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## Estimation and prediction in regression models with random explanatory variables

Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 317 wydano: 1992
Zawartość
Warianty tytułu
Abstrakty
EN
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
$(Q)E(Y'|x) = ∑^k_{i=1} b_{i}(x)θ_{i} I_{S_{i}}(x), (Q)D(Y|x) = ∑^k_{i=1} d_{i}(x)Σ_{i}I_{S_{i}}(x)$
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 $θ' = (θ'_{1}⋮ ⋯⋮ θ'_{k})$ 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.
EN

CONTENTS
Preface..................................................................................................................................................................................................................5
I. A data transformation preserving the conditional distribution and localizing the explanatory variable.................................................................6
1. Introduction........................................................................................................................................................................................................6
2. Theorems on data transformation......................................................................................................................................................................7
3. Proofs of the theorems.......................................................................................................................................................................................9
4. Interpretation of the theorems..........................................................................................................................................................................14
II. Conditional linear models and estimation of regression parameters.................................................................................................................17
5. Introduction......................................................................................................................................................................................................17
6. Conditional generalized least squares estimators (CGLSE).............................................................................................................................19
7. Conditional estimability.....................................................................................................................................................................................25
8. Properties of the CGLSE..................................................................................................................................................................................29
III. Prediction of the response variable.................................................................................................................................................................34
9. Introduction......................................................................................................................................................................................................35
10. Predictors connnected wi.th the CGLSE........................................................................................................................................................35
11. Properties of CGLS predictors.......................................................................................................................................................................38
References..........................................................................................................................................................................................................43
Słowa kluczowe
Tematy
Kategoryzacja MSC:
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 317
Liczba stron
42
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXVII
Daty
wydano
1992
otrzymano
1990-06-11
poprawiono
1991-04-19
Twórcy
autor
• Department of Mathematics, University of HoChiMinh City, 227 Nguyen-van-Cu, Q.5, VietNam
Bibliografia
• [1] N. Bac -Van, On the statistical analysis of a random number of observations, Acta Math. Vietnam. 13 (1) (1988), 55-61.
• [2] D. R. Cox, Some aspects of conditional and asymptotic inference: a review, Sankhya Ser. A 50 (1988), 314-337.
• [3] H. Drygas, Best quadratic unbiased estimation in variance-covariance component models, Math. Operationsforsch. Statist. Ser. Statist. 8 (1977), 211-231.
• [4] K. M. S. Humak, Statistische Methoden der Modellbildung, Band I, Akademie-Verlag, Berlin 1977, Band III, Akademie-Verlag, Berlin 1984.
• [5] E. L. Lehmann and H. Scheffé, Completeness, similar regions, and unbiased estimation. I , Sankhya 10 (1950), 305-340.
• [6] Y. P. Mack and H.-G. Müller, Derivative estimation in nonparametric regression with random predictor variable, Sankhya A 51 (1989), 59-72.
• [7] K. V. Mardi a, Statistics of directional data, Academic Press, London 1972.
• [8] C. R. Rao, Methodology based on the L1-norm, in statistical inference, Sankhya Ser. A 50 (1988), 289-313.
Języki publikacji
 EN
Uwagi
1991 Mathematics Subject Classification: Primary 62J02; Secondary 62F11.