Breiman, Leo; Friedman, Jerome H. Estimating optimal transformations for multiple regression and correlation. (English) Zbl 0594.62044 J. Am. Stat. Assoc. 80, 580-619 (1985). 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 Cited in 11 ReviewsCited in 248 Documents MSC: 62G05 Nonparametric estimation 62J02 General nonlinear regression Keywords:smoothing; random designs in regression; autoregressive schemes in; stationary ergodic time series; controlled designs in regression; alternating conditional expectations (ACE) algorithm; existence of optimal transformations; consistency; product moment correlation coefficient; maximal correlation PDFBibTeX XMLCite \textit{L. Breiman} and \textit{J. H. Friedman}, J. Am. Stat. Assoc. 80, 580--619 (1985; Zbl 0594.62044) Full Text: DOI