C.R. Rao and H. Toutenburg, Linear Models: Least Squares and Alternatives, Springer-Verlag, Berlin, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....yields 1=# n # n 0=# n n . These equations are solved by n = M 1 X and # n =1 X. With these substitutions we find that # i,n indeed equals # i,n defined in (9) It is also known (see, for example, Hesterberg and Nelson [12] or equation (7. 10) of Rao and Toutenburg [20] in the regression setting) that with some algebraic rearrangement using (2) the linear control variable estimator (1) can be expressed as a weighted average, # i,n Y i , using the same # i,n . Thus, the two estimators coincide in this case. # It is worth noting that although they ....

Rao, C.R., and Toutenburg, H. (1995) Linear Models: Least Squares and Alternatives, Springer-Verlag, New York.

....case. However, we will see that even a weaker requirement is su#cient in order to make eqn. 17) hold. Proposition: If E [#] cI holds, the requirement (17) assuring unbiasedness holds as well. In order to prove this proposition, we first note that EigVec (A cI) EigVec (A) for any matrix A ([15]) Now we define # # = # cI. Obviously E [# # ] 0 holds, thus eqn. 13) can be applied and we obtain: EigVec (A 0 A 0 cI) # # 0 A 0 cI . Therefore, we have to ensure that E [#] E [# 1 ] E [# 2 ] E E holds. From E [D] 0 follows E [# 1 ] 0, and ....

Rao, C.R.; Toutenburg, H.: Linear models: Least Squares and alternatives. Springer Series in Statistics, 1995.

.... M = M 22 M 11 M 12 M 1 22 M 21 . A.4) If # # N (0, 1) then E[e t # 2 ] 1 # 1 2t , t 1 2. A.5) Recognize equation (A.2) as a property of the moment generating function of the normal distribution with covariance matrix M 1 ; equation (A. 4) is standard (see, e.g. Rao and Toutenburg 1995, p. 289) equation (A.5) is the moment generating function of a chi square random variable with 1 degree of freedom. Conditional on u # Z = a, the vector Z has distribution N (au, I uu # ) Consider first the special case u = e 1 = 1, 0, 0) # , and let X have distribution N (ae 1 , ....

RAO, C. R., and H. TOUTENBURG (1995): Linear Models: Least Squares and Alternatives. New York: Springer-Verlag.

....squares type of linear regression can be performed using Principal Component Analysis (PCA) in model tting. This approach allows measurement errors also in inputs while the usual least squares approach assumes that the input variables are accurate and there is error in the output variables only (Rao and Toutenburg, 1995). These two modeling methods combined take advantage of the nonlinear elasticity of the SOM as well as the local eOEciency of the PCA. Also the topology preservation property of the SOM projection can be incorporated by allowing neighboring data sets and models to interact in some way. Ritter et ....

Rao, C. R. and Toutenburg, H. (1995). Linear models: least squares and alternatives. SpringerVerlag, New York.

....squares type of linear regression can be performed using Principal Component Analysis (PCA) in model tting. This approach allows measurement errors also in inputs while the usual least squares approach assumes that the input variables are accurate and there is error in the output variables only [12]. Combining these two modeling methods takes advantage of the nonlinear elasticity of the SOM as well as the local eOEciency of the PCA. Also, the topology preservation property of the SOM projection can be incorporated by allowing neighboring data sets and models to interact in some way. Ritter ....

C. R. Rao and H. Toutenburg. Linear models: least squares and alternatives. Springer-Verlag, New York Inc., 1995.

....the sweep operator on row and column. 3 Figure 1: The main program 4 The valuation criterias for the estimators We decided to choose the mean square error to compare the three estimators. This choice is theoretical based on the matrix valued MSE criterion as discussed in full detail e.g. in Rao and Toutenburg (1999). The MSE will cover as well bias as variance of the estimates. So we will not assess an estimator higher than another when he has little bias but too strong variance and the other way round. Let b be the estimated and b the real parameter vector of the model. To calculate the MSE we used the ....

Rao, C. R. and Toutenburg, H. (1999). Linear Models: Least Squares and Alternatives, 2 edn, Springer, New York.

....squares type of linear regression can be performed using Principal Component Analysis (PCA) in model tting. This approach allows measurement errors also in inputs while the usual least squares approach assumes that the input variables are accurate and there is error in the output variables only [25]. Combination of these two modeling methods takes advantage of the nonlinear elasticity of the SOM as well as the local eOEciency of the PCA. Also, the topology preservation property of the SOM projection can be incorporated by allowing neighboring data sets and models to interact in some way. ....

C. R. Rao and H. Toutenburg. Linear models: least squares and alternatives. SpringerVerlag, New York, 1995.

.... models, considerable attention has been devoted to analyze the comparative performance of amputation and imputation strategies when missingness of observations pertains to either the study variable or some explanatory variables; see, for example, Little (1992) Little and Rubin (1987) and Rao and Toutenburg (1995) for an interesting account. Realistic situations may often necessitate us to assume that there are some cases in which values of some explanatory variables as well as the 1 study variable are missing simultaneously. Such a framework is considered in this paper and the estimation of regression ....

.... variable on the remaining K explanatory variables using only the n 1 complete observations and then utilizing the estimated relationship for finding the predicted values of the missing observations; see, e.g. Afifi and Elashoff (1967) Dagenais (1973) Gourieroux and Monfort (1981) and Rao and Toutenburg (1995). This yields the following imputed values for x 2 and x 4 : x 2 = X 2 (X 0 1 X 1 ) Gamma1 X 0 1 x 1 (2.8) x 4 = X 4 (X 0 1 X 1 ) Gamma1 X 0 1 x 1 (2.9) In the same spirit, if we run the regression of the study variable on the K explanatory variables utilizing the n 1 ....

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Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

....Between this intervall (namely 0.2, 0.4, 0.6, 0.8) missingness will depend on Y 1 and Y 2 in di erent weights. We decided to choose the mean square error to compare the three estimators. This choice is theoretical based on the matrix valued MSE criterion as discussed in full detail e.g. Rao and Toutenburg (1999). The MSE will cover as well bias as variance of the estimates. So we will not assess an estimator higher than another when he has little bias but too strong variance and the other way round. To calculate the MSE we used the equation: E B B B B 0 ; where B is the ....

Rao, C. R. and Toutenburg, H. (1999). Linear Models: Least Squares and Alternatives, 2 edn, Springer, New York.

....some observations on the study variable are missing. Now the estimation 1 of regression coefficients by least squares method using all the observations provides essentially the same estimators as obtained by an application of the least squares method to complete observations alone; see, e.g. Rao and Toutenburg (1995, Chap. 8) Thus incomplete observations play absolutely no role, and no gain in the efficiency is achieved despite their use is in estimation procedure. This result may take a pleasent turn when some additional information about the model is available. This is the point of investigation here. Let ....

.... past experience of similar investigations and or from the exhibition of stability of estimates of regression coefficients in repeated studies and or from some extraneous sources and or from some theoretical considerations; see, e.g. Judge, Griffiths, Hill, Lutkepohl and Lee (1985, Chap.3) and Rao and Toutenburg (1995, Chap. 5) Incorporation of such a prior information into the estimation procedure, it is well documented, leads to generally efficient estimation of regression coefficients provided that there is no missing observation in the data set. This article examines the truthfulness of this result when ....

Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

.... Under the additional condition fi 0 Bfi r (14) with a positive definite (K Theta K) matrix B and a constant r 0, the minimaxlinear estimation (MMLE) is of the form b = r Gamma1 oe 2 B S) Gamma1 X 0 y = D Gamma1 X 0 y (15) where D = I Gamma1 oe 2 B S) cf. e.g. Rao and Toutenburg(1995, Theorem 3.9) This estimator is biased: bias(b ; fi) E(b ) Gamma fi = D Gamma1 S Gamma I)fi = Gammar Gamma1 oe 2 D Gamma1 I : 16) The covariance matrix oe 2 V is oe 2 V = E [ b Gamma E(b ) b Gamma E(b ) 0 ] oe 2 D Gamma1 SD Gamma1 : ....

Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer.

....cc X c fi oe 2 X fi (2. 4) Sigma c Sigma Gamma1 cc (Y c Gamma fi 0 X 0 c Sigma Gamma1 cc Y c fi 0 X 0 c Sigma Gamma1 cc X c fi oe 2 X c fi) fi 0 X c Y c fi 0 X 0 c X c fi oe 2 (1 Gamma ) X fi J 0 n Y c 1 (n Gamma 1) Jm see, e.g. Rao and Toutenburg (1995, Sec. 6.5) The predictor (2.4) has no practical utility due to involvement of unknown quantities oe 2 and fi. A simple way to obtain a feasible version is to replace them by their unbiased estimators. Thus substituting b c in place of fi and s 2 = 1 n Gamma K (Y c Gamma X c b c ) ....

....4 ) is D(P 2 ; P ) E(P 2 Gamma Y mis ) P 2 Gamma Y mis ) 0 Gamma E( P Gamma Y mis ) P Gamma Y mis ) 0 (3. 6) oe 4 (1 Gamma ) 2 fi 0 X 0 c X c fi X QX where Q = 2(X 0 c X c ) Gamma1 Gamma 5(n Gamma K) 2 (n Gamma K)fi 0 X 0 c X c fi fifi 0 : Using Rao and Toutenburg (1995, Theorem A.7, p.303) it is observed that Q cannot be a nonnegative definite matrix so that it follows from (3.6) that P does not dominate P 2 with respect to the criterion of mean squared error matrix to the given order of approximation. Similarly, using Rao and Toutenburg (1995, A.59, p. ....

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Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

....solution is perhaps to amputate the incomplete observations and to restrict attention to complete observations only for the purpose of statistical analysis. Alternatively, one may employ some imputation method for finding the substitutes of missing observations; see, e.g. Little and Rubin (1987) Rao and Toutenburg (1995) and Rubin (1987) for an interesting account. Treating these imputed values as true observations, one may conduct the statistical analysis using the standard procedures developed for data without any missing observation. Such a practice, it is well recognized, may tend to invalidate the inferences ....

Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

....mean imputation, also known as zero order regression (ZOR) and first introduced by Wilks (1932) is also a common approach to missing values. Each missing value of a regressor is replaced by the sample mean of the observed values of the regressor computed from the complete cases (see e.g. Rao and Toutenburg (1999, ch. 8) Its usage however strictly regarded requires an adaption to the non continuous scaling. Except for treating ordinally scaled variables as continuous the median has to be imputed for ordinally scaled data and the mode for nominally scaled data instead of the mean. Because of rather ....

....treating ordinally scaled variables as continuous the median has to be imputed for ordinally scaled data and the mode for nominally scaled data instead of the mean. Because of rather secondary attention this short illustration should be sufficient, for more information about this topic see e.g. Rao and Toutenburg (1999, ch. 8) 3 Methods based on the conditional mean imputation The conditional mean imputation is also known as first order regression (FOR) or Buck s method (see (Buck, 1960) An auxiliary regression based on the complete cases of the incompletely observed variable incorporates the structure of ....

Rao, C. R. and Toutenburg, H. (1999). Linear Models: Least Squares and Alternatives, 2 edn, Springer, New York.

....model and presents an estimation strategy based on hypothesis testing. 1 Introduction When some observations on some of the explanatory variables in a linear regression model are missing, there are several imputation procedures to obtain their substitutes; see, e.g. Little and Rubin (1987) and Rao and Toutenburg (1995) for an interesting account. Among them, a popular procedure is the method of first order regression. It essentially amounts to running an auxiliary regression of each explanatory variable (on which the observations are missing) on the remaining explanatory variables (on which no observation is ....

Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

....is to plug in imputed values for missing observations and then to carry out the regression analysis. Such imputed values can be obtained in several ways; see, e.g. Little and Rubin (1987) for basic considerations and Little (1992) for a detailed discussion of missing X values in regression, and Rao and Toutenburg (1995) for a detailed account of MDE superiority investigations for imputation methods. When these imputed values are non stochastic, the application of the least squares procedure for the estimation of regression coefficients generally yields biased and inconsistent estimators; see, e.g. Toutenburg, ....

....kind of complete discard is obviously not always a satisfactory proposition and may often have misleading implications. An alternative solution is to employ some imputation method so that missing values of the last explanatory variable can be replaced. There are several ways to do this; see, e.g. Rao and Toutenburg (1995, Chap. 8) Among them, an interesting procedure known as first order regression method (FOR) is to run an auxiliary regression of the variable in x c on the remaining (K Gamma 1) variables in Z c and to use the estimated equation for finding the predicted values of missing observations, viz. ....

[Article contains additional citation context not shown here]

Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

....help of complete observations alone and obtaining the predicted values for the missing observations. These predicted values are then substituted in order to get a repaired or completed data set which is finally used for the estimation of parameters; see, e.g. Little and Rubin (1987, Chap. 2) and Rao and Toutenburg (1995, Chap. 8) for an interesting exposition. This strategy is adopted for the estimation of parameters in a linear regression model with some missing observations on the study variable. For the following it is assumed that missingness of the study variable y is independent of the value y itself and ....

....between the mean squared error matrix of fi and the variance covariance matrix of b c can be easily obtained. As b c is identically equal to the unbiased estimator b c based on complete observations, its variance covariance matrix can be estimated in the traditional manner; see, e.g. Rao and Toutenburg (1995, Chap. 8) for the details. 5 Some Concluding Remarks Recognizing that the use of least squares predictions for some missing values of study variable in a linear regression model does not lead to improved estimation of regression coefficients, we have considered the application of shrinkage ....

Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

....If the disturbances are correlated according to E(uu 0 ) oe 2 W, then this information leads to solutions y = X fi u (cp. Goldberger, 1962) Remark (ii) The two alternative prediction problems the Xfi superiority and the y superiority, respectively are discussed in full detail in Rao and Toutenburg (1995, Chapter 6) As a central result, we have the fact that the superiority (in the Loewner ordering of definite matrices) of one predictor over another predictor can change if the criterion is changed. This was one of the motivations to define a target as in (1.2) that combines these two risks. In ....

....An interesting problem in all regression models relates to missing data. In general, we may assume the following structure of data: 0 y obs y mis y obs 1 A = 0 X obs X obs X mis 1 A fi u : 1. 25) Estimation of y mis corresponds to the prediction problem discussed in Chapter 6 of Rao and Toutenburg (1995) in full detail. We may therefore confine ourselves to the structure y obs = X obs X mis fi u (1.26) and change the notation as follows: y y = X X fi u u ; u u Gamma 0; oe 2 I Delta : 1.27) The submodel y = Xfi u (1.28) presents the ....

[Article contains additional citation context not shown here]

Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

....is to plug in imputed values for missing observations and then to carry out the regression analysis. Such imputed values can be obtained in several ways; see, e.g. Little and Rubin (1987) for basic considerations and Little (1992) for a detailed discussion of missing X values in regression, and Rao and Toutenburg (1995) for a detailed account of MSE superiority investigations for imputation methods. When these imputed values are nonstochastic, the application of least squares procedure for the estimation of regression coefficients generally yields biased and inconsistent estimators; see, e.g. Toutenburg, ....

....kind of complete discard is obviously not always a satisfactory proposition and may often have misleading implications. An alternative solution is to employ some imputation method so that missing values of the last explanatory variable can be replaced. There are several ways to do it; see, e.g. Rao and Toutenburg (1995, Chap. 8) Among them, an interesting procedure known as first order regression method is to run an auxiliary regression of the variable in x c on the remaining (K Gamma 1) variables in Z c and to use the estimated equation for finding the predicted values of missing observations, viz. xR = ....

Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

....missing, there are two alternatives. One is to use complete observations alone and to discard the incomplete data set while the other alternative is to substitute estimated values for missing observations and to use the thus repaired data set; see, e.g. Little (1992) Little and Rubin (1987) and Rao and Toutenburg (1995) for an interesting account. The substitutions are generally constructed on the basis of regression analysis of complete observations. If a weakly unbiased substitution is utilized for missing observations and least squares procedure is applied to the model with the repaired data set, the ....

....a weakly unbiased substitution X # # c for missing observations in (2.5) i.e. E(X # # c ) E(Y mis ) 2.6) the resulting operational version of # is nothing but the estimator # c itself. This is a celebrated result due to Yates; see, for instance, Little and Rubin (1987, chap. 2) or Rao and Toutenburg (1995, chap. 8) Let us next consider biased substitutions for the vector of missing observations. If we consider substitutions of the type A c Y c with matrix A c of order m # m c for the replacement of X # # in (2.5) and choose A c such that the risk under a general quadratic loss function is ....

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Rao, C. R. and Toutenburg, H. (1995). Linear Models: Least Squares and Alternatives, Springer, New York.

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C.R. Rao and H. Toutenburg, Linear Models: Least Squares and Alternatives, Springer-Verlag, Berlin, 1995.

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C.R. Rao and H. Toutenburg, Linear Models: Least Squares and Alternatives, Springer-Verlag, Berlin, 1995.

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C. R. Rao and H. Toutenburg, Linear Models: Least Squares and Alternatives. Berlin, Germany: Springer-Verlag, 1995.

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Rao, C. R., and Toutenburg, H. Linear Models -- Least Squares and Alternatives. Springer, New York, 1995.

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Rao, C.R., and H. Toutenburg, Linear Models: Least Squares and Alternatives, SpringerVerlag, New York, 1995.

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