Mean square error is our measure of the quality of unbiased estimators, so the following definitions are natural. The gamma distribution is often used to model random times and certain other types of positive random variables, and is studied in more detail in the chapter on Special Distributions. GX = X. best linear unbiased prediction __\). Farebrother Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. We also assume that \[ \frac{d}{d \theta} \E_\theta\left(h(\bs{X})\right) = \E_\theta\left(h(\bs{X}) L_1(\bs{X}, \theta)\right) \] This is equivalent to the assumption that the derivative operator \(d / d\theta\) can be interchanged with the expected value operator \(\E_\theta\). In addition, because E n n1 S2 = n n1 E ⇥ S2 ⇤ = n n1 n1 n 2 = 2 and S2 u = n n1 S2 = 1 n1 Xn i=1 (X i X¯)2 is an unbiased estimator for 2. Statistical Science, 6, 15--32. The normal distribution is used to calculate the prediction intervals. This follows immediately from the Cramér-Rao lower bound, since \(\E_\theta\left(h(\bs{X})\right) = \lambda\) for \(\theta \in \Theta\). Suppose now that \(\sigma_i = \sigma\) for \(i \in \{1, 2, \ldots, n\}\) so that the outcome variables have the same standard deviation. VARIANCE COMPONENT ESTIMATION & BEST LINEAR UNBIASED PREDICTION (BLUP) V.K. In the rest of this subsection, we consider statistics \(h(\bs{X})\) where \(h: S \to \R\) (and so in particular, \(h\) does not depend on \(\theta\)). The linear regression model is “linear in parameters.”A2. Of course, the Cramér-Rao Theorem does not apply, by the previous exercise. [ "article:topic", "license:ccby", "authorname:ksiegrist" ], \(\newcommand{\R}{\mathbb{R}}\) \(\newcommand{\N}{\mathbb{N}}\) \(\newcommand{\Z}{\mathbb{Z}}\) \(\newcommand{\E}{\mathbb{E}}\) \(\newcommand{\P}{\mathbb{P}}\) \(\newcommand{\var}{\text{var}}\) \(\newcommand{\sd}{\text{sd}}\) \(\newcommand{\cov}{\text{cov}}\) \(\newcommand{\cor}{\text{cor}}\) \(\newcommand{\bias}{\text{bias}}\) \(\newcommand{\MSE}{\text{MSE}}\) \(\newcommand{\bs}{\boldsymbol}\), 7.6: Sufficient, Complete and Ancillary Statistics, If \(\var_\theta(U) \le \var_\theta(V)\) for all \(\theta \in \Theta \) then \(U\) is a, If \(U\) is uniformly better than every other unbiased estimator of \(\lambda\), then \(U\) is a, \(\E_\theta\left(L^2(\bs{X}, \theta)\right) = n \E_\theta\left(l^2(X, \theta)\right)\), \(\E_\theta\left(L_2(\bs{X}, \theta)\right) = n \E_\theta\left(l_2(X, \theta)\right)\), \(\sigma^2 = \frac{a}{(a + 1)^2 (a + 2)}\). The following steps summarize the construction of the Best Linear Unbiased Estimator (B.L.U.E) Define a linear estimator. Suppose the the true parameters are N(0, 1), they can be arbitrary. For best linear unbiased predictions of only the random effects, see ranef. Except for Linear Model case, the optimal MVU estimator might: 1. not even exist 2. be difficult or impossible to find ⇒ Resort to a sub-optimal estimate BLUE is one such sub-optimal estimate Idea for BLUE: 1. Note that the OLS estimator b is a linear estimator with C = (X 0X) 1X : Theorem 5.1. •The vector a is a vector of constants, whose values we will design to meet certain criteria. linear regression model, the ordinary least squares estimator (OLSE) is the best linear unbiased estimator of the regression coefficient when measurement errors are absent. Mixed linear models are assumed in most animal breeding applications. Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a sequence of observable real-valued random variables that are uncorrelated and have the same unknown mean \(\mu \in \R\), but possibly different standard deviations. The mimimum variance is then computed. Linear regression models have several applications in real life. The BLUPs for these models will therefore be equal to the usual fitted values, that is, those obtained with fitted.rma and predict.rma. Moreover, recall that the mean of the Bernoulli distribution is \(p\), while the variance is \(p (1 - p)\). Unbiasedness of two-stage estimation and prediction procedures for mixed linear models. Suppose now that \(\lambda = \lambda(\theta)\) is a parameter of interest that is derived from \(\theta\). The best answers are voted up and rise to the top Sponsored by. The variance of \(Y\) is \[ \var(Y) = \sum_{i=1}^n c_i^2 \sigma_i^2 \], The variance is minimized, subject to the unbiased constraint, when \[ c_j = \frac{1 / \sigma_j^2}{\sum_{i=1}^n 1 / \sigma_i^2}, \quad j \in \{1, 2, \ldots, n\} \]. Kackar, R. N., & Harville, D. A. Best Linear Unbiased Estimator •simplify ﬁning an estimator by constraining the class of estimators under consideration to the class of linear estimators, i.e. This follows from the fundamental assumption by letting \(h(\bs{x}) = 1\) for \(\bs{x} \in S\). Puntanen, Simo and Styan, George P. H. (1989). We will consider estimators of \(\mu\) that are linear functions of the outcome variables. Let \(\bs{\sigma} = (\sigma_1, \sigma_2, \ldots, \sigma_n)\) where \(\sigma_i = \sd(X_i)\) for \(i \in \{1, 2, \ldots, n\}\). The basic assumption is satisfied with respect to both of these parameters. We now define unbiased and biased estimators. It must have the property of being unbiased. Restrict estimate to be unbiased 3. Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the Bernoulli distribution with unknown success parameter \(p \in (0, 1)\). The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Best linear unbiased prediction (BLUP) is a standard method for estimating random effects of a mixed model. How to calculate the best linear unbiased estimator? Thus, if we can find an estimator that achieves this lower bound for all \(\theta\), then the estimator must be an UMVUE of \(\lambda\). If unspecified, no transformation is used. In our specialized case, the probability density function of the sampling distribution is \[ g_a(x) = a \, x^{a-1}, \quad x \in (0, 1) \]. Encyclopedia. If \(\lambda(\theta)\) is a parameter of interest and \(h(\bs{X})\) is an unbiased estimator of \(\lambda\) then. A linear estimator is one that can be written in the form e = Cy where C is a k nmatrix of xed constants. When the measurement errors are present in the data, the same OLSE becomes biased as well as inconsistent estimator of regression coefficients. The following theorem gives the general Cramér-Rao lower bound on the variance of a statistic. Beta distributions are widely used to model random proportions and other random variables that take values in bounded intervals, and are studied in more detail in the chapter on Special Distributions. In more precise language we want the expected value of our statistic to equal the parameter. If this is the case, then we say that our statistic is an unbiased estimator of the parameter. The mean and variance of the distribution are. \(\frac{2 \sigma^4}{n}\) is the Cramér-Rao lower bound for the variance of unbiased estimators of \(\sigma^2\). The Cramér-Rao lower bound for the variance of unbiased estimators of \(a\) is \(\frac{a^2}{n}\). This exercise shows how to construct the Best Linear Unbiased Estimator (BLUE) of \(\mu\), assuming that the vector of standard deviations \(\bs{\sigma}\) is known. Download PDF . In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. For \(\bs{x} \in S\) and \(\theta \in \Theta\), define \begin{align} L_1(\bs{x}, \theta) & = \frac{d}{d \theta} \ln\left(f_\theta(\bs{x})\right) \\ L_2(\bs{x}, \theta) & = -\frac{d}{d \theta} L_1(\bs{x}, \theta) = -\frac{d^2}{d \theta^2} \ln\left(f_\theta(\bs{x})\right) \end{align}. Recall also that the mean and variance of the distribution are both \(\theta\). Ask Question Asked 6 years ago. Suppose that \(\theta\) is a real parameter of the distribution of \(\bs{X}\), taking values in a parameter space \(\Theta\). If the appropriate derivatives exist and if the appropriate interchanges are permissible then \[ \E_\theta\left(L_1^2(\bs{X}, \theta)\right) = \E_\theta\left(L_2(\bs{X}, \theta)\right) \]. The equality of the ordinary least squares estimator and the best linear unbiased estimator [with comments by Oscar Kempthorne and by Shayle R. Searle and with "Reply" by the authors]. unbiased-polarized relay: gepoltes Relais {n} ohne Vorspannung: 4 Wörter: stat. Note: True Bias = … Journal of Educational Statistics, 10, 75--98. The following theorem gives the second version of the Cramér-Rao lower bound for unbiased estimators of a parameter. The American Statistician, 43, 153--164. The function calculates best linear unbiased predictions (BLUPs) of the study-specific true outcomes by combining the fitted values based on the fixed effects and the estimated contributions of the random effects for objects of class "rma.uni". \(\frac{b^2}{n k}\) is the Cramér-Rao lower bound for the variance of unbiased estimators of \(b\). \(L^2\) can be written in terms of \(l^2\) and \(L_2\) can be written in terms of \(l_2\): The following theorem gives the second version of the general Cramér-Rao lower bound on the variance of a statistic, specialized for random samples. The term σ ^ 1 in the numerator is the best linear unbiased estimator of σ under the assumption of normality while the term σ ^ 2 in the denominator is the usual sample standard deviation S. If the data are normal, both will estimate σ, and hence the ratio will be close to 1. The Cramér-Rao lower bound for the variance of unbiased estimators of \(\mu\) is \(\frac{a^2}{n \, (a + 1)^4}\). Note first that \[\frac{d}{d \theta} \E\left(h(\bs{X})\right)= \frac{d}{d \theta} \int_S h(\bs{x}) f_\theta(\bs{x}) \, d \bs{x}\] On the other hand, \begin{align} \E_\theta\left(h(\bs{X}) L_1(\bs{X}, \theta)\right) & = \E_\theta\left(h(\bs{X}) \frac{d}{d \theta} \ln\left(f_\theta(\bs{X})\right) \right) = \int_S h(\bs{x}) \frac{d}{d \theta} \ln\left(f_\theta(\bs{x})\right) f_\theta(\bs{x}) \, d \bs{x} \\ & = \int_S h(\bs{x}) \frac{\frac{d}{d \theta} f_\theta(\bs{x})}{f_\theta(\bs{x})} f_\theta(\bs{x}) \, d \bs{x} = \int_S h(\bs{x}) \frac{d}{d \theta} f_\theta(\bs{x}) \, d \bs{x} = \int_S \frac{d}{d \theta} h(\bs{x}) f_\theta(\bs{x}) \, d \bs{x} \end{align} Thus the two expressions are the same if and only if we can interchange the derivative and integral operators. Viechtbauer, W. (2010). Once again, the experiment is typically to sample \(n\) objects from a population and record one or more measurements for each item. Die obige Ungleichung besagt, dass nach dem Satz von Gauß-Markow , ein bester linearer erwartungstreuer Schätzer, kurz BLES (englisch Best Linear Unbiased Estimator, kurz: BLUE) bzw. Recall also that the fourth central moment is \(\E\left((X - \mu)^4\right) = 3 \, \sigma^4\). Of course, a minimum variance unbiased estimator is the best we can hope for. The object is a list containing the following components: The "list.rma" object is formatted and printed with print.list.rma. Unbiased and Biased Estimators . Robinson, G. K. (1991). If the appropriate derivatives exist and the appropriate interchanges are permissible) then \[ \var_\theta\left(h(\bs{X})\right) \ge \frac{\left(d\lambda / d\theta\right)^2}{n \E_\theta\left(l_2(X, \theta)\right)} \]. \(\var_\theta\left(L_1(\bs{X}, \theta)\right) = \E_\theta\left(L_1^2(\bs{X}, \theta)\right)\). … This follows since \(L_1(\bs{X}, \theta)\) has mean 0 by the theorem above. In particular, this would be the case if the outcome variables form a random sample of size \(n\) from a distribution with mean \(\mu\) and standard deviation \(\sigma\). Linear estimation • seeking optimum values of coefﬁcients of a linear ﬁlter • only (numerical) values of statistics of P required (if P is random), i.e., linear \(\frac{M}{k}\) attains the lower bound in the previous exercise and hence is an UMVUE of \(b\). Estimate the best linear unbiased prediction (BLUP) for various effects in the model. Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the uniform distribution on \([0, a]\) where \(a \gt 0\) is the unknown parameter. \(\sigma^2 / n\) is the Cramér-Rao lower bound for the variance of unbiased estimators of \(\mu\). For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. In statistics, best linear unbiased prediction (BLUP) is used in linear mixed models for the estimation of random effects. Best Linear Unbiased Prediction: an Illustration Based on, but Not Limited to, Shelf Life Estimation @inproceedings{Ptukhina2015BestLU, title={Best Linear Unbiased Prediction: an Illustration Based on, but Not Limited to, Shelf Life Estimation}, author={Maryna Ptukhina and W. Stroup}, year={2015} } Note that the bias is equal to Var(X¯). In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. By best we mean the estimator in the The result then follows from the basic condition. Why do the estimated values from a Best Linear Unbiased Predictor (BLUP) differ from a Best Linear Unbiased Estimator (BLUE)? Active 1 year, 4 months ago. icon-arrow-top icon-arrow-top. Recall that the normal distribution plays an especially important role in statistics, in part because of the central limit theorem. If \(\mu\) is unknown, no unbiased estimator of \(\sigma^2\) attains the Cramér-Rao lower bound above. The reason that the basic assumption is not satisfied is that the support set \(\left\{x \in \R: g_a(x) \gt 0\right\}\) depends on the parameter \(a\). Best Linear Unbiased Estimator | The SAGE Encyclopedia of Social Science Research Methods Search form. 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 do not need to know the value of the standard deviation. The distinction between biased and unbiased estimates was something that students questioned me on last week, so it’s what I’ve tried to walk through here.) First we need to recall some standard notation. In 302, we teach students that sample means provide an unbiased estimate of population means. Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the distribution of a real-valued random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\). We now consider a somewhat specialized problem, but one that fits the general theme of this section. Recall that the Bernoulli distribution has probability density function \[ g_p(x) = p^x (1 - p)^{1-x}, \quad x \in \{0, 1\} \] The basic assumption is satisfied. integer specifying the number of decimal places to which the printed results should be rounded (if unspecified, the default is to take the value from the object). Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Communications in Statistics, Theory and Methods, 10, 1249--1261. Best Linear Unbiased Estimators We now consider a somewhat specialized problem, but one that fits the general theme of this section. ein minimalvarianter linearer erwartungstreuer Schätzer ist, das heißt in der Klasse der linearen erwartungstreuen Schätzern ist er derjenige Schätzer, der die kleinste Varianz bzw. I have 130 bread wheat lines, which evaluated during two years under water-stressed and well-watered environments. Best unbiased estimators from a minimum variance viewpoint for mean, variance and standard deviation for independent Gaussian data samples are … BLUP was derived by Charles Roy Henderson in 1950 but the term "best linear unbiased predictor" (or "prediction") seems not to have been used until 1962. The OLS estimator is the best (in the sense of smallest variance) linear conditionally unbiased estimator (BLUE) in this setting. Note that the expected value, variance, and covariance operators also depend on \(\theta\), although we will sometimes suppress this to keep the notation from becoming too unwieldy. Life will be much easier if we give these functions names. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Find the best one (i.e. Raudenbush, S. W., & Bryk, A. S. (1985). Then \[ \var_\theta\left(h(\bs{X})\right) \ge \frac{(d\lambda / d\theta)^2}{n \E_\theta\left(l^2(X, \theta)\right)} \]. Search form. For predicted/fitted values that are based only on the fixed effects of the model, see fitted.rma and predict.rma. Suppose now that \(\lambda(\theta)\) is a parameter of interest and \(h(\bs{X})\) is an unbiased estimator of \(\lambda\). Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the gamma distribution with known shape parameter \(k \gt 0\) and unknown scale parameter \(b \gt 0\). The following theorem give the third version of the Cramér-Rao lower bound for unbiased estimators of a parameter, specialized for random samples. Using the deﬁnition in (14.1), we can see that it is biased downwards. (1981). The basic assumption is satisfied with respect to \(a\). The OLS estimator bis the Best Linear Unbiased Estimator (BLUE) of the classical regresssion model. Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the beta distribution with left parameter \(a \gt 0\) and right parameter \(b = 1\). with minimum variance) This follows from the result above on equality in the Cramér-Rao inequality. 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. The conditions under which the minimum variance is computed need to be determined. This method was originally developed in animal breeding for estimation of breeding values and is now widely used in many areas of research. [11] Puntanen, Simo; Styan, George P. H. and Werner, Hans Joachim (2000). The following version gives the fourth version of the Cramér-Rao lower bound for unbiased estimators of a parameter, again specialized for random samples. Best Linear Unbiased Predictions for 'rma.uni' Objects. The last line uses (14.2). A Best Linear Unbiased Estimator of Rβ with a Scalar Variance Matrix - Volume 6 Issue 4 - R.W. Moreover, the mean and variance of the gamma distribution are \(k b\) and \(k b^2\), respectively. Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the normal distribution with mean \(\mu \in \R\) and variance \(\sigma^2 \in (0, \infty)\). This then needs to be put in the form of a vector. The standard errors are then set equal to NA and are omitted from the printed output. Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a sequence of observable real-valued random variables that are uncorrelated and have the same unknown mean \(\mu \in \R\), but possibly different standard deviations. Menu. The probability density function is \[ g_b(x) = \frac{1}{\Gamma(k) b^k} x^{k-1} e^{-x/b}, \quad x \in (0, \infty) \] The basic assumption is satisfied with respect to \(b\). BLUP Best Linear Unbiased Prediction-Estimation References Searle, S.R. Corresponding standard errors and prediction interval bounds are also provided. Legal. Opener. Viewed 14k times 22. Fixed-effects models (with or without moderators) do not contain random study effects. Use the method of Lagrange multipliers (named after Joseph-Louis Lagrange). electr. First note that the covariance is simply the expected value of the product of the variables, since the second variable has mean 0 by the previous theorem. Suppose that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the Poisson distribution with parameter \(\theta \in (0, \infty)\). The distinction arises because it is conventional to talk about estimating fixe… Not Found. Bhatia I.A.S.R.I., Library Avenue, New Delhi- 11 0012 vkbhatia@iasri.res.in Introduction Variance components are commonly used in formulating appropriate designs, establishing quality control procedures, or, in statistical genetics in estimating heritabilities and genetic We will use lower-case letters for the derivative of the log likelihood function of \(X\) and the negative of the second derivative of the log likelihood function of \(X\). The sample mean \(M\) does not achieve the Cramér-Rao lower bound in the previous exercise, and hence is not an UMVUE of \(\mu\). Home Questions Tags Users ... can u guys give some hint on how to prove that tilde beta is a linear estimator and that it is unbiased? The function calculates best linear unbiased predictions (BLUPs) of the study-specific true outcomes by combining the fitted values based on the fixed effects and the estimated contributions of the random effects for objects of class "rma.uni".Corresponding standard errors and prediction interval bounds are also provided. A lesser, but still important role, is played by the negative of the second derivative of the log-likelihood function. The sample mean is \[ M = \frac{1}{n} \sum_{i=1}^n X_i \] Recall that \(\E(M) = \mu\) and \(\var(M) = \sigma^2 / n\). Suppose now that \(\lambda(\theta)\) is a parameter of interest and \(h(\bs{X})\) is an unbiased estimator of \(\lambda\). For conditional residuals (the deviations of the observed outcomes from the BLUPs), see rstandard.rma.uni with type="conditional". An unbiased linear estimator Gy for Xβ is deﬁned to be the best linear unbiased estimator, BLUE, for Xβ under M if cov(Gy) ≤ L cov(Ly) for all L: LX = X, where “≤ L” refers to the Lo¨wner partial ordering. There is a random sampling of observations.A3. 1971 Linear Models, Wiley Schaefer, L.R., Linear Models and Computer Strategies in Animal Breeding Lynch and Walsh Chapter 26. Recall that if \(U\) is an unbiased estimator of \(\lambda\), then \(\var_\theta(U)\) is the mean square error. For \(x \in R\) and \(\theta \in \Theta\) define \begin{align} l(x, \theta) & = \frac{d}{d\theta} \ln\left(g_\theta(x)\right) \\ l_2(x, \theta) & = -\frac{d^2}{d\theta^2} \ln\left(g_\theta(x)\right) \end{align}. The lower bound is named for Harold Cramér and CR Rao: If \(h(\bs{X})\) is a statistic then \[ \var_\theta\left(h(\bs{X})\right) \ge \frac{\left(\frac{d}{d \theta} \E_\theta\left(h(\bs{X})\right) \right)^2}{\E_\theta\left(L_1^2(\bs{X}, \theta)\right)} \]. Have questions or comments? Show page numbers . Kovarianzmatrix … Suppose now that \(\bs{X} = (X_1, X_2, \ldots, X_n)\) is a random sample of size \(n\) from the distribution of a random variable \(X\) having probability density function \(g_\theta\) and taking values in a set \(R\). Suppose that \(U\) and \(V\) are unbiased estimators of \(\lambda\). The sample variance \(S^2\) has variance \(\frac{2 \sigma^4}{n-1}\) and hence does not attain the lower bound in the previous exercise. Sections . \(\theta / n\) is the Cramér-Rao lower bound for the variance of unbiased estimators of \(\theta\). The sample mean \(M\) attains the lower bound in the previous exercise and hence is an UMVUE of \(\mu\). optional argument specifying the name of a function that should be used to transform the predicted values and interval bounds (e.g., transf=exp; see also transf). Since W satisﬁes the relations ( 3), we obtain from Theorem Farkas-Minkowski ([5]) that N(W) ⊂ E⊥ De nition 5.1. If normality does not hold, σ ^ 1 does not estimate σ, and hence the ratio will be quite different from 1. In the usual language of reliability, \(X_i = 1\) means success on trial \(i\) and \(X_i = 0\) means failure on trial \(i\); the distribution is named for Jacob Bernoulli. best linear unbiased estimator bester linearer unverzerrter Schätzer {m} stat. Given unbiased estimators \( U \) and \( V \) of \( \lambda \), it may be the case that \(U\) has smaller variance for some values of \(\theta\) while \(V\) has smaller variance for other values of \(\theta\), so that neither estimator is uniformly better than the other. From the Cauchy-Scharwtz (correlation) inequality, \[\cov_\theta^2\left(h(\bs{X}), L_1(\bs{X}, \theta)\right) \le \var_\theta\left(h(\bs{X})\right) \var_\theta\left(L_1(\bs{X}, \theta)\right)\] The result now follows from the previous two theorems. Missed the LibreFest? Empirical Bayes meta-analysis. We can now give the first version of the Cramér-Rao lower bound for unbiased estimators of a parameter. The quantity \(\E_\theta\left(L^2(\bs{X}, \theta)\right)\) that occurs in the denominator of the lower bounds in the previous two theorems is called the Fisher information number of \(\bs{X}\), named after Sir Ronald Fisher. The sample mean \(M\) attains the lower bound in the previous exercise and hence is an UMVUE of \(\theta\). Not Found. __

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