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. There is a random sampling of observations.A3. If this is the case, then we say that our statistic is an unbiased estimator of the parameter. This shows that S 2is a biased estimator for . To summarize, we have four versions of the Cramér-Rao lower bound for the variance of an unbiased estimate of \(\lambda\): version 1 and version 2 in the general case, and version 1 and version 2 in the special case that \(\bs{X}\) is a random sample from the distribution of \(X\). In this section we will consider the general problem of finding the best estimator of \(\lambda\) among a given class of unbiased estimators. GX = X. 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]. The Poisson distribution is named for Simeon Poisson and has probability density function \[ g_\theta(x) = e^{-\theta} \frac{\theta^x}{x! 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. The derivative of the log likelihood function, sometimes called the score, will play a critical role in our anaylsis. Generally speaking, the fundamental assumption will be satisfied if \(f_\theta(\bs{x})\) is differentiable as a function of \(\theta\), with a derivative that is jointly continuous in \(\bs{x}\) and \(\theta\), and if the support set \(\left\{\bs{x} \in S: f_\theta(\bs{x}) \gt 0 \right\}\) does not depend on \(\theta\). The result then follows from the basic condition. \(\E_\theta\left(L_1(\bs{X}, \theta)\right) = 0\) for \(\theta \in \Theta\). The BLUPs for these models will therefore be equal to the usual fitted values, that is, those obtained with fitted.rma and predict.rma. \(\var_\theta\left(L_1(\bs{X}, \theta)\right) = \E_\theta\left(L_1^2(\bs{X}, \theta)\right)\). Suppose now that \(\lambda = \lambda(\theta)\) is a parameter of interest that is derived from \(\theta\). A lesser, but still important role, is played by the negative of the second derivative of the log-likelihood function. Consider again the basic statistical model, in which we have a random experiment that results in an observable random variable \(\bs{X}\) taking values in a set \(S\). [ "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, \(\E_\theta\left(L^2(\bs{X}, \theta)\right) = n \E_\theta\left(l^2(X, \theta)\right)\), \(\E_\theta\left(L_2(\bs{X}, \theta)\right) = n \E_\theta\left(l_2(X, \theta)\right)\), \(\sigma^2 = \frac{a}{(a + 1)^2 (a + 2)}\). rma.uni, predict.rma, fitted.rma, ranef.rma.uni. Kovarianzmatrix … 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. 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 \(b = 1\). 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 \(b\). \(p (1 - p) / n\) is the Cramér-Rao lower bound for the variance of unbiased estimators of \(p\). In our specialized case, the probability density function of the sampling distribution is \[ g_a(x) = a \, x^{a-1}, \quad x \in (0, 1) \]. 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. 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 \(M\) attains the lower bound in the previous exercise and hence is an UMVUE of \(\theta\). If the appropriate derivatives exist and if the appropriate interchanges are permissible then \[ \E_\theta\left(L_1^2(\bs{X}, \theta)\right) = \E_\theta\left(L_2(\bs{X}, \theta)\right) \]. 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 \(\sigma^2\). Opener. 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. If unspecified, no transformation is used. \(\frac{2 \sigma^4}{n}\) is the Cramér-Rao lower bound for the variance of unbiased estimators of \(\sigma^2\). Kackar, R. N., & Harville, D. A. Thus \(S = R^n\). numerical value between 0 and 100 specifying the prediction interval level (if unspecified, the default is to take the value from the object). Robinson, G. K. (1991). 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 conditions under which the minimum variance is computed need to be determined. with minimum variance) The sample mean \(M\) attains the lower bound in the previous exercise and hence is an UMVUE of \(\mu\). Of course, the Cramér-Rao Theorem does not apply, by the previous exercise. 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. The OLS estimator is the best (in the sense of smallest variance) linear conditionally unbiased estimator (BLUE) in this setting. Convenient methods for computing BLUE of the estimable linear functions of the fixed elements of the model and for computing best linear unbiased predictions of the random elements of the model have been available. The standard errors are then set equal to NA and are omitted from the printed output. The lower bound is named for Harold Cramér and CR Rao: If \(h(\bs{X})\) is a statistic then \[ \var_\theta\left(h(\bs{X})\right) \ge \frac{\left(\frac{d}{d \theta} \E_\theta\left(h(\bs{X})\right) \right)^2}{\E_\theta\left(L_1^2(\bs{X}, \theta)\right)} \]. Raudenbush, S. W., & Bryk, A. S. (1985). 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. linear regression model, the ordinary least squares estimator (OLSE) is the best linear unbiased estimator of the regression coefficient when measurement errors are absent. DOI: 10.4148/2475-7772.1091 Corpus ID: 55273875. 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. Suppose now that \(\lambda(\theta)\) is a parameter of interest and \(h(\bs{X})\) is an unbiased estimator of \(\lambda\). Thus, the probability density function of the sampling distribution is \[ g_a(x) = \frac{1}{a}, \quad x \in [0, a] \]. 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. Unbiased and Biased Estimators . [11] Puntanen, Simo; Styan, George P. H. and Werner, Hans Joachim (2000). 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 \(\sigma\). The conditional mean should be zero.A4. electr. The normal distribution is widely used to model physical quantities subject to numerous small, random errors, and has probability density function \[ g_{\mu,\sigma^2}(x) = \frac{1}{\sqrt{2 \, \pi} \sigma} \exp\left[-\left(\frac{x - \mu}{\sigma}\right)^2 \right], \quad x \in \R\]. This variance is smaller than the Cramér-Rao bound in the previous exercise. If \(\mu\) is unknown, no unbiased estimator of \(\sigma^2\) attains the Cramér-Rao lower bound above. "Best linear unbiased predictions" (BLUPs) of random effects are similar to best linear unbiased estimates (BLUEs) (see Gauss–Markov theorem) of fixed effects. Corresponding standard errors and prediction interval bounds are also provided. The last line uses (14.2). Show page numbers . Recall that this distribution is often used to model the number of random points in a region of time or space and is studied in more detail in the chapter on the Poisson Process. Empirical Bayes meta-analysis. In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. Note that the OLS estimator b is a linear estimator with C = (X 0X) 1X : Theorem 5.1. Thus, if we can find an estimator that achieves this lower bound for all \(\theta\), then the estimator must be an UMVUE of \(\lambda\). The following theorem gives the general Cramér-Rao lower bound on the variance of a statistic. 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 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. 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 \(R\). Why do the estimated values from a Best Linear Unbiased Predictor (BLUP) differ from a Best Linear Unbiased Estimator (BLUE)? Linear regression models have several applications in real life. The Cramér-Rao lower bound for the variance of unbiased estimators of \(\mu\) is \(\frac{a^2}{n \, (a + 1)^4}\). Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. If \(\mu\) is known, then the special sample variance \(W^2\) attains the lower bound above and hence is an UMVUE of \(\sigma^2\). The sample mean \(M\) does not achieve the Cramér-Rao lower bound in the previous exercise, and hence is not an UMVUE of \(\mu\). Note that the Cramér-Rao lower bound varies inversely with the sample size \(n\). Note: True Bias = … Puntanen, Simo and Styan, George P. H. (1989). Conducting meta-analyses in R with the metafor package. 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. Menu. }, \quad x \in \N \] The basic assumption is satisfied. Search form. By best we mean the estimator in the Mixed linear models are assumed in most animal breeding applications. The normal distribution is used to calculate the prediction intervals. Best Linear Unbiased Predictions for 'rma.uni' Objects. Journal of Statistical Software, 36(3), 1--48. https://www.jstatsoft.org/v036/i03. Life will be much easier if we give these functions names. Have questions or comments? Viechtbauer, W. (2010). I would build a simulation model at first, For example, X are all i.i.d, Two parameters are unknown. Opener. 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. In this case the variance is minimized when \(c_i = 1 / n\) for each \(i\) and hence \(Y = M\), the sample mean. Recall that the normal distribution plays an especially important role in statistics, in part because of the central limit theorem. The Cramér-Rao lower bound for the variance of unbiased estimators of \(a\) is \(\frac{a^2}{n}\). 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 \(\E_\theta\). This follows from the fundamental assumption by letting \(h(\bs{x}) = 1\) for \(\bs{x} \in S\). We now define unbiased and biased estimators. The following theorem gives the second version of the Cramér-Rao lower bound for unbiased estimators of a parameter. Best Linear Unbiased Estimators We now consider a somewhat specialized problem, but one that fits the general theme of this section. Using the deﬁnition in (14.1), we can see that it is biased downwards. When the model was fitted with the Knapp and Hartung (2003) method (i.e., test="knha" in the rma.uni function), then the t-distribution with \(k-p\) degrees of freedom is used. b(2)= n1 n 2 2 = 1 n 2. Recall that if \(U\) is an unbiased estimator of \(\lambda\), then \(\var_\theta(U)\) is the mean square error. 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 \(\sigma^2 \in (0, \infty)\). \(L^2\) 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 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 \[ c_j = \frac{1 / \sigma_j^2}{\sum_{i=1}^n 1 / \sigma_i^2}, \quad j \in \{1, 2, \ldots, n\} \]. 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 \(a\). De nition 5.1. When the measurement errors are present in the data, the same OLSE becomes biased as well as inconsistent estimator of regression coefficients. Recall that \(V = \frac{n+1}{n} \max\{X_1, X_2, \ldots, X_n\}\) is unbiased and has variance \(\frac{a^2}{n (n + 2)}\). The sample mean \(M\) (which is the proportion of successes) attains the lower bound in the previous exercise and hence is an UMVUE of \(p\). \(\theta / n\) is the Cramér-Rao lower bound for the variance of unbiased estimators of \(\theta\). For best linear unbiased predictions of only the random effects, see ranef. Best linear unbiased estimators in growth curve models PROOF.Let (A,Y ) be a BLUE of E(A,Y ) with A ∈ K. Then there exist A1 ∈ R(W) and A2 ∈ N(W) (the null space of the operator W), such that A = A1 +A2. It must have the property of being unbiased. The mean and variance of the distribution are. Communications in Statistics, Theory and Methods, 10, 1249--1261. Specifically, we will consider estimators of the following form, where the vector of coefficients \(\bs{c} = (c_1, c_2, \ldots, c_n)\) is to be determined: \[ Y = \sum_{i=1}^n c_i X_i \]. Page; Site; Advanced 7 of 230. 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. Viewed 14k times 22. The mimimum variance is then computed. icon-arrow-top icon-arrow-top. A Best Linear Unbiased Estimator of Rβ with a Scalar Variance Matrix - Volume 6 Issue 4 - R.W. 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}. Recall also that the fourth central moment is \(\E\left((X - \mu)^4\right) = 3 \, \sigma^4\). Let \(f_\theta\) denote the probability density function of \(\bs{X}\) for \(\theta \in \Theta\). How to calculate the best linear unbiased estimator? This follows since \(L_1(\bs{X}, \theta)\) has mean 0 by the theorem above. 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. Mean square error is our measure of the quality of unbiased estimators, so the following definitions are natural. Ask Question Asked 6 years ago. Of course, a minimum variance unbiased estimator is the best we can hope for. Restrict estimate to be unbiased 3. We will consider estimators of \(\mu\) that are linear functions of the outcome variables. It does not, however, seem to have gained the same popularity in plant breeding and variety testing as it has in animal breeding. This then needs to be put in the form of a vector. 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 sample variance \(S^2\) has variance \(\frac{2 \sigma^4}{n-1}\) and hence does not attain the lower bound in the previous exercise. In statistics, best linear unbiased prediction (BLUP) is used in linear mixed models for the estimation of random effects. An estimator of \(\lambda\) that achieves the Cramér-Rao lower bound must be a uniformly minimum variance unbiased estimator (UMVUE) of \(\lambda\). That BLUP is a good thing: The estimation of random effects. Best linear unbiased prediction (BLUP) is a standard method for estimating random effects of a mixed model. (1981). 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} } 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 following theorem gives an alternate version of the Fisher information number that is usually computationally better. # S3 method for rma.uni (Of course, \(\lambda\) might be \(\theta\) itself, but more generally might be a function of \(\theta\).) The OLS estimator bis the Best Linear Unbiased Estimator (BLUE) of the classical regresssion model. 1971 Linear Models, Wiley Schaefer, L.R., Linear Models and Computer Strategies in Animal Breeding Lynch and Walsh Chapter 26. For conditional residuals (the deviations of the observed outcomes from the BLUPs), see rstandard.rma.uni with type="conditional". For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. 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 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. The basic assumption is satisfied with respect to \(a\). best linear unbiased prediction __\). In this case, the observable random variable has the form \[ \bs{X} = (X_1, X_2, \ldots, X_n) \] where \(X_i\) is the vector of measurements for the \(i\)th item. Sections . Encyclopedia. The basic assumption is satisfied with respect to both of these parameters. The following steps summarize the construction of the Best Linear Unbiased Estimator (B.L.U.E) Define a linear estimator. The distinction arises because it is conventional to talk about estimating fixe… Let \(\bs{\sigma} = (\sigma_1, \sigma_2, \ldots, \sigma_n)\) where \(\sigma_i = \sd(X_i)\) for \(i \in \{1, 2, \ldots, n\}\). Suppose now that \(\sigma_i = \sigma\) for \(i \in \{1, 2, \ldots, n\}\) so that the outcome variables have the same standard deviation. \(Y\) is unbiased if and only if \(\sum_{i=1}^n c_i = 1\). Active 1 year, 4 months ago. In more precise language we want the expected value of our statistic to equal the parameter. Watch the recordings here on Youtube! \(\sigma^2 / n\) is the Cramér-Rao lower bound for the variance of unbiased estimators of \(\mu\). The best answers are voted up and rise to the top Sponsored by. 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. We will apply the results above to several parametric families of distributions. Recall also that the mean and variance of the distribution are both \(\theta\). 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. 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). … best linear unbiased estimator bester linearer unverzerrter Schätzer {m} stat. The object is a list containing the following components: The "list.rma" object is formatted and printed with print.list.rma. Then \[ \var_\theta\left(h(\bs{X})\right) \ge \frac{\left(d\lambda / d\theta\right)^2}{\E_\theta\left(L_1^2(\bs{X}, \theta)\right)} \]. Missed the LibreFest? The sample mean is \[ M = \frac{1}{n} \sum_{i=1}^n X_i \] Recall that \(\E(M) = \mu\) and \(\var(M) = \sigma^2 / n\). Estimate the best linear unbiased prediction (BLUP) for various effects in the model. This follows immediately from the Cramér-Rao lower bound, since \(\E_\theta\left(h(\bs{X})\right) = \lambda\) for \(\theta \in \Theta\). 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. If the appropriate derivatives exist and the appropriate interchanges are permissible) then \[ \var_\theta\left(h(\bs{X})\right) \ge \frac{\left(d\lambda / d\theta\right)^2}{n \E_\theta\left(l_2(X, \theta)\right)} \]. 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 Estimator | The SAGE Encyclopedia of Social Science Research Methods Search form. First we need to recall some standard notation. 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). Moreover, recall that the mean of the Bernoulli distribution is \(p\), while the variance is \(p (1 - p)\). Best Linear Unbiased Estimator In: The SAGE Encyclopedia of Social Science Research Methods. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. The linear regression model is “linear in parameters.”A2. For predicted/fitted values that are based only on the fixed effects of the model, see fitted.rma and predict.rma. An object of class "list.rma". If \(\lambda(\theta)\) is a parameter of interest and \(h(\bs{X})\) is an unbiased estimator of \(\lambda\) then. We can now give the first version of the Cramér-Rao lower bound for unbiased estimators of a parameter. Suppose that \(\theta\) is a real parameter of the distribution of \(\bs{X}\), taking values in a parameter space \(\Theta\). If normality does not hold, σ ^ 1 does not estimate σ, and hence the ratio will be quite different from 1. Equality holds in the Cauchy-Schwartz inequality if and only if the random variables are linear transformations of each other. VARIANCE COMPONENT ESTIMATION & BEST LINEAR UNBIASED PREDICTION (BLUP) V.K. __

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