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Mortality modeling and regression with matrix distributions. (English) Zbl 1515.62096

The authors study the applicability of inhomogeneous phase-type distributions for the modeling of human mortality.
Let \((J(t))_{t \geq 0}\) denote a time-inhomogeneous Markov jump process on the state-space \(\{1,\dots , p + 1\}\), where states \(\{1,\dots, p\}\) are transient and state \(p + 1\) is absorbing, with intensity matrix \[ \Lambda(t) = \left(\begin{array}{cc}\mathbf{T}(t) & \mathbf{t}(t) \\ 0 & 0\end{array}\right) \in \mathbb{R}^{(p+1)\times(p+1)} ~ (t \geq 0) \] where the \(p \times p\) matrix function \(\mathbf{T}\) is of the form \(\mathbf{T}(t) = \lambda(t) \cdot T\) with non-negative real function \(\lambda\), and \(\mathbf{t}(t) = -\mathbf{T}(t)\cdot (1,\dots,1)^{\top}\). Let \(\mathbf{\pi} = (\pi_{1},\dots,\pi_{p},0)\) denote the distribution of \(J(0)\). Then \(\tau = \inf\{t \geq 0\colon J(t) = p+1\}\) has inhomogeneous phase-type distribution \(IPH(\mathbf{\pi},\lambda, T)\).
For estimating the model, the authors assume \(\lambda\) is a parametric function which depends on the predictor variables \(\mathbf{X}\) proportionally, that is \(\lambda(t|\mathbf{X}) = \ell(t)\cdot m(\mathbf{X})\). Various specifications of the model are estimated using an expectation maximization algorithm for
Danish female mortality data;
US and Japanese female mortality data using the country as predictor variable;
Danish female mortality data using the time of observation as predictor variable;
veterans’ lung cancer mortality data.
The ability of the model to fit empirical data is contrasted to the Lee-Carter model in [R. D. Lee and L. R. Carter, J. Am. Stat. Assoc. 87, No. 419, 659–675 (1992; Zbl 1351.62186)]. The authors note that the IPH model does not and is not expected to outperform any individual survival regression model specification in the literature, but it has significantly less parameters and enjoys favorable estimation and closed-form mathematical formulas for the resulting lifetime distribution, which is important to build realistic insurance products which are to be analyzed explicitly.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62N02 Estimation in survival analysis and censored data
60J28 Applications of continuous-time Markov processes on discrete state spaces
91D20 Mathematical geography and demography
91G05 Actuarial mathematics

Citations:

Zbl 1351.62186

Software:

EMpht; matrixdist
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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