## 4 Methods of Estimation and Test
### The Nikulin-Rao-Robson Test  
The probability distribution used to describe any phenomenon is very important in this analysis. Since the twentieth century, researchers have continued to develop techniques to validate the different models. In addition to the classical procedures, we considered in this work a criteria test statistic based on the Nikulin-Rao-Robson statistic (NRR) to fit data to the PZD. The great interest of the NRR statistic is that it recovers all the information lost while grouping data. Based on the MLE on initial data, this statistic \(Y^{2}\) introduced by [34; 35] is a modification of the well-known Chi-square Pearson statistic \(T^{2}\). For testing the null hypothesis \(H_{0}\), given by  
\[H_{0}=P(X_{i}\leq x)=F(x,\Psi),\quad x\geq 0.\]  
According to the above, the sample \(X_{1},X_{2},\ldots,\ X_{n}\) belongs to the parametric family \(F(x,\Psi)\) with unknown parameters, and \(Y^{2}\) consists in grouping data into \(r\) equiprobable intervals \(I_{1}\), \(I_{2}\),..., \(I_{r}\), where  
\[I_{j}=\left\lvert\,a_{j-1},a_{j}\,\right\rvert;\quad I_{i}\cap I_{j}=\phi\quad i \
eq j;\quad\overset{r}{\underset{j=1}{\cup}}I_{j}=R^{1},\]  
such as  
\[p_{j}=\int_{a_{j-1}}^{a_{j}}f(x,\Psi)dx=\frac{1}{r},\ j=1,2,...,r.\]  
If \(v=(v_{1},v_{2},...,v_{r})^{T}\) represents the number of observed \(X_{i}\) grouping into these intervals \(I_{j}\), and the vector \(T_{n}(\Psi)\) is given by  
\[T_{n}\big{(}\dot{\Psi}\big{)}=\Bigg{(}\frac{v_{1}-np_{1}\big{(}\dot{\Psi} \big{)}}{\sqrt{np_{1}\big{(}\dot{\Psi}\big{)}}},\frac{v_{2}-np_{2}\big{(}\dot{ \Psi}\big{)}}{\sqrt{np_{2}\big{(}\dot{\Psi}\big{)}}},\ldots,\frac{v_{r}-np_{r} \big{(}\dot{\Psi}\big{)}}{\sqrt{np_{r}\big{(}\dot{\Psi}\big{)}}}\Bigg{)}^{t}.\]  
The NRR statistic \(Y^{2}\) is defined as  
\[Y^{2}_{n}\big{(}\dot{\Psi}\big{)}=T^{2}_{n}\big{(}\dot{\Psi}\big{)}+\frac{1}{n }l^{T}\big{(}\dot{\Psi}\big{)}\big{(}I\big{(}\dot{\Psi}\big{)}-J\big{(}\dot{ \Psi}\big{)}\big{)}^{-1}l\big{(}\dot{\Psi}\big{)},\]  
where \(I\big{(}\dot{\Psi}\big{)}\) and \(J\big{(}\dot{\Psi}\big{)}\) are the estimated information matrices on non-grouped and grouped data, respectively, and \(\Psi\) is the vector of the ML estimators on initial data. The components of the vector \(l\big{(}\dot{\Psi}\big{)}=\big{(}l_{s}\big{(}\dot{\Psi}\big{)}\big{)}_{1\times s}\) and the matrix \(J\big{(}\dot{\Psi}\big{)}\) are  
\[l_{s}\big{(}\dot{\Psi}\big{)}=\sum_{i=1}^{r}\frac{v_{j}}{p_{j}}\frac{\partial p _{j}\big{(}\dot{\Psi}\big{)}}{\partial\dot{\Psi}_{s}}\,\]  
\[J\big{(}\dot{\Psi}\big{)}=\Bigg{(}\frac{1}{\sqrt{p_{j}}}\frac{\partial p_{j} \big{(}\dot{\Psi}\big{)}}{\partial\dot{\Psi}_{k}}\Bigg{)}_{r\times s}^{t} \Bigg{(}\frac{1}{\sqrt{p_{j}}}\frac{\partial p_{j}\big{(}\dot{\Psi}\big{)}}{ \partial\dot{\Psi}_{k}}\Bigg{)}_{r\times s},j=1,2,\ldots,r\ \text{and}\ k=1,2,\ldots,s,\]  
where \(s\) represents the number of parameters. The distribution of \(Y^{2}\big{(}\dot{\Psi}\big{)}\) is a Chi-square with \(r-1\) degrees of freedom. To construct the test statistic \(Y^{2}\) corresponding to the PZD \((\dot{\Psi})\) we calculated the ML estimators \(\dot{\Psi}=(\dot{\beta},\dot{\theta})^{t}\), the limit intervals \(a_{j}\), and the derivatives \(\frac{\partial p_{j}\big{(}\dot{\Psi}\big{)}}{\partial\dot{\Psi}_{k}}\) for our model.  
\[\frac{\partial p_{j}\big{(}\dot{\Psi}\big{)}}{\partial\dot{\theta}}=\varphi_{1 }\big{(}x_{j},\dot{\Psi}\big{)}-\varphi_{1}\big{(}x_{j-1},\dot{\Psi}\big{)}\,\frac{\partial p_{j}\big{(}\dot{\Psi}\big{)}}{\partial\dot{\beta}}=\varphi_{2 }\big{(}x_{j},\dot{\Psi}\big{)}-\varphi_{2}\big{(}x_{j-1},\dot{\Psi}\big{)}.\]Then, after computing the estimated information matrices \(I(\hat{\Psi})\) and \(J(\hat{\Psi})\), we obtained the statistic \(\gamma^{2}(\hat{\Psi})\) for the \(PZD\) distribution. This method is used to construct GOF test statistics for some generalized models; for examples see [36; 37; 38; 39].