# Estimation and prediction in regression models with random explanatory variables

• Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1992

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## Abstract

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The regression model X(t),Y(t);t=1,...,n with random explanatory variable X is transformed by prescribing a partition ${S}_{1},...,{S}_{k}$ of the given domain S of X-values and specifying$X\left(1\right),...,X\left(n\right)\cap {S}_{i}={X}_{i1},...,{X}_{i\alpha \left(i\right)},i=1,...,k.$Through the conditioning$\alpha \left(i\right)=a\left(i\right),i=1,...,k,{X}_{i1},...,{X}_{i\alpha \left(i\right)};i=1,...,k={x}_{11},...,{x}_{ka\left(k\right)}$the initial model with i.i.d. pairs (X(t),Y(t)),t=1,...,n, becomes a conditional fixed-design $\left({x}_{11},...,{x}_{ka\left(k\right)}\right)$ model${Y}_{ij},i=1,...,k;j=1,...,a\left(i\right)$where the response variables ${Y}_{ij}$ are independent and distributed according to the mixed conditional distribution $Q\left(·,{x}_{ij}\right)$ of Y given X at the observed value ${x}_{ij}$.Afterwards, we investigate the case$\left(Q\right)E\left({Y}^{\text{'}}|x\right)={\sum }_{i=1}^{k}{b}_{i}\left(x\right){\theta }_{i}{I}_{{S}_{i}}\left(x\right),\left(Q\right)D\left(Y|x\right)={\sum }_{i=1}^{k}{d}_{i}\left(x\right){\Sigma }_{i}{I}_{{S}_{i}}\left(x\right)$which arises when the conditional distribution law of Y given X changes as X passes from a domain ${S}_{i}$ to another, whence Y follows a mixture of distributions. Then the general transformation gives the equivalent reduction to a conditional multivariate Behrens-Fisher model. We construct conditional generalized least squares estimators of ${\theta }^{\text{'}}=\left({\theta }_{1}^{\text{'}}⋮\cdots ⋮{\theta }_{k}^{\text{'}}\right)$ and predictors of Y(n+1) given X(n+1) = x ∈ S. Through some condition imposed on the range of θ, the CGLS estimator and predictor are shown to enjoy local and global optimality.CONTENTSPreface..................................................................................................................................................................................................................5I. A data transformation preserving the conditional distribution and localizing the explanatory variable.................................................................61. Introduction........................................................................................................................................................................................................62. Theorems on data transformation......................................................................................................................................................................73. Proofs of the theorems.......................................................................................................................................................................................94. Interpretation of the theorems..........................................................................................................................................................................14II. Conditional linear models and estimation of regression parameters.................................................................................................................175. Introduction......................................................................................................................................................................................................176. Conditional generalized least squares estimators (CGLSE).............................................................................................................................197. Conditional estimability.....................................................................................................................................................................................258. Properties of the CGLSE..................................................................................................................................................................................29III. Prediction of the response variable.................................................................................................................................................................349. Introduction......................................................................................................................................................................................................3510. Predictors connnected wi.th the CGLSE........................................................................................................................................................3511. Properties of CGLS predictors.......................................................................................................................................................................38References..........................................................................................................................................................................................................431991 Mathematics Subject Classification: Primary 62J02; Secondary 62F11.

## How to cite

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Nguyen Bac-Van. Estimation and prediction in regression models with random explanatory variables. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1992. <http://eudml.org/doc/219330>.

@book{NguyenBac1992,
abstract = {The regression model X(t),Y(t);t=1,...,n with random explanatory variable X is transformed by prescribing a partition $S_\{1\},...,S_\{k\}$ of the given domain S of X-values and specifying$\{X(1),...,X(n)\} ∩ S_\{i\} = \{X_\{i1\},...,X_\{iα(i)\} \}, i=1,...,k.$Through the conditioning$\{α(i)=a(i), i=1,...,k\}, \{X_\{i1\},...,X_\{iα(i)\}; i=1,...,k\} = \{x_\{11\},...,x_\{ka(k)\}\}$the initial model with i.i.d. pairs (X(t),Y(t)),t=1,...,n, becomes a conditional fixed-design $(x_\{11\},...,x_\{ka(k)\})$ model$\{Y_\{ij\},i=1,...,k;j=1,...,a(i)\}$where the response variables $Y_\{ij\}$ are independent and distributed according to the mixed conditional distribution $Q(·,x_\{ij\})$ of Y given X at the observed value $x_\{ij\}$.Afterwards, we investigate the case$(Q)E(Y^\{\prime \}|x) = ∑^k_\{i=1\} b_\{i\}(x)θ_\{i\} I_\{S_\{i\}\}(x), (Q)D(Y|x) = ∑^k_\{i=1\} d_\{i\}(x)Σ_\{i\}I_\{S_\{i\}\}(x)$which arises when the conditional distribution law of Y given X changes as X passes from a domain $S_\{i\}$ to another, whence Y follows a mixture of distributions. Then the general transformation gives the equivalent reduction to a conditional multivariate Behrens-Fisher model. We construct conditional generalized least squares estimators of $θ^\{\prime \} = (θ^\{\prime \}_\{1\}⋮ ⋯⋮ θ^\{\prime \}_\{k\})$ and predictors of Y(n+1) given X(n+1) = x ∈ S. Through some condition imposed on the range of θ, the CGLS estimator and predictor are shown to enjoy local and global optimality.CONTENTSPreface..................................................................................................................................................................................................................5I. A data transformation preserving the conditional distribution and localizing the explanatory variable.................................................................61. Introduction........................................................................................................................................................................................................62. Theorems on data transformation......................................................................................................................................................................73. Proofs of the theorems.......................................................................................................................................................................................94. Interpretation of the theorems..........................................................................................................................................................................14II. Conditional linear models and estimation of regression parameters.................................................................................................................175. Introduction......................................................................................................................................................................................................176. Conditional generalized least squares estimators (CGLSE).............................................................................................................................197. Conditional estimability.....................................................................................................................................................................................258. Properties of the CGLSE..................................................................................................................................................................................29III. Prediction of the response variable.................................................................................................................................................................349. Introduction......................................................................................................................................................................................................3510. Predictors connnected wi.th the CGLSE........................................................................................................................................................3511. Properties of CGLS predictors.......................................................................................................................................................................38References..........................................................................................................................................................................................................431991 Mathematics Subject Classification: Primary 62J02; Secondary 62F11.},
author = {Nguyen Bac-Van},
keywords = {least squares procedure; random explanatory variable regression problems; conditional fixed-design model; data transformation; conditioning; asymptotic estimability; conditional unbiasedness; conditional generalized least squares estimates; prediction problem},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Estimation and prediction in regression models with random explanatory variables},
url = {http://eudml.org/doc/219330},
year = {1992},
}

TY - BOOK
AU - Nguyen Bac-Van
TI - Estimation and prediction in regression models with random explanatory variables
PY - 1992
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - The regression model X(t),Y(t);t=1,...,n with random explanatory variable X is transformed by prescribing a partition $S_{1},...,S_{k}$ of the given domain S of X-values and specifying${X(1),...,X(n)} ∩ S_{i} = {X_{i1},...,X_{iα(i)} }, i=1,...,k.$Through the conditioning${α(i)=a(i), i=1,...,k}, {X_{i1},...,X_{iα(i)}; i=1,...,k} = {x_{11},...,x_{ka(k)}}$the initial model with i.i.d. pairs (X(t),Y(t)),t=1,...,n, becomes a conditional fixed-design $(x_{11},...,x_{ka(k)})$ model${Y_{ij},i=1,...,k;j=1,...,a(i)}$where the response variables $Y_{ij}$ are independent and distributed according to the mixed conditional distribution $Q(·,x_{ij})$ of Y given X at the observed value $x_{ij}$.Afterwards, we investigate the case$(Q)E(Y^{\prime }|x) = ∑^k_{i=1} b_{i}(x)θ_{i} I_{S_{i}}(x), (Q)D(Y|x) = ∑^k_{i=1} d_{i}(x)Σ_{i}I_{S_{i}}(x)$which arises when the conditional distribution law of Y given X changes as X passes from a domain $S_{i}$ to another, whence Y follows a mixture of distributions. Then the general transformation gives the equivalent reduction to a conditional multivariate Behrens-Fisher model. We construct conditional generalized least squares estimators of $θ^{\prime } = (θ^{\prime }_{1}⋮ ⋯⋮ θ^{\prime }_{k})$ and predictors of Y(n+1) given X(n+1) = x ∈ S. Through some condition imposed on the range of θ, the CGLS estimator and predictor are shown to enjoy local and global optimality.CONTENTSPreface..................................................................................................................................................................................................................5I. A data transformation preserving the conditional distribution and localizing the explanatory variable.................................................................61. Introduction........................................................................................................................................................................................................62. Theorems on data transformation......................................................................................................................................................................73. Proofs of the theorems.......................................................................................................................................................................................94. Interpretation of the theorems..........................................................................................................................................................................14II. Conditional linear models and estimation of regression parameters.................................................................................................................175. Introduction......................................................................................................................................................................................................176. Conditional generalized least squares estimators (CGLSE).............................................................................................................................197. Conditional estimability.....................................................................................................................................................................................258. Properties of the CGLSE..................................................................................................................................................................................29III. Prediction of the response variable.................................................................................................................................................................349. Introduction......................................................................................................................................................................................................3510. Predictors connnected wi.th the CGLSE........................................................................................................................................................3511. Properties of CGLS predictors.......................................................................................................................................................................38References..........................................................................................................................................................................................................431991 Mathematics Subject Classification: Primary 62J02; Secondary 62F11.
LA - eng
KW - least squares procedure; random explanatory variable regression problems; conditional fixed-design model; data transformation; conditioning; asymptotic estimability; conditional unbiasedness; conditional generalized least squares estimates; prediction problem
UR - http://eudml.org/doc/219330
ER -

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