A bivariate extension of three-parameter generalized crack distribution for loss severity modelling

Abstract

In this paper, we introduce a bivariate extension of three-parameter generalized crack distribution for modelling loss data. Some basic properties such as the conditional distribution and the measures of association are discussed, and a method of parameter estimation is offered. A simulation-based approach to compute bivariate value-at-risk under the model is also discussed. The proposed model and estimation method are illustrated with a model fitting exercise on a real catastrophic loss data set.

Appendix

1.1 Proof of Proposition 1

For notational convenience, we drop parameters in the argument of functions, and write $\bar{G}(\cdot ) = 1 - G(\cdot )$ and $b(t_i) = \alpha _i^{-1}\left( \sqrt{\beta _i/t_i}- \sqrt{t_i/\beta _i}\right) ,\, i = 1, 2$.

By (10), the joint distribution of $(T_1, T_2)$ having $\text {BVGCR}(\varvec{\alpha },\varvec{\beta },\varvec{p};g)$ is

$$\begin{aligned}&F(t_1, t_2) = p_{11} F_{IS}(t_1)F_{IS}(t_2) + p_{12} F_{IS}(t_1)F_{LB}(t_2)\\&\quad + p_{21} F_{LB}(t_1)F_{IS}(t_2) + p_{22} F_{LB}(t_1)F_{LB}(t_2), \end{aligned}$$

and the marginal density of $T_i, \, i =1, 2$, is

$$\begin{aligned} f_{T_i}(t_i) = p_i f_{IS}(t_i) + q_i f_{LB}(t_i), \end{aligned}$$

where $p_1 = p_{11}+p_{12}$, $p_2 = p_{11}+p_{21}$, $q_i = 1-p_i$, $i =1,2$. By expanding the integrand in (12) using these expressions and taking the double integration, we have

$$\begin{aligned}&\int _0^{\infty }\int _0^{\infty }F(t_1, t_2) f_{T_1}(t_1)f_{T_2}(t_2)\,dt_1 dt_2\\&\quad = p_{11}\left\{ p_1 p_2 \text{ E }[F_{S_1}(S_1)]\text{ E }[F_{S_2}(S_2)] + p_1 q_1 \text{ E }[F_{S_1}(S_1)] \zeta _2 + q_1 p_2 \zeta _1 \text{ E }[F_{S_2}(S_2)] + q_1 q_2 \zeta _1 \zeta _2\right\} \\&\qquad + p_{12}\left\{ p_1 p_2 \text{ E }[F_{S_1}(S_1)]\eta _2 + p_1 q_1 \text{ E }[F_{S_1}(S_1)] \text{ E }[F_{V_2}(V_2)] + q_1 p_2 \zeta _1 \eta _2 + q_1 q_2 \zeta _1 \text{ E }[F_{V_2}(V_2)] \right\} \\&\qquad + p_{21}\left\{ p_1 p_2 \eta _1 \text{ E }[F_{S_2}(S_2)] + p_1 q_1 \eta _1 \zeta _2 + q_1 p_2\text{ E }[F_{V_1}(V_1)] \text{ E }[F_{S_2}(S_2)] + q_1 q_2 \text{ E }[F_{V_1}(V_1)]\zeta _2 \right\} \\&\, + p_{22}\left\{ p_1 p_2 \eta _1 \eta _2 + p_1 q_1 \eta _1 \text{ E }[F_{V_2}(V_2)] + q_1 p_2\text{ E }[F_{V_1}(V_1)] \eta _2 + q_1 q_2 \text{ E }[F_{V_1}(V_1)]\text{ E }[F_{V_2}(V_2)] \right\} , \end{aligned}$$

where, for each $i = 1,2$,

$$\begin{aligned} \text{ E }[F_{S_i}(S_i)]= & {} \int _0^{\infty } F_{IS}(t_i)f_{IS}(t_i) dt_i = \frac{1}{2}\\ \text{ E }[F_{V_i}(V_i)]= & {} \int _0^{\infty } F_{LB}(t_i)f_{LB}(t_i) dt_i = \frac{1}{2}\\ \zeta _i:= & {} \int _0^{\infty } F_{IS}(t_i)f_{LB}(t_i) dt_i \\= & {} \int _0^{\infty } \left[ F_{LB}(t_i) + 2 H(t_i)\right] f_{LB}(t_i) dt_i\\= & {} \text{ E }[F_{V_i}(V_i)] + 2 \int _0^{\infty } H(t_i)f_{LB}(t_i) dt_i = \frac{1}{2} + 2 \int _0^{\infty } H(t_i)f_{LB}(t_i) dt_i\\ \eta _i:= & {} \int _0^{\infty } F_{LB}(t_i)f_{IS}(t_i) dt_i \\= & {} \int _0^{\infty } \left[ F_{IS}(t_i) - 2 H(t_i)\right] f_{IS}(t_i) dt_i\\= & {} \text{ E }[F_{S_i}(S_i)] - 2 \int _0^{\infty } H(t_i)f_{IS}(t_i) dt_i = \frac{1}{2} - 2 \int _0^{\infty } H(t_i)f_{IS}(t_i) dt_i. \end{aligned}$$

That is, the evaluation of the double integral in (12) reduces to that of the following two integrals: $\int _0^{\infty } H(t)f_{IS}(t) dt$ and $\int _0^{\infty } H(t)f_{LB}(t) dt$. Due to the expression $F_{IS}(t) = 1- G(b(t)) + H(t)$ and

$$\begin{aligned}&\int _{-\infty }^{\infty } H(b^{-1}(s)) \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds \\&\quad = \int _{-\infty }^{\infty } \left( \int _{s}^{\infty } \frac{z}{\sqrt{z^2+4/\alpha ^2}}\,g(z)dz\right) \, \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds = 0, \end{aligned}$$

and changing the order of integrations, we obtain

$$\begin{aligned} \int _0^{\infty } H(t)f_{IS}(t) dt= & {} \int _0^{\infty } \int _{b(t)}^{\infty }\frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\, f_{IS}(t)dt\\= & {} \int _{-\infty }^{\infty } \left( \int _{b^{-1}(s)}^{\infty } f_{IS}(t)dt\right) \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\\= & {} \int _{-\infty }^{\infty } \left( 1- F_{IS}(b^{-1}(s))\right) \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\\= & {} \int _{-\infty }^{\infty } \left( G(s) - H(b^{-1}(s)) \right) \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\\= & {} \int _{-\infty }^{\infty } G(s)\frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds - \int _{-\infty }^{\infty } H(b^{-1}(s)) \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\\= & {} \int _{-\infty }^{\infty } \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,G(s)g(s)ds = \gamma . \end{aligned}$$

Similarly, due to the expression $F_{LB}(t) = 1- G(b(t)) - H(t)$, we obtain

$$\begin{aligned} \int _0^{\infty } H(t)f_{LB}(t) dt= & {} \int _0^{\infty } \int _{b(t)}^{\infty }\frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\, f_{LB}(t)dt\\= & {} \int _{-\infty }^{\infty } \left( \int _{b^{-1}(s)}^{\infty } f_{LB}(t)dt\right) \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\\= & {} \int _{-\infty }^{\infty } \left( 1- F_{LB}(b^{-1}(s))\right) \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\\= & {} \int _{-\infty }^{\infty } \left( G(s) + H(b^{-1}(s)) \right) \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\\= & {} \int _{-\infty }^{\infty } G(s)\frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\\&\quad + \int _{-\infty }^{\infty } H(b^{-1}(s)) \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,g(s)ds\\= & {} \int _{-\infty }^{\infty } \frac{s}{\sqrt{s^2+4/\alpha ^2}}\,G(s)g(s)ds = \gamma . \end{aligned}$$

Combining these results and some simplifications give the expression (13).

1.2 Proof of Proposition 2

As in the proof of Proposition 1, expanding the integrand in (14) using the joint cdf and pdf of the BVGCR and taking double integrals give

$$\begin{aligned}&\int _0^{\infty }\int _0^{\infty }F(t_1, t_2) f(t_1, t_2)\,dt_1 dt_2\\&\quad = (p_{11})^2 \text{ E }[F_{S_1}(S_1)]\text{ E }[F_{S_2}(S_2)] + p_{11} p_{12} \text{ E }[F_{S_1}(S_1)] \zeta _2 + p_{11} p_{21} \zeta _1 \text{ E }[F_{S_2}(S_2)] \\&\qquad + p_{11} p_{22} \zeta _1 \zeta _2\\&\qquad + p_{12} p_{11} \text{ E }[F_{S_1}(S_1)]\eta _2 + (p_{12})^2 \text{ E }[F_{S_1}(S_1)] \text{ E }[F_{V_2}(V_2)] \\&\qquad + p_{12} p_{21} \zeta _1 \eta _2 + p_{12} p_{22} \zeta _1 \text{ E }[F_{V_2}(V_2)] \\&\qquad + p_{21} p_{11}\eta _1 \text{ E }[F_{S_2}(S_2)] + p_{21} p_{12} \eta _1 \zeta _2 \\&\qquad + (p_{21})^2\text{ E }[F_{V_1}(V_1)] \text{ E }[F_{S_2}(S_2)] + p_{21} p_{22} \text{ E }[F_{V_1}(V_1)]\zeta _2\\&\qquad + p_{22} p_{11} \eta _1 \eta _2 + p_{21} p_{11} \eta _1 \text{ E }[F_{V_2}(V_2)] + p_{22} p_{21}\text{ E }[F_{V_1}(V_1)] \eta _2 \\&\qquad + (p_{22})^2 \text{ E }[F_{V_1}(V_1)]\text{ E }[F_{V_2}(V_2)]. \end{aligned}$$

With the results derived in the proof of Proposition 1, i.e., $\text{ E }[F_{S_i}(S_i)] = \text{ E }[F_{V_i}(V_i)] = 1/2$, $\eta _i = 1/2 + 2 \gamma _i$, and $\zeta _i = 1/2 - 2 \gamma _i$ for each $i = 1,2$, and after some simplification, the above integral can be expressed as

$$\begin{aligned} \int _0^{\infty }\int _0^{\infty }F(t_1, t_2) f(t_1, t_2)\,dt_1 dt_2 = \frac{1}{4} + 8(p_{11}p_{22} - p_{12}p_{21})\gamma _1\gamma _2, \end{aligned}$$

which gives the desired expression (15) for Kendall’s tau.

1.3 Calculation of the coefficient of upper tail dependence

The following gives the calculation of coefficient of upper tail dependence under the GVGCR distribution. Specifically, the upper tail dependence is measured by the coefficient $\lambda$ defined as

$$\begin{aligned} \lambda = \lim _{u \rightarrow 1}\text{ P }(T_2> F^{-1}_2(u)\,\big {|}\, T_1> F^{-1}_1(u)) = \lim _{u \rightarrow 1} \frac{\text{ P }(T_1> F^{-1}_1(u), T_2 > F^{-1}_2(u))}{1-u}, \end{aligned}$$

(20)

where $F_1(\cdot )$ and $F_2(\cdot )$ are the marginal cdfs of $T_1$ and $T_2$, respectively. By the expressions (5) and (6), the numerator in (20) can be written as follows:

$$\begin{aligned}&\text{ P }(T_1> F^{-1}_1(u), T_2 > F^{-1}_2(u)) \\&\quad = G(b_1(F^{-1}_1(u)))G(b_2(F^{-1}_2(u)))\\&\qquad + (p_{12}+p_{22}-p_{11}-p_{21})G(b_1(F^{-1}_1(u)))H_2(F^{-1}_2(u))\\&\qquad + (p_{21}+p_{22}-p_{11}-p_{12})G(b_2(F^{-1}_2(u)))H_1(F^{-1}_1(u))\\&\qquad + (p_{11} +p_{22}-p_{12}-p_{21})H_1(F^{-1}_1(u))H_2(F^{-1}_2(u)), \end{aligned}$$

where $b_i(t) = \alpha _i^{-1}\left( \sqrt{\beta _i/t}-\sqrt{t/\beta _i}\right)$, $i = 1, 2$. Direct applications of L’hopital’s rule give

$$\begin{aligned} \lim _{u \rightarrow 1} \frac{H_i(F^{-1}_i(u))}{G(b_i(F^{-1}_i(u)))} = 1,\quad \lim _{u \rightarrow 1} \frac{G(b_i(F^{-1}_i(u)))}{1- u} = \frac{1}{2 q_i}, \end{aligned}$$

where $q_1 = 1- p_1 = p_{21} + p_{22}$ and $q_2 = 1- p_2 = p_{12} + p_{22}$. Using these and $\lim _{u\rightarrow 1}G(b_i(F^{-1}_i(u))) = \lim _{u\rightarrow 1}H(F^{-1}_i(u)) = 0$, we can easily show

$$\begin{aligned} \lambda= & {} \lim _{u \rightarrow 1} \left\{ \frac{G(b_1(F^{-1}_1(u)))}{1-u}\right\} G(b_2(F^{-1}_2(u))) + (p_{12}+p_{22}-p_{11}-p_{21})\\&\lim _{u \rightarrow 1}\left\{ \frac{G(b_1(F^{-1}_1(u)))}{1-u}\right\} H_2(F^{-1}_2(u))\\&+ (p_{21}+p_{22}-p_{11}-p_{12})\lim _{u \rightarrow 1} \left\{ \frac{G(b_2(F^{-1}_2(u)))}{1-u}\right\} H_1(F^{-1}_1(u))\\&+ (p_{21}+p_{22}-p_{11}-p_{12}) \lim _{u \rightarrow 1} \left\{ \frac{H_1(F^{-1}_1(u))}{G(b_1(F^{-1}_1(u)))}\right\} \left\{ \frac{H_2(F^{-1}_2(u))}{G(b_2(F^{-1}_2(u)))}\right\} \\&\left\{ \frac{G(b_1(F^{-1}_1(u)))}{1-u}\right\} G(b_2(F^{-1}_2(u))) = 0. \end{aligned}$$