The \(Q_i\)'s estimate the underlying distribution \(f_X\). Therefore, it is possible to build \(f_X\) through performing nonparametric regression between the \(Q_i\)'s and the center points of the intervals \(I_i\). However, an issue arises because the variances of the \(Q_i\)'s are not equal (\(Var\left(Q_i\right)\ =\ np_i\)). This would violate the major assumption of homoscedasticity required by many nonparametric regression techniques \cite{solutions} \cite{2006}. In order to combat the heteroscedasticity of the \(Q_i\)'s, a transformation \(g\left(Q_i\right)\) is applied to \(Q_i\) that would fulfill two major objectives:
1. Minimize Bias. The expected value of \(g\left(Q_i\right)\) should be deterministically related to the expected value of \(Q_i\). This is essential since \(g\left(Q_i\right)\) would enable the estimation of mean of \(Q_i\) which in turn leads to an estimate of the unknown pdf, as shown above. In other words, we need to find a transformation that would minimize the bias of the estimate of \(f_X\).
2. Approximate Homoscedasticity. All \(g\left(Q_i\right)\) random variables should have the same constant variance effectively turning the problem into a homoscedastic regression problem.
Brown \cite{Brown_2009} found that the transformation \(\sqrt{Q_i+c}\) achieves both of those just goals. We present the result here for completeness: