Estimation of a regression function corresponding to latent variables

Abstract The problem of estimation of a univariate regression function from latent variables given an independent and identically distributed sample of the observable variables in the corresponding common factor analysis model is considered. Nonparametric least squares estimates of the regression function are defined. The strong consistency of the estimates is shown for subgaussian random variables whose characteristic function vanishes nowhere. This consistency result does not require any assumptions on the structure or the smoothness of the regression function.

[1]  K. Bollen.  Latent variables in psychology and the social sciences. , 2002, Annual review of psychology.

[2]  S. Lang.  Complex Analysis , 1977 .

[3]  G. S. Watson,et al.  Smooth regression analysis , 1964 .

[4]  George A. Marcoulides,et al.  Interaction and Nonlinear Effects in Structural Equation Modeling , 1998 .

[5]  E. Nadaraya.  On Estimating Regression , 1964 .

[6]  K. Holzinger,et al.  A study in factor analysis : the stability of a bi-factor solution , 1939 .

[7]  R. Tibshirani,et al.  "Preconditioning" for feature selection and regression in high-dimensional problems , 2007, math/0703858.

[8]  Radakovič.  The theory of approximation , 1932 .

[9]  L. Devroye,et al.  An equivalence theorem for L1 convergence of the kernel regression estimate , 1989 .

[10]  H. Marsh,et al.  Structural equation models of latent interaction and quadratic effects. , 2013 .

[11]  Adam Krzyzak,et al.  A Distribution-Free Theory of Nonparametric Regression , 2002, Springer series in statistics.

[12]  S. Rabe-Hesketh,et al.  Latent Variable Modelling: A Survey * , 2007 .

[13]  Adam Krzyzak,et al.  Nonparametric estimation of a latent variable model , 2017, J. Multivar. Anal..

[14]  G. Lugosi,et al.  On the Strong Universal Consistency of Nearest Neighbor Regression Function Estimates , 1994 .

[15]  E. Nadaraya.  Remarks on Non-Parametric Estimates for Density Functions and Regression Curves , 1970 .

[16]  A. Meister.  Deconvolution Problems in Nonparametric Statistics , 2009 .

[17]  L. Zhao.  Exponential bounds of mean error for the nearest neighbor estimates of regression functions*1 , 1987 .

[18]  Gábor Lugosi,et al.  Nonparametric estimation via empirical risk minimization , 1995, IEEE Trans. Inf. Theory.

[19]  L. Györfi,et al.  On the asymptotic normality of the L2-error in partitioning regression estimation , 1998 .

[20]  L. Devroye.  Necessary and sufficient conditions for the pointwise convergence of nearest neighbor regression function estimates , 1982 .

[21]  Heinz Bauer,et al.  Probability Theory , 2021, Foundations of Constructive Probability Theory.

[22]  L. Devroye,et al.  Distribution-Free Consistency Results in Nonparametric Discrimination and Regression Function Estimation , 1980 .

[23]  Y. Mack,et al.  Local Properties of k-NN Regression Estimates , 1981 .

[24]  A. Nobel.  Histogram regression estimation using data-dependent partitions , 1996 .

[25]  Adam Krzyzak,et al.  Nonparametric regression estimation using penalized least squares , 2001, IEEE Trans. Inf. Theory.

[26]  E. Ronchetti,et al.  Goodness of Fit for Generalized Linear Latent Variables Models , 2010 .

[27]  C. J. Stone,et al.  Consistent Nonparametric Regression , 1977 .