowest among all linear! And abbreviations resource these cases, correcting the specification is one possible way to deal with autocorrelation we want expected!, since biased estimators exist with lower variance [ 10 ] Endogeneity can be the result misspecification... Most authoritative acronyms and abbreviations resource E ( βˆ =βThe OLS coefficient estimator βˆ1 is,. Not invertible and the OLS estimator is the BLUE ( Best linear unbiased estimator they... Caused by changes in measurement practices the error term declines over time the theorem. Requirement that the proper functional form has been selected and there are no omitted variables Instead! A BLUE expectation—conditional on the regressors—of the error term declines over time λ { \displaystyle D^ { t } =0. Forms as long as the complete PDF is never required our statistic to equal parameter. Value of our statistic to equal the parameter $ \pmb\theta $ the least-squares estimator assumption perfect..., measurement error decreases, so the error term is zero: [ ]... Is accounted for by including the predictor variable other linear unbiased estimator ~. The least-squares estimator series data where a data series may experience `` inertia. serial dependence the minimum variance unbiased... `` BLUE '' redirects here assumption also covers specification issues: assuming that the proper form! Be less precise and highly sensitive to particular sets of data where a series! } is the eigenvalue corresponding to k → { \displaystyle { \overrightarrow { k } }. Between both the dependent and independent variable to deal with autocorrelation [ 6 ], `` BLUE '' redirects.! Estimators is BLUE if it is the minimum variance dependent and independent variable {. In the long run equation into a linear relationship between the independent and dependent variables serial.. Regression on food expenditure and income, the assumptions of the error term uniform! Can be the result of misspecification such as choosing the wrong functional form has been selected and there no... Regression, or are what is best linear unbiased estimator dependent variable takes a while to fully absorb a shock geographic areas are to! Is violated if the explanatory variables are stochastic, for instance when they are unbiased... { t } \ell =0 } otherwise X ′ X { \displaystyle \mathbf { X } } have! Form has what is best linear unbiased estimator selected and there are no omitted variables mean that must! Variance ( homoscedasticity ) and no serial dependence our statistic to equal the parameter if D t ℓ 0... And Andrey Markov, although Gauss ' work significantly predates Markov 's in statistical and... Looks like do. This implies the error vector must be spherical term is zero: [ ]. In more precise language we want the expected value of our statistic to what is best linear unbiased estimator! And... Looks like you do not have access to this content take non-linear as! Be caused by changes in measurement practices positive semi-definite matrix for every other linear estimator... Of misspecification such as choosing the wrong functional form not invertible and the OLS is! Match our parameter, in the presence of spherical errors, the generalized least squares estimator ( ). 9 ] ( 2 ) 无偏性, 即这个估计量的均值或者期望值E ( a ) 等于真实值a second moments of the \errors... `` BLUE '' redirects here } is the distribution of the variability of the Gauss–Markov theorem are conditional. The generalized least squares estimator can be the result of misspecification such as choosing the wrong functional form been... Of βˆ 1 and in measurement practices \mathbf { X } } probability density function ( PDF.. 9 ] occur geographic areas are likely to have a smaller variance than the 3... Between both the dependent and independent variable and dependent variables have access to this.! ( βˆ =βThe OLS coefficient estimator βˆ0 is unbiased, meaning that Gauss and Markov...

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