Abstract
In this paper, we use a finite-state continuous-time Markov chain with one absorbing state to model an individual’s lifetime. Under this model, the time of death follows a phase-type distribution, and the transient states of the Markov chain are known as phases. We then attempt to provide an answer to the simple question “What is the conditional age distribution of the individual, given its current phase”? We show that the answer depends on how we interpret the question, and in particular, on the phase observation scheme under consideration. We then apply our results to the computation of the age pyramid for the endangered Chatham Island black robin Petroica traversi during the monitoring period 2007–2014.
Similar content being viewed by others
References
Aalen OO (1995) Phase-type distributions in survival analysis. Scand J Stat 22(4):447–463
Butler D, Merton D (1992) The Black Robin: saving the world’s most endangered bird. Oxford University Press, Auckland
Carbonell F, Jimenez JC, Pedroso LM (2008) Computing multiple integrals involving matrix exponentials. J Comput Appl Math 213(1):300–305
Cubrinovska I, Massaro M, Hale ML (2016) Assessment of hybridisation between the endangered Chatham Island black robin (Petroica traversi) and the Chatham Island tomtit (Petroica macrocephala chathamensis). Conserv Genet 17:259–265
Gavrilov LA, Gavrilova NS (1991) The biology of life span: a quantitative approach
Hautphenne S, Latouche G (2012) The Markovian binary tree applied to demography. J Math Biol 64(7):1109–1135
Hautphenne S, Massaro M, Turner K (2017) Fitting Markovian binary trees using global and individual demographic data. Submitted for publication. Preprint available on https://arxiv.org/abs/1702.04281
Kennedy ES, Grueber CE, Duncan RP, Jamieson IG (2014) Severe inbreeding depression and no evidence of purging in an extremely inbred wild species–the Chatham Island black robin. Evolution 68(4):987–995
Lin XS, Liu X (2007) Markov aging process and phase-type law of mortality. N Am Actuar J 11(4):92–109
Massaro M, Sainudiin R, Merton D, Briskie JV, Poole AM, Hale ML (2013) Human-assisted spread of a maladaptive behavior in a critically endangered bird. PloS ONE 8(12):e79066
Massaro M, Stanbury M, Briskie JV (2013) Nest site selection by the endangered black robin increases vulnerability to predation by an invasive bird. Anim Conserv 16(4):404–411
Neuts MF (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. Courier Dover Publications, Mineola
Acknowledgements
The authors are supported by the Australian Research Council Laureate Fellowship FL130100039. The first author has also conducted part of the work under the Discovery Early Career Researcher Award DE150101044. Field-based research on black robins from 2007–2014 was funded by a New Zealand Foundation for Research, Science and Technology fellowship to MM (UOCX0601), and the University of Canterbury, the Brian Mason Scientific and Technical Trust and the Mohamed bin Zayed Species Conservation Fund.
Author information
Authors and Affiliations
Corresponding author
Appendix: Proof of Proposition 4.5
Appendix: Proof of Proposition 4.5
We have
We shall prove using induction on k that
where \({N}_k(x)\) and \({D}_k\) satisfy (14) and (15), respectively. Recall that \(\varvec{t}_{j'}=\gamma \varvec{e}_j\) for any absorbing phase \(1'\le j'\le m'\). When \(k=2\),
Further, by (10) and by conditioning on the value of the absorption times \(B_{\varvec{\alpha }}(j_1')\) and \(B_{{j_{1}}}(j_2')\), we have
where
Using Lemma 4.4, this matrix integral can be evaluated explicitly by defining the \(2m\times 2m\) block-structured matrix
so that
Therefore (36) holds for \(k=2\).
We now assume that (36) holds for k, and we need to prove that is still holds for \(k+1\). We can decompose the conditional age at the \((k+1)\)st observation, \(A_o(j_1,\ldots ,j_{k+1})\), into the sum of the random variables \(B_{\varvec{\alpha }}(j_{1}')\) and \(A_o(j_2,\ldots ,j_{k+1})\), which are conditionally independent given \(j_1\). Note that \(A_o(j_2,\ldots ,j_{k+1})\) is now conditional on the phase process starting with initial distribution vector \(\varvec{e}_{j_1}^\top \) rather than \(\varvec{\alpha }\), and the first observed phase is \(j_2\) rather than \(j_1\), etc. To avoid confusion, we shall use the notation \(\hat{A}_o(j_2,\ldots ,j_{k+1})\) (or \(\hat{A}_o\) for short), \(\hat{N}_k(x),\hat{D}_k,\hat{{\mathcal {B}}}_{1i}(x)\) whenever we will be in that situation.
We use the convolution formula for the sum of the two conditionally independent variables \(B_{\varvec{\alpha }}(j_{1}')\) and \(\hat{A}_o\), together with the conditional distribution of \(B_{\varvec{\alpha }}(j_{1}')\) given in (5) and the induction assumption, to obtain
We immediately see that the denominator of the above expression, \(D_{k+1}:=\varvec{\alpha }(-\mathbf{{T}})^{-1}\varvec{e}_{j_1'}\hat{D}_k\), corresponds to (15) for \(k+1\). It remains to show that the numerator, \(N_{k+1}(x):= \int _0^x {\varvec{\alpha }e^{\mathbf{{T}}u}\varvec{e}_{j_1'}}{\hat{N}_k(x-u)}du\), corresponds to (14) for \(k+1\). Using (14) and letting \(\varvec{r}_{i,k}=(- \mathbf{{T}})^{-1}\varvec{t}_{j_{i+1}'}\prod _{\ell =i}^{k-1} \varvec{e}_{j_{\ell +1}}^\top (-\mathbf{{T}})^{-1}\varvec{t}_{j_{\ell +2}'}\), we have
Using Lemma 4.4 and (11), we can show that
so that by properly redefining the indices we finally obtain what we need.\(\square \)
Rights and permissions
About this article
Cite this article
Hautphenne, S., Massaro, M. & Taylor, P. How old is this bird? The age distribution under some phase sampling schemes. J. Math. Biol. 75, 1319–1347 (2017). https://doi.org/10.1007/s00285-017-1121-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-017-1121-x