Estimation and prediction in regression models with random explanatory variables

Nguyen Bac-Van

GDML_Books, (1992), p.

Résumé

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..........................................................................................................................................................................................................43

1991 Mathematics Subject Classification: Primary 62J02; Secondary 62F11.

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Zbl 0766.62037

EUDML-ID : urn:eudml:doc:219330