Bivariate Sarmanov Phase-Type Distributions for Joint Lifetimes Modeling

Abstract

In this paper, we are interested in the dependence between lifetimes based on a joint survival model. This model is built using the bivariate Sarmanov distribution with Phase-Type marginal distributions. Capitalizing on these two classes of distributions’ mathematical properties, we drive some useful closed-form expressions of distributions and quantities of interest in the context of multiple-life insurance contracts. The dependence structure that we consider in this paper is based on a general form of kernel function for the Bivariate Sarmanov distribution. The introduction of this new kernel function allows us to improve the attainable correlation range.

Appendix

1.1 Substochastic Matrix

Definition 5.1

A substochastic matrix is a square matrix with nonnegative entries so that every row adds up to at most 1

1.2 Matrix Exponential

Definition 5.2

Let A be a square matrix of order n, then we call matrix exponential, denoted by e^A, the matrix

$$ e^{A} = \sum\limits_{k=0}^{\infty} \frac{1}{k!}A^{k}, $$

with $e^{0_{n}} = I_{n}$, where 0_n is the zero matrix of order n.

It is well known that f(x) = e^x is the only function to have the property that f(x + y) = f(x)f(y), i.e. e^xe^y = e^x+y, where $x,y \in \mathbb {R}$. However, this is not true for matrix exponential.

Proposition 5.1

Let A and B be square matrices of order n. Then,

$$ e^{tA}e^{tB} = e^{t(A+B)}, t \in \mathbb{R}, $$

if ABB = BA.

The derivative of matrix exponential function is given in the following proposition.

Proposition 5.1

The derivative of a matrix exponential function is given by

$$ \frac{d}{dx}e^{Ax}=Ae^{Ax} = e^{Ax}A. $$

1.3 Kronecker Product and Kronecker Sum

The Kronecker product and the Kronecker sum are defined as follows

Definition 5.3

Let A = (a_ij) and B be (n × m) and (l × k) real matrices, respectively. Then the Kronecker product $A \otimes B \in \mathbb {R}^{nl*mk}$ is the partitioned matrix

$$ A \otimes B= \left( \begin{aligned} a_{11} B & a_{12} B & {\dots} & a_{1m} B \\ a_{21} B & a_{22} B & {\dots} & a_{2m} B \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ a_{n1} B & a_{n2} B & {\dots} & a_{nm} B \end{aligned}\right). $$

Definition 5.4

Let A and B be square matrices of orders n and m respectively. Then the Kronecker sumA ⊕ B is a square matrix of order nm and is given by

$$ A \oplus B = A \otimes I_{m} + I_{n} \otimes B. $$

Definition 5.5

Consider a matrix A. For k ≥ 1, the k th Kronecker power, $A^{\otimes _{k+1}}$, and the k th Kronecker sum, $A^{\oplus _{k+1}}$, are defined inductively by

• $A^{\otimes _{1}}=A$ and $A^{\otimes _{k}}= A\otimes A^{\otimes _{(k-1)}}$ for $k=2,3,\dots$.

• $A^{\oplus _{1}}=A$ and $A^{\oplus _{k}}= A\oplus A^{\oplus _{(k-1)}}$ for $k=2,3,\dots$.

1.4 Some Properties of Kronecker Product and Kronecker Sum

Proposition 5.3

Define Let M_l,j denote the space of i × j real (or complex) matrices. Let A ∈ M_m,n, B ∈ M_p,q, C ∈ M_n,k, and D ∈ M_q,r. Then

$$ (A \otimes B)(C \otimes D)=(A C) \otimes(B D). $$

Proposition 5.4

Define Let M_l the space of square real (or complex) matrices. Consider two matrices A ∈ M_p and B ∈ M_q

(i):

Assume that μ is an eigenvalue for A with corresponding eigenvector x, and ξ is an eigenvalue for B with corresponding eigenvector y. Then μ + ξ is an eigenvalue of A ⊕ B with corresponding eigenvector y ⊗ x.

(ii):

Any eigenvalue of A ⊕ B arises as such a sum of eigenvalues of A and B

We refer the readers to Horn and Johnson (1991) for more details on Matrix Mathematics and proofs of the previous propositions.