  In this dissertation, I derive a new method to estimate the
Vapnik-Chervonenkis Dimension (VCD) for the class of linear functions. This
method is inspired by the technique developed by Vapnik et al. Vapnik et al.
(1994). My contribution rests on the approximation of the expected maximum
difference between two empirical Losses (EMDBTEL). In fact, I use a
cross-validated form of the error to compute the EMDBTEL, and I make the bound
on the EMDBTEL tighter by minimizing a constant in of its right upper bound. I
also derive two bounds for the true unknown risk using the additive (ERM1) and
the multiplicative (ERM2) Chernoff bounds. These bounds depend on the estimated
VCD and the empirical risk. These bounds can be used to perform model selection
and to declare with high probability, the chosen model will perform better
without making strong assumptions about the data generating process (DG).
  I measure the accuracy of my technique on simulated datasets and also on
three real datasets. The model selection provided by VCD was always as good as
if not better than the other methods under reasonable conditions.
