}$, $F(\boldsymbol{y}|\boldsymbol{x}_{hi})$ and $G(\boldsymbol{y}|\boldsymbol{x}_{hi})$ are the distribution functions corresponding to the densities $f(\boldsymbol{y}|\boldsymbol{x}_{hi})$ and $g(\boldsymbol{y}|\boldsymbol{x}_{hi})$, respectively, and$$d_{\lambda }^{\ast }\left( g(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi}),f_{\boldsymbol{\beta }}(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})\right) =E[f_{\boldsymbol{\beta }}^{\lambda }(\boldsymbol{Y}_{hij}|\boldsymbol{x}_{hi})]-\frac{\lambda +1}{\lambda }\int\nolimits_{\mathbb{Y}_{hi}}f_{\boldsymbol{\beta }}^{\lambda }(\boldsymbol{y}|\boldsymbol{x}_{hi})dG(\boldsymbol{y}|\boldsymbol{x}_{hi})$$is the kernel of $d_{\lambda }\left( g(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})\text{, }f_{\boldsymbol{\beta }}(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})\right) $. In practice, since $G$ is unknown, it must be estimated from the sample, which is in this case a single individual, so$$d_{\lambda }^{\ast }\left( \widehat{g}(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi}),f_{\boldsymbol{\beta }}(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})\right) =E[f_{\boldsymbol{\beta }}^{\lambda }(\boldsymbol{Y}_{hij}|\boldsymbol{x}_{hi})]-\frac{\lambda +1}{\lambda }f_{\boldsymbol{\beta }}^{\lambda }(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi}).$$Based on Ghosh and Basu (2013), the kernel of the ordinary DPD between the probability mass functions $\widehat{g}(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})$ and $f_{\boldsymbol{\beta }}(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})$ for the whole sample is defined as a total discrepancy given by $$d_{\lambda }^{\ast }(\widehat{g},f_{\boldsymbol{\beta }})=\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\sum_{j=1}^{m_{hi}}\left( E\left[ f_{\boldsymbol{\beta }}^{\lambda }(\boldsymbol{Y}_{hij}|\boldsymbol{x}_{hi})\right] -\frac{\lambda
+1}{\lambda }f_{\boldsymbol{\beta }}^{\lambda }(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})\right) ,\quad \text{for }\lambda >0.$$Since the support of $\boldsymbol{Y}_{hij}|\boldsymbol{x}_{hi}$, is the set of column vectors of identity matrix $\boldsymbol{I}_{d+1}$, it holds $$E\left[ f_{\boldsymbol{\beta }}^{\lambda }(\boldsymbol{Y}_{hij}|\boldsymbol{x}_{hi})\right] =E\left[ \boldsymbol{\pi }_{hi}^{\lambda ,T}\left( 
\boldsymbol{\beta }\right) \boldsymbol{Y}_{hij}\right] =\sum_{l=1}^{d+1}\pi
_{hil}^{\lambda +1}\left( \boldsymbol{\beta }\right) .$$

In the current paper it is defined for the first time the kernel of the quasi weighted DPD between the probability mass functions $\widehat{g}(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})$ and $f_{\boldsymbol{\beta }}(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})$ in the whole sample as a weighted sum$$d_{\lambda }^{\ast }(\widehat{g},f_{\boldsymbol{\beta }},w)=\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\sum_{j=1}^{m_{hi}}w_{hi}\left( E\left[ f_{\boldsymbol{\beta }}^{\lambda }(\boldsymbol{Y}_{hij}|\boldsymbol{x}_{hi})\right] -\frac{\lambda +1}{\lambda }f_{\boldsymbol{\beta }}^{\lambda }(\boldsymbol{y}_{hij}|\boldsymbol{x}_{hi})\right) ,\quad 