  Additive regression models are actively researched in the statistical field
because of their usefulness in the analysis of responses determined by
non-linear relationships with multivariate predictors. In this kind of
statistical models, the response depends linearly on unknown functions of
predictor variables and typically, the goal of the analysis is to make
inference about these functions.
  In this paper, we consider the problem of Additive Regression with random
designs from a novel viewpoint: we propose an estimator based on an orthogonal
projection onto a multiresolution space using empirical wavelet coefficients
that are fully data driven. In this setting, we derive a mean-square consistent
estimator based on periodic wavelets on the interval $[0,1]$. For construction
of the estimator, we assume that the joint distribution of predictors is
non-zero and bounded on its support; We also assume that the functions belong
to a Sobolev space and integrate to zero over the $[0,1]$ interval, which
guarantees model identifiability and convergence of the proposed method.
Moreover, we provide the $\mathbb{L}_{2}$ risk analysis of the estimator and
derive its convergence rate.
  Theoretically, we show that this approach achieves good convergence rates
when the dimensionality of the problem is relatively low and the set of unknown
functions is sufficiently smooth. In this approach, the results are obtained
without the assumption of an equispaced design, a condition that is typically
assumed in most wavelet-based procedures.
  Finally, we show practical results obtained from simulated data,
demonstrating the potential applicability of our method in the problem of
additive regression models with random designs.
