Univariate and Bivariate Compound Models Based on Random Sum of Variates with Application to the Insurance Losses Data

Abstract

In this article, we propose a new three-parameter model called compound zero-truncated Poisson gamma model. The corresponding variate of this model represents the zero-truncated Poisson sum of independent and identically distributed gamma random variables. Several mathematical properties of the proposed model are discussed. The proposed model can be unimodal as well as multimodal, moreover, gamma distribution can be obtained as a special case. The model parameters are estimated using expectation–maximization (EM)-type algorithm, and it is observed that it is quite convenient to be implemented in practice. Furthermore, an extension of the model is made to consider four-parameter bivariate model, and derived estimation of unknown parameters and confidence intervals based on EM-type algorithm. For validation purposes, some numerical simulation experiments are conducted to check how the proposed EM-type algorithm performs. Finally, the analysis of real data set has been presented to show the flexibility of the proposed models.

Appendices

Appendices:

Here, we present some derivations for the Hessian matrix based on ZTP-EXP and BZTP-EXP models.

Appendix A: Hessian Matrix B and Vector S Components for UZTP-GA Model

$$\begin{aligned} B= \left( \begin{array}{ccc} \dfrac{\partial ^{2} l_{s}}{\partial \alpha ^{2}} \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial \beta } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial \theta } \\ \dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial \alpha } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \beta ^{2}} \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial \theta } \\ \dfrac{\partial ^{2} l_{s}}{\partial \theta \ \partial \alpha } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \theta \ \partial \beta } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \theta ^{2}} \end{array} \right) \ \ \ \ \ \hbox {and} \ \ \ \ \ S=\bigg [\dfrac{\partial l_{s}}{\partial \alpha } \ \ \ \ \ \ \dfrac{\partial l_{s}}{\partial \beta } \ \ \ \ \ \ \dfrac{\partial l_{s}}{\partial \theta } \bigg ]^{T}, \end{aligned}$$

with

$$\begin{aligned} \dfrac{\partial l_{s}}{\partial \alpha }= & {} \displaystyle \sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}}) \ln y_{i}-\displaystyle \sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}})\Psi (\alpha E_{i}({\varvec{\Omega ^{(j)}}}))\\&+\,\ln \bigg ( \alpha \dfrac{\sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}})}{\sum _{i=1}^{n}y_{i}} \bigg )\displaystyle \sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}}),\\ \dfrac{\partial l_{s}}{\partial \beta }= & {} \dfrac{\alpha }{\beta }\displaystyle \sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}})-\displaystyle \sum _{i=1}^{n}y_{i},\\ \dfrac{\partial l_{s}}{\partial \theta }= & {} \dfrac{\sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}})}{\theta }-\dfrac{ne^{\theta }}{e^{\theta }-1},\\ \dfrac{\partial ^{2} l_{s}}{\partial \alpha ^{2}}= & {} \dfrac{1}{\alpha }\displaystyle \sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}})-\displaystyle \sum _{i=1}^{n}(E_{i}({\varvec{\Omega ^{(j)}}}))^{2}\Psi ^{'}(\alpha E_{i}({\varvec{\Omega ^{(j)}}})),\\ \dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial \beta }= & {} \dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial \alpha }=\dfrac{1}{\beta }\displaystyle \sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}}), \\ \dfrac{\partial ^{2} l_{s}}{\partial \beta ^{2}}= & {} -\dfrac{\alpha }{\beta ^{2}}\displaystyle \sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}}), \ \ \, \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \theta ^{2}}=-\dfrac{\sum _{i=1}^{n}E_{i}({\varvec{\Omega ^{(j)}}})}{\theta ^2}+\dfrac{ne^{-\theta }}{(1-e^{-\theta })^{2}}, \end{aligned}$$

and

$$\begin{aligned} \dfrac{\partial ^{2} l_{s}}{\partial \theta \ \partial \alpha }=\dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial \theta }=\dfrac{\partial ^{2} l_{s}}{\partial \theta \ \partial \beta }=\dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial \theta }=0. \end{aligned}$$

Appendix B: Hessian Matrix B and Vector S Components for BZTP-GA Model

$$\begin{aligned} B= \left( \begin{array}{cccc} \dfrac{\partial ^{2} l_{s}}{\partial \alpha ^{2}} \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial \beta } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial \lambda } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial p} \\ \dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial \alpha } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \beta ^{2}} \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial \lambda } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial p} \\ \dfrac{\partial ^{2} l_{s}}{\partial \lambda \ \partial \alpha } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \lambda \ \partial \beta } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \lambda ^{2}} \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \lambda \ \partial p} \\ \dfrac{\partial ^{2} l_{s}}{\partial p \ \partial \alpha } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial p \partial \beta } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial p \ \partial \lambda } \ \ \ &{} \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial p^{2}} \end{array} \right) \ \ \ \ \ \hbox {and} \ \ \ \ \ S=\bigg [\dfrac{\partial l_{s}}{\partial \alpha } \ \ \ \ \ \ \dfrac{\partial l_{s}}{\partial \beta } \ \ \ \ \ \ \dfrac{\partial l_{s}}{\partial \lambda } \ \ \ \ \ \dfrac{\partial l_{s}}{\partial p} \bigg ]^{T}, \end{aligned}$$

with

$$\begin{aligned} \dfrac{\partial l_{s}}{\partial \alpha }= & {} \displaystyle \sum _{i=1}^{n}m_{i}^{(j)} \ln z_{i}-\displaystyle \sum _{i=1}^{n}m_{i}^{(j)}\Psi (\alpha m_{i}^{(j)})+\ln \bigg ( \alpha \dfrac{\sum _{i=1}^{n}m_{i}^{(j)}}{\sum _{i=1}^{n}z_{i}} \bigg )\displaystyle \sum _{i=1}^{n}m_{i}^{(j)},\\ \dfrac{\partial l_{s}}{\partial \beta }= & {} \dfrac{\alpha }{\beta }\displaystyle \sum _{i=1}^{n}m_{i}^{(j)}-\displaystyle \sum _{i=1}^{n}z_{i}, \ \ \ \ \ \dfrac{\partial l_{s}}{\partial \lambda }=\dfrac{1}{\lambda } \displaystyle \sum _{i=1}^{n}v_{i}-\dfrac{n e^{\lambda }}{e^{\lambda }-1},\\ \dfrac{\partial l_{s}}{\partial p}= & {} \dfrac{1}{p}\displaystyle \sum _{i=1}^{n} m_{i}^{(j)}-\dfrac{1}{1-p}\displaystyle \sum _{i=1}^{n}(v_{i}-m_{i}^{(j)})-\displaystyle \sum _{i=1}^{n} \bigg [\dfrac{v_{i}(1-p)^{v_{i}-1}}{1-(1-p)^{v_{i}}}\bigg ],\\ \dfrac{\partial ^{2} l_{s}}{\partial \alpha ^{2}}= & {} \dfrac{1}{\alpha }\displaystyle \sum _{i=1}^{n}m_{i}^{(j)}-\displaystyle \sum _{i=1}^{n}(m_{i}^{(j)})^{2}\Psi ^{'}(\alpha m_{i}^{(j)}), \ \ \ \ \ \dfrac{\partial ^{2} l_{s}}{\partial \beta ^{2}}= -\dfrac{\alpha }{\beta ^{2}}\displaystyle \sum _{i=1}^{n}m_{i}^{(j)},\\ \dfrac{\partial ^{2} l_{s}}{\partial \lambda ^{2}}= & {} -\dfrac{1}{\lambda ^{2}}\displaystyle \sum _{i=1}^{n}v_{i}+\dfrac{ne^{-\lambda }}{(1-e^{-\lambda })^{2}}, \\ \dfrac{\partial ^{2} l_{s}}{\partial p^{2}}= & {} -\dfrac{1}{p^{2}}\displaystyle \sum _{i=1}^{n}m_{i}^{(j)}- \dfrac{1}{(1-p)^{2}} \bigg (\displaystyle \sum _{i=1}^{n}(v_{i}-m_{i}^{(j)})\bigg )\\&+\,\displaystyle \sum _{i=1}^{n}\bigg [\dfrac{v_{i}(v_{i}-1)(1-p)^{v_{i}-2}+(1-p)^{2v_{i}-2}}{(1-(1-p)^{v_{i}})^{2}} \bigg ],\\ \dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial \beta }= & {} \dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial \alpha }=\dfrac{1}{\beta }\displaystyle \sum _{i=1}^{n}m_{i}^{(j)}, \end{aligned}$$

and

$$\begin{aligned} \dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial \lambda }= & {} \dfrac{\partial ^{2} l_{s}}{\partial \lambda \ \partial \beta }=\dfrac{\partial ^{2} l_{s}}{\partial \beta \ \partial p}=\dfrac{\partial ^{2} l_{s}}{\partial p \ \partial \beta }= \dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial \lambda }\\= & {} \dfrac{\partial ^{2} l_{s}}{\partial \lambda \ \partial \alpha }= \dfrac{\partial ^{2} l_{s}}{\partial \alpha \ \partial p}=\dfrac{\partial ^{2} l_{s}}{\partial p \ \partial \alpha }=\dfrac{\partial ^{2} l_{s}}{\partial \lambda \ \partial p}=\dfrac{\partial ^{2} l_{s}}{\partial p \ \partial \lambda }=0. \end{aligned}$$