## 5 Two-Parameter Family of Densities with Different Degrees of Smoothness
### Densities with Different Degrees of Smoothness and Related Distribution Functions  
Let \(\xi\) be the random variable defined by the density \(g_{v_{1},v_{2}}\):  
\[g_{v_{1},v_{2}}(x)=\gamma_{v_{1},v_{2}}\left\{ \begin{array}{ll}1-(-x)^{v_{1 }}&\mbox{for }x\in[-1,0),\\ 1-x^{v_{2}}&\mbox{for }x\in[0,1],\\ 0&\mbox{otherwise},\end{array} \right. \tag{30}\]  
where \(v_{1}\) and \(v_{2}\) are parameters (natural numbers) and \(\gamma_{v_{1},v_{2}}\) is a normalizing constant, with  
\[\frac{1}{\gamma_{v_{1},v_{2}}}=\int_{-1}^{0}\bigl{(}1-(-x)^{v_{1}}\bigr{)}dx+ \int_{0}^{1}(1-x^{v_{2}})dx=\frac{v_{1}}{v_{1}+1}+\frac{v_{2}}{v_{2}+1},\]  
because  
\[\int_{-1}^{0}\bigl{(}1-(-x)^{v}\bigr{)}dx=\int_{0}^{1}(1-x^{v})dx=\left.\left( x-\frac{x^{v+1}}{v+1}\right)\right|_{0}^{1}=1-\frac{1}{v+1}=\frac{v}{v+1}.\]  
Further, we assume that \(v_{1}\
eq v_{2}\). The function \(g_{v_{1},v_{2}}\) has the following properties:  
(a) \(g_{v_{1},v_{2}}\) is continuous on the set of real numbers;  
(b) The support of \(g_{v_{1},v_{2}}\) is the set \(\Omega=[-1,1]\);  
(c) The normalization condition holds:  
\[\int_{-\infty}^{+\infty}g_{v_{1},v_{2}}(x)dx=\int_{-1}^{1}g_{v_{1},v_{2}}(x) dx=1;\](d) \(g_{v_{1},v_{2}}\) is differentiable at \(x=0\) only \(r\) times:  
\[g_{v_{1},v_{2}}\in C^{r}(\Omega),\ \ \ r=\min\{v_{1},v_{2}\}-1;\]  
(e) The derivative \(g_{v_{1},v_{2}}^{(r+1)}\) exists almost everywhere on \(\Omega\); if the derivative is understood in a generalized sense, then  
\[g_{v_{1},v_{2}}^{(s)}\in L_{2}(\Omega),\ \ \ g_{v_{1},v_{2}}\in W_{2}^{s}(\Omega),\ \ s=r+1=\min\{v_{1},v_{2}\}.\]  
Next, we determine the distribution function \(f_{v_{1},v_{2}}\) for the random variable \(\xi\) with density (30):  
\[f_{v_{1},v_{2}}(x)=\int_{-\infty}^{x}g_{v_{1},v_{2}}(\theta)d\theta. \tag{31}\]  
It is easy to see that \(f_{v_{1},v_{2}}(x)=0\) for \(x<-1\) and \(f_{v_{1},v_{2}}(x)=1\) for \(x>1\). Moreover,  
\[\int_{-1}^{x}\bigl{(}1-(-\theta)^{v}\bigr{)}d\theta=\biggl{(} \theta+(-1)^{v+1}\frac{\theta^{v+1}}{v+1}\biggr{)}\bigg{|}_{-1}^{x}=\frac{v}{v +1}+x+\frac{(-x)^{v+1}}{v+1},\ \ \ x\in[-1,0],\] \[\int_{0}^{x}(1-\theta^{v})d\theta=\biggl{(}\theta-\frac{\theta^{v +1}}{v+1}\biggr{)}\bigg{|}_{0}^{x}=x-\frac{x^{v+1}}{v+1},\ \ \ x\in[0,1],\]  
and consequently,  
\[f_{v_{1},v_{2}}(x)=\left\{ \begin{array}{ll}0&\mbox{for $x<-1$},\\ \gamma_{v_{1},v_{2}}\biggl{(}\frac{v_{1}}{v_{1}+1}+x+\frac{(-x)^{v_{1}+1}}{v_{ 1}+1}\biggr{)}&\mbox{for $x\in[-1,0)$},\\ \gamma_{v_{1},v_{2}}\biggl{(}\frac{v_{1}}{v_{1}+1}+x-\frac{x^{v_{2}+1}}{v_{ 2}+1}\biggr{)}&\mbox{for $x\in[0,1]$},\\ 1&\mbox{for $x>1$}.\end{array} \right. \tag{32}\]  
The function \(f_{v_{1},v_{2}}\) is differentiable \(r+1\) times at \(x=0\) since \(f_{v_{1},v_{2}}^{(r+1)}=g_{v_{1},v_{2}}^{(r)}\) due to relationship (31). Thus,  
\[f_{v_{1},v_{2}}\in C^{r+1}(\Omega),\ \ \ f_{v_{1},v_{2}}^{(s+1)}\in L_{2}( \Omega),\ \ \ f_{v_{1},v_{2}}\in W_{2}^{s+1}(\Omega).\]  
If we do not restrict ourselves to the space \(W_{2}^{s}(\Omega)\) with natural \(s\) (see Remark 1), then we can show that \(s<\min\{v_{1},v_{2}\}+1/2\) and \(\sup s=\min\{v_{1},v_{2}\}+1/2\). Consider the case \(v_{1}>v_{2}\). Then, the generalized derivative \(g_{v_{1},v_{2}}^{(v_{2})}\) is represented as a linear combination of functions \(\eta(x)\) and \(x^{v_{1}-v_{2}}\eta(-x)\), where \(\eta\) is the indicator of \((0,\infty)\). It suffices to find a condition for parameter \(\sigma\) which ensures that \(\eta\in W_{2}^{s}(\Omega)\), where \(W_{2}^{\sigma}(\Omega)\) is defined by Formula (11) and \(\sigma\in(0,1)\). If \(x\) and \(y\) have the same sign, then \(|\eta(x)-\eta(y)|=0\), and if \(x\) and \(y\) have different signs, then \(|\eta(x)-\eta(y)|=1\).