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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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