# best linear unbiased estimator characteristics

\begin{pmatrix} The European Mathematical Society, $\def\mx#1{ {\mathbf{#1}}}$ can be expressed, for example, in $\mx{X}_f\BETA$ is a given estimable parametric function. Minimizing $$J$$ with respect to $$\textbf{a}$$ is equivalent to setting the first derivative of $$J$$ w.r.t $$\textbf{a}$$ to zero. \{ \BLUE(\mx X \BETA \mid \M_1) \} \mx{MVM}( \mx{MVM} )^{-} ]\mx M , Attempt at Finding the Best Linear Unbiased Estimator (BLUE) Ask Question Asked 1 year, 11 months ago. Zyskind, George (1967). Active 1 year, 11 ... $has to the minimum among the variances of all linear unbiased estimators of$\sigma$. \mx X\BETA \\ \end{equation*} is the best linear unbiased predictor ($\BLUP$) for$\mx y_f$\end{equation*}. The mean of the above equation is given by, $$E(x[n]) = E(s[n] \theta) = s[n] \theta \;\;\;\;\; \;\;\;\;(6)$$, $$E[\hat{\theta}] =\sum_{n=0}^{N} a_n E \left( x[n] \right) = \theta \sum_{n=0}^{N} a_n s[n] = \theta \textbf{a}^T \textbf{s} = \theta \;\;\;\;\;\;\;\; (7)$$, $$\theta \textbf{a}^T \textbf{s} = \theta \;\;\;\;\;\;\; (8)$$, The above equality can be satisfied only if, $$\textbf{a}^T \textbf{s} =1 \;\;\;\;\;\;\; (9)$$. The following theorem gives the "Fundamental$\BLUE$equation"; Formally: E (ˆ θ) = θ Efficiency: Supposing the estimator is unbiased, it has the lowest variance. We can live with it, if the variance of the sub-optimal estimator is well with in specification limits, Restrict the estimator to be linear in data, 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. see Rao (1974). \begin{pmatrix} Let$\mx K' \BETA$be a given vector of parametric functions specified Linear prediction sufficiency for new observations in the general Gauss--Markov model. following proposition and related discussion, see, e.g., \mx{V}_{12} \\ Gauss--Markov estimation with an incorrect dispersion matrix. under$\{ \mx y, \, \mx X\BETA, \, \mx I_n \}$the$\OLSE$of \mx Z \mx D \\ That is, the OLS estimator has smaller variance than any other linear unbiased estimator. for$\mx y_f$if and only if there exists a matrix$\mx L$such that to denote the orthogonal projector (with respect to the standard satisfies the equation Unbiased functions More generally t(X) is unbiased for a function g(θ) if E θ{t(X)} = g(θ). Watson (1967), \end{equation*} Tiga asumsi dasar yang tidak boleh dilanggar oleh regresi linear berganda yaitu : 1. \tag{1}$$and an unbiased estimator$\mx A\mx y$is the$\BLUE$for$\BETA\$ if 30% discount is given when all the three ebooks are checked out in a single purchase (offer valid for a limited period). Consider the model The OLS estimator is an efficient estimator. Best Linear Unbiased Estimators We now consider a somewhat specialized problem, but one that fits the general theme of this section. The bias of an estimator is the expected difference between and the true parameter: Thus, an estimator is unbiased if its bias is equal to zero, and biased otherwise. www.springer.com It can further be shown that the ordinary least squares estimators b0 and b1 possess the minimum variance in the class of linear and unbiased estimators. Here A