## 6 Numerical Experiments
Table 2 demonstrates higher accuracy of distribution function estimation._  
_In our calculations with the formulae for errors, the following values are used (squared norms of the functions \(g_{1,2}\) and \(f_{1,2}\)):_  
\[\int_{-1}^{1}g_{1,2}^{2}(x)dx=\frac{156}{245},\ \ \ \int_{-1}^{1}f_{1,2}^{2}(x)dx= \frac{235}{343}.\]_For clarity, Figure 1 contains the approximation errors in graphical form. One axis shows \(k=0,1,\ldots,4\), which determines the projection expansion length, \(n=2^{k+2}\), and another axis \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(m\) & \(n=4\) & \(n=8\) & \(n=16\) & \(n=32\) & \(n=64\) \\ \hline \(*\) & 0.024614 & 0.009800 & 0.003838 & 0.001442 & 0.000527 \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 1: The approximation errors \(\epsilon_{g}^{(n)}\) (in the row \({}^{*}*^{\prime\prime}\)) and \(\tilde{\epsilon}_{g}^{(n)}\) (in the remaining rows) for sample sizes \(N=2^{m+9}\) (Example 1). shows \(m=0,1,\ldots,18\), which determines the sample size, \(N=2^{m+9}\). The vertical axis corresponds to the density approximation errors._  
_The example under consideration corresponds to the left part of Figure 1, which presents two surfaces. The first one (red) corresponds to the obtained computational error, it is formed on data from Table 1. The second one (blue, with marked nodes) corresponds to the theoretical error according to Formula (28) with \(s=3/2\):_  
\[e_{g}^{(n)}\approx\sqrt{\frac{C_{1}^{S}n}{N}+\frac{C_{2}^{S}}{n^{3}}}.\]  
_Constants \(C_{1}^{S}=0.885\) and \(C_{2}^{S}=0.276\) are approximately determined based on the condition of a minimum sum of squared deviations between the theoretical error and computational error._  
**Example 2**. : _Let \(g_{\
u_{1},\
u_{2}}\) be the density from a two-parameter family (30), and let \(f_{\
u_{1},\
u_{2}}\) be the related distribution function described by Formula (32), with the following parameters: \(\
u_{1}=\
u+1,\
u_{2}=\
u\) for \(\
u=2\). The modeling formula is given in Proposition 2._  
_The function \(g_{3,2}\) is continuous, differentiable everywhere on \(\Omega\), and its second-order derivative exists almost everywhere on \(\Omega\): \(g_{3,2}\in C^{1}(\Omega)\), \(g_{3,2}^{(2)}\in L_{2}(\Omega)\); hence, \(g_{3,2}\in W_{2}^{2}(\Omega)\). Considering the space \(W_{2}^{2}(\Omega)\) with real non-negative \(s\) (see Remark 1), we affirm that \(g_{3,2}\in W_{2}^{2}(\Omega)\) for \(s<\min\{\
u_{1},\
u_{2}\}+1/2=5/2\) (\(\sup s=5/2\)). The degree of smoothness of \(f_{3,2}\) is greater, and \(f_{3,2}\in W_{2}^{s+1}(\Omega)\). This corresponds to Formula (40) when substituting given parameters \(\
u_{1}\) and \(\
u_{2}\):_  
\[\varepsilon_{g}^{(n)}\approx\frac{C^{S}}{n^{5/2}},\quad\varepsilon_{f}^{(n)} \approx\frac{C^{f}}{n^{7/2}}.\]  
_Theorem 1 implies that in order to achieve the required approximation accuracy \(B(g_{3,2},g_{3,2}^{(n)})=\gamma\leqslant cN^{-5/12}\), \(c>0\), with the limit value \(s=5/2\), the conditionally optimal parameters should satisfy the relationship \(\mathrm{N_{opt}}\asymp n_{\mathrm{opt}}^{6}\), and this is illustrated by the statistical modeling results from Tables 3 and 4. In the row \"\(*\)\" of Table 3, if the projection expansion length \(n\) is doubled, then the deterministic component of error decreases by approximately \(2^{5/2}\approx 5.6\) times (in Table 4, by approximately \(2^{7/2}\approx 11.2\) times). In the rest of this table, errors corresponding to optimal parameters \(n_{\mathrm{opt}}\) and \(N_{\mathrm{opt}}\) are shown in bold, and they are consistent with the relationship \(N_{\mathrm{opt}}\asymp n_{\mathrm{opt}}^{6}\). Table 4 shows higher accuracy of distribution function estimation._  
Figure 1: The approximation errors for densities (the left part for \(g_{1,2}\) from Example 1 and the right part for \(g_{3,2}\) from Example 2).