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An adjusted iterative algorithmic approach for maximum likelihood estimation of the pseudo-copula regression: P-MBP

  • Mathematical methods in data science
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Abstract

In this study, a pseudo-maximization by parts method is introduced by developing the maximization by parts algorithm for the parameter estimation of pseudo-copula regression models. Sub- and main score equations are obtained from the pairwise log-likelihood function and solved by the proposed iterative algorithm. The pseudo-maximization by parts algorithm is an iterative algorithm to avoid having to calculate the second-order derivative of the full log-likelihood function as maximization by parts algorithm. Instead of the Gaussian copula function in maximization by parts algorithm, the pseudo-Gaussian copula function is included in the new algorithm. The mean square errors of the estimators found by the maximization by parts algorithm and the pseudo-maximization by parts algorithm are compared using real Turkish comprehensive insurance data taken from the Turkish Insurance Information and Monitoring Center for the year 2017, and it is notable that the proposed algorithm provided better results in terms of having lower errors.

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Funding

The author declares that no funds, grants, or other support were received during the preparation of this manuscript.

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Authors and Affiliations

  1. Department of Actuarial Science, Faculty of Science, Hacettepe University, Ankara, Turkey

    Övgücan Karadağ Erdemir

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  1. Övgücan Karadağ Erdemir
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Appendix

Appendix

$$\frac{\partial dep\left({{\varvec{\theta}}}_{1},{{\varvec{\theta}}}_{2}\right)}{\partial{\varvec{\beta}}}=\frac{\partial }{\partial{\varvec{\beta}}}ln\prod_{i=1}^{n}\left\{{f}_{{{X}_{1}X}_{2}}\left({x}_{i1},{x}_{i2}|{\mu }_{i},{\nu }_{i}^{2},{\lambda }_{i}\right)\right\}=\sum_{i=1}^{n}\frac{\partial }{\partial{\varvec{\beta}}}ln\left\{{D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)-{D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)\right\}=\sum_{i=1}^{n}\frac{1}{\left\{{D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)-{D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)\right\}}\left(\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)}{\partial{\varvec{\beta}}}-\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)}{\partial{\varvec{\beta}}}\right)$$

Let \(\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)}{\partial{\varvec{\beta}}}\) be multiplied by \(\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}\) and \(\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)}{\partial{\varvec{\beta}}}\) be multiplied by \(\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}-1\right)}{\partial {F}_{{X}_{2}}\left({x}_{i2}-1\right)}\).

$$=\sum_{i=1}^{n}\frac{1}{\left\{{D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)-{D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)\right\}}\left(\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)}{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial{\varvec{\beta}}} -\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)}{\partial {F}_{{X}_{2}}\left({x}_{i2}-1\right)}\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}-1\right)}{\partial{\varvec{\beta}}}\right)$$

Partial integrals \(\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)}{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}\), \(\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial{\varvec{\beta}}}\), \(\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)}{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}\) and \(\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}-1\right)}{\partial{\varvec{\beta}}}\) are expressed one by one below.

$$\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)}{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}=\frac{\partial }{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}\Phi \left(\frac{{\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}\right)\right)-{\rho }_{12}^{*}{\Phi }^{-1}\left({F}_{{X}_{1}}\left({x}_{i1}\right)\right)}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}\right)\frac{1}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}\frac{1}{\left({\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}\right)\right)\right)}$$
$$\frac{\partial }{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}\Phi \left(\frac{{\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}\right)\right)-{\rho }_{12}^{*}{\Phi }^{-1}\left({F}_{{X}_{1}}\left({x}_{i1}\right)\right)}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}\right)=\phi \left(\frac{{\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}\right)\right)-{\rho }_{12}^{*}{\Phi }^{-1}\left({F}_{{X}_{1}}\left({x}_{i1}\right)\right)}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}\right)={d}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)$$
$$\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)}{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}={d}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)\frac{1}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}\frac{1}{\left({\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}\right)\right)\right)}$$
(1)

Let \(\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial{\varvec{\beta}}}\) be multiplied by \(\frac{\partial {\lambda }_{i}}{\partial {\lambda }_{i}}\). For a Poisson GLM \({\lambda }_{i}={e}^{{{\varvec{z}}}_{{\varvec{i}}2}^{\boldsymbol{^{\prime}}}{\varvec{\beta}}}\), \(\frac{\partial {\lambda }_{i}}{\partial{\varvec{\beta}}}={{\varvec{z}}}_{{\varvec{i}}2}^{\boldsymbol{^{\prime}}}{e}^{{{\varvec{z}}}_{{\varvec{i}}2}^{\boldsymbol{^{\prime}}}{\varvec{\beta}}}={{\varvec{z}}}_{{\varvec{i}}2}^{\boldsymbol{^{\prime}}}{\lambda }_{i}\) is written.

$$\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial{\varvec{\beta}}}=\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial {\lambda }_{i}}\frac{\partial {\lambda }_{i}}{\partial{\varvec{\beta}}}=\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial {\lambda }_{i}}{{\varvec{z}}}_{{\varvec{i}}2}^{\boldsymbol{^{\prime}}}{\lambda }_{i}$$

\(\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial {\lambda }_{i}}\) is expressed as \(\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial {\lambda }_{i}}=\frac{\partial }{\partial {\lambda }_{i}}\sum_{k=0}^{{x}_{i2}}\frac{1}{k!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}=\sum_{k=0}^{{x}_{i2}}\frac{\partial }{\partial {\lambda }_{i}}\left(\frac{1}{k!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}\right)\).

Since \(0\le k\le {x}_{i2}\), when deriving at the point k = 0, the expression \(\sum_{k=0}^{{x}_{i2}}\frac{1}{(k-1)!}\dots .\) is encountered. The range is separated as \(k=0\) and \(1\le k\le ,\) since the factorial operation is undefined for negative numbers.

$$=\frac{\partial }{\partial {\lambda }_{i}}\left(\frac{1}{0!}{{\lambda }_{i}}^{0}{e}^{-{\lambda }_{i}}\right)+\sum_{k=1}^{{x}_{i2}}\frac{\partial }{\partial {\lambda }_{i}}\left(\frac{1}{k!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}\right)$$
$$=\frac{\partial }{\partial {\lambda }_{i}}\left({e}^{-{\lambda }_{i}}\right)+\sum_{k=1}^{{x}_{i2}}\left(\frac{1}{k!}k{{\lambda }_{i}}^{k-1}{e}^{-{\lambda }_{i}}-\frac{1}{k!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}\right)=-{e}^{-{\lambda }_{i}}+\sum_{k=1}^{{x}_{i2}}\left(\frac{1}{\left(k-1\right)!}{{\lambda }_{i}}^{k-1}{e}^{-{\lambda }_{i}}-\frac{1}{k!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}\right)=-{e}^{-{\lambda }_{i}}+\left(\sum_{k=1}^{{x}_{i2}}\frac{1}{\left(k-1\right)!}{{\lambda }_{i}}^{k-1}{e}^{-{\lambda }_{i}}\right)-\left(\sum_{k=1}^{{x}_{i2}}\frac{1}{k!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}\right)$$
$$=-{e}^{-{\lambda }_{i}}+\left(\sum_{k=0}^{{x}_{i2}-1}\frac{1}{\left(k\right)!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}\right)-\left(\sum_{k=1}^{{x}_{i2}}\frac{1}{k!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}\right)=-{e}^{-{\lambda }_{i}}+{e}^{-{\lambda }_{i}}+\left(\sum_{k=1}^{{x}_{i2}-1}\frac{1}{\left(k\right)!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}\right)-\left(\sum_{k=1}^{{x}_{i2}-1}\frac{1}{k!}{{\lambda }_{i}}^{k}{e}^{-{\lambda }_{i}}\right)-\frac{1}{{x}_{i2}!}{{\lambda }_{i}}^{{x}_{i2}}{e}^{-{\lambda }_{i}}=-\frac{1}{{x}_{i2}!}{{\lambda }_{i}}^{{x}_{i2}}{e}^{-{\lambda }_{i}}=-{f}_{{X}_{2}}\left({x}_{i2}|{\lambda }_{i}\right)$$
$$\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}{\partial{\varvec{\beta}}}=-{f}_{{X}_{2}}\left({x}_{i2}|{\lambda }_{i}\right){{\varvec{z}}}_{{\varvec{i}}2}^{\boldsymbol{^{\prime}}}{\lambda }_{i}$$
(2)

\(\frac{\partial {D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)}{\partial {F}_{{X}_{2}}\left({x}_{i2}\right)}\) and \(\frac{\partial {F}_{{X}_{2}}\left({x}_{i2}-1\right)}{\partial{\varvec{\beta}}}\) are the discrete forms of Eqs. (1) and (2), respectively.

$$ \frac{{\partial D_{{\rho_{12}^{*} }} \left( {F_{{X_{1} }} \left( {x_{i1} } \right),F_{{X_{2} }} \left( {x_{i2} - 1} \right)} \right)}}{{\partial F_{{X_{2} }} \left( {x_{i2} } \right)}} = d_{{\rho_{12}^{*} }} \left( {F_{{X_{1} }} \left( {x_{i1} } \right),F_{{X_{2} }} \left( {x_{i2} - 1} \right)} \right)\frac{1}{{\sqrt {\left( {1 - \left( {\rho_{12}^{*} } \right)^{2} } \right)} }} $$
(3)

\(\frac{{\partial F_{{X_{2} }} \left( {x_{i2} - 1} \right)}}{{\partial {\varvec{\beta}}}} = \frac{{\partial F_{{X_{2} }} \left( {x_{i2} - 1} \right)}}{{\partial \lambda_{i} }}\frac{{\partial \lambda_{i} }}{{\partial {\varvec{\beta}}}} = \frac{{\partial F_{{X_{2} }} \left( {x_{i2} - 1} \right)}}{{\partial \lambda_{i} }}{\varvec{z}}_{{{\varvec{i}}2}}^{\user2{^{\prime}}} \lambda_{i}\)

$$ \frac{{\partial F_{{X_{2} }} \left( {x_{i2} - 1} \right)}}{{\partial \lambda_{i} }} = \frac{\partial }{{\partial \lambda_{i} }}\mathop \sum \limits_{k = 0}^{{x_{i2} - 1}} \frac{1}{k!}\lambda_{i}^{k} e^{{ - \lambda_{i} }} = - f_{{X_{2} }} \left( {x_{i2} - 1{|}\lambda_{i} } \right) $$
$$ \frac{{\partial F_{{X_{2} }} \left( {x_{i2} - 11} \right)}}{{\partial {\varvec{\beta}}}} = - f_{{X_{2} }} \left( {x_{i2} - 1{|}\lambda_{i} } \right){\varvec{z}}_{{{\varvec{i}}2}}^{\user2{^{\prime}}} \lambda_{i} $$
(4)

All components for the solution of the second main score equation are obtained by Eqs. (1)–(4), and \(\frac{\partial {l}_\mathrm{dep}\left({{\varvec{\theta}}}_{1},{{\varvec{\theta}}}_{2}\right)}{\partial{\varvec{\beta}}}\) is written in Eq. (5) with the definition of \({d}_{{\rho }_{12}^{*}}\left(.,.\right)\).

$$\frac{\partial {l}_\mathrm{dep}\left({{\varvec{\theta}}}_{1},{{\varvec{\theta}}}_{2}\right)}{\partial{\varvec{\beta}}}=\sum_{i=1}^{n}\frac{1}{\left({D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)-{D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)\right)}*\left[\phi \left(\frac{{\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}\right)\right)-{\rho }_{12}^{*}{\Phi }^{-1}\left({F}_{{X}_{1}}\left({x}_{i1}\right)\right)}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}\right)*\frac{1}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}\frac{1}{\phi \left({\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}\right)\right)\right)}\left(-{f}_{{X}_{2}}\left({x}_{i2}|{\lambda }_{i}\right){{\varvec{z}}}_{{\varvec{i}}2}^{\boldsymbol{^{\prime}}}{\lambda }_{i} \right)-\phi \left(\frac{{\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)-{\rho }_{12}^{*}{\Phi }^{-1}\left({F}_{{X}_{1}}\left({x}_{i1}\right)\right)}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}\right)\frac{1}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}\frac{1}{\phi \left({\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)\right)}\left(-{f}_{{X}_{2}}\left({x}_{i2}-1|{\lambda }_{i}\right){{\varvec{z}}}_{{\varvec{i}}2}^{\boldsymbol{^{\prime}}}{\lambda }_{i}\right)\right]$$
$$\frac{\partial {l}_\mathrm{dep}\left({{\varvec{\theta}}}_{1},{{\varvec{\theta}}}_{2}\right)}{\partial{\varvec{\beta}}}=\sum_{i=1}^{n}\frac{1}{\left({D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)-{D}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)\right)} * \left[\frac{{d}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)}{\phi \left({\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}-1\right)\right)\right)}-\frac{{d}_{{\rho }_{12}^{*}}\left({F}_{{X}_{1}}\left({x}_{i1}\right),{F}_{{X}_{2}}\left({x}_{i2}\right)\right)}{\phi \left({\Phi }^{-1}\left({F}_{{X}_{2}}\left({x}_{i2}\right)\right)\right)}\right]*\frac{{{\varvec{z}}}_{{\varvec{i}}2}^{\mathbf{^{\prime}}}{\lambda }_{i}}{\sqrt{\left(1-{\left({\rho }_{12}^{*}\right)}^{2}\right)}}$$
(5)

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Erdemir, Ö.K. An adjusted iterative algorithmic approach for maximum likelihood estimation of the pseudo-copula regression: P-MBP. Soft Comput 27, 11227–11241 (2023). https://doi.org/10.1007/s00500-023-08722-8

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