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Estimating optimal transformations for multiple regression and correlation. (English) Zbl 0594.62044
For regression problems with the response variable Y and the predictor variables $$X_ 1,...,X_ p$$, the authors propose a procedure, alternating conditional expectations (ACE) algorithm, for estimating the function $$\theta$$ (Y) and $$\phi_ 1(X_ 1),...,\phi_ p(X_ p)$$ that minimize $e^ 2=E[\theta (Y)-\sum^{p}_{j=1}\phi_ j(X_ j)]^ 2/Var(\theta (Y)),$ given only a sample $$\{(Y_ k,x_{k1},...,x_{kp})$$; $$k=1,...,N\}$$ and making minimal assumptions concerning the data distribution or the form of the solution functions.
The article is presented in two distinct parts. Sections 1 through 4 give a fairly nontechnical overview of the ACE and discuss its application to data. Section 5 and Appendix A provide some theoretical foundation for the algorithm. They involve the existence of optimal transformations and consistency results.
For the bivariate case, $$p=1$$, the optimal transformations $$\theta^*,\phi^*$$ satisfy $$\rho^*=\rho (\theta^*,\phi^*)=\max_{\theta,\phi}\rho (\theta (Y),\phi (X))$$, where $$\rho$$ is the product moment correlation coefficient and $$\rho^*$$ is the maximal correlation between X and Y.
Reviewer: Songgui Wang

##### MSC:
 62G05 Nonparametric estimation 62J02 General nonlinear regression
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