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How old is this bird? The age distribution under some phase sampling schemes

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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.

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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.

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Correspondence to Sophie Hautphenne.

Appendix: Proof of Proposition 4.5

Appendix: Proof of Proposition 4.5

We have

$$\begin{aligned}&{\text {P}[A_o\le x\,|\,B_{\varvec{\alpha }}(j_1')<\infty , B_{{j_1}}(j_2')<\infty ,\ldots ,B_{{j_{k-1}}}(j_k')<\infty ]}\\&\quad = \dfrac{\text {P}[A_o\le x, \,B_{\varvec{\alpha }}(j_1')<\infty , \ldots ,B_{{j_{k-1}}}(j_k')<\infty ]}{\text {P}[B_{\varvec{\alpha }}(j_1')<\infty , B_{{j_1}}(j_2')<\infty ,\ldots ,B_{{j_{k-1}}}(j_k')<\infty ]}\\&\quad =:\dfrac{\bar{N}_k(x)}{\bar{D}_k}. \end{aligned}$$

We shall prove using induction on k that

$$\begin{aligned} \dfrac{\bar{N}_k(x)}{\bar{D}_k}= \dfrac{{N}_k(x)}{{D}_k}, \end{aligned}$$
(36)

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 \(k=2\),

$$\begin{aligned} \bar{D}_2= & {} \text {P}[B_{\varvec{\alpha }}(j_1')<\infty , B_{{j_1}}(j_2')<\infty ]\nonumber \\= & {} \varvec{\alpha }(-\mathbf{{T}})^{-1} \varvec{t}_{j_1'}\,\varvec{e}_{j_1}^\top (-\mathbf{{T}})^{-1} \varvec{t}_{j_2'}\nonumber \\= & {} \gamma ^2 D_2. \end{aligned}$$
(37)

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

$$\begin{aligned} \bar{N}_2(x)= & {} \text {P}[(B_{\varvec{\alpha }}(j_1')+B_{{j_{1}}}(j_2'))\le x, \,B_{\varvec{\alpha }}(j_1')<\infty , B_{{j_1}}(j_2')<\infty ]\nonumber \\= & {} \int _{u=0}^x \varvec{\alpha }e^{\mathbf{{T}}u}\varvec{t}_{j_1'}\int _{v=0}^{x-u} \varvec{e}_{j_1}^\top e^{\mathbf{{T}}v}\varvec{t}_{j_2'}\, dv \,du\nonumber \\= & {} \gamma ^2\int _{u=0}^x \varvec{\alpha }e^{\mathbf{{T}}u}\varvec{e}_{j_1'}\varvec{e}_{j_1}^\top (\mathbf{{I}}-e^{\mathbf{{T}}(x-u)})(-\mathbf{{T}})^{-1}\varvec{e}_{j_2'} \,du\nonumber \\= & {} \gamma ^2\{\varvec{\alpha }(\mathbf{{I}}-e^{\mathbf{{T}}x})(-\mathbf{{T}})^{-1}\varvec{e}_{j_1'}\varvec{e}_{j_1}^\top (-\mathbf{{T}})^{-1}\varvec{e}_{j_2'}-\varvec{\alpha }{\mathcal {B}}_{12}(x)(-\mathbf{{T}})^{-1}\varvec{e}_{j_2'}\}\qquad \nonumber \\= & {} \gamma ^2\,N_2(x), \end{aligned}$$
(38)

where

$$\begin{aligned} {\mathcal {B}}_{12}(x)= & {} \int _{u=0}^x e^{\mathbf{{T}}u} \varvec{e}_{j_1'}\varvec{e}_{j_1}^\top e^{\mathbf{{T}}(x-u)}du. \end{aligned}$$

Using Lemma 4.4, this matrix integral can be evaluated explicitly by defining the \(2m\times 2m\) block-structured matrix

$$\begin{aligned} \mathbf{{A}}^{(2)}=\left[ \begin{array}{cc} \mathbf{{T}} &{}\quad \varvec{e}_{j_1'}\varvec{e}_{j_1}^\top \\ \mathbf{{0}} &{}\quad \mathbf{{T}}\end{array}\right] , \end{aligned}$$

so that

$$\begin{aligned} {\mathcal {B}}_{12}(x)=(\varvec{f}_{2,1}^\top \otimes \mathbf{{I}}_m) e^{\mathbf{{A}}^{(2)}x}(\varvec{f}_{2,2}\otimes \mathbf{{I}}_m). \end{aligned}$$

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

$$\begin{aligned}&{\text {P}[(B_{\varvec{\alpha }}(j_{1'})+\hat{A}_o)\le x\,|\,B_{\varvec{\alpha }}(j_1')<\infty , B_{{j_1}}(j_2')<\infty ,\ldots ,B_{{j_{k}}}(j_{k+1}')<\infty ]}\\&\quad =\int _0^x \dfrac{\varvec{\alpha }e^{ \mathbf{{T}}u}\varvec{t}_{j_1'}}{\varvec{\alpha }(-\mathbf{{T}})^{-1}\varvec{t}_{j_1'}}\text {P}[\hat{A}_o\le x-u|B_{{j_1}}(j_2')<\infty ,\ldots ,B_{{j_{k}}}(j_{k+1}')<\infty ]\,du\\&\quad = \int _0^x \dfrac{\varvec{\alpha }e^{\mathbf{{T}}u}\varvec{e}_{j_1'}}{\varvec{\alpha }(-\mathbf{{T}})^{-1}\varvec{e}_{j_1'}}\dfrac{\hat{N}_k(x-u)}{\hat{D}_k}du. \end{aligned}$$

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

$$\begin{aligned}&{\int _0^x {\varvec{\alpha }e^{\mathbf{{T}}u}\varvec{t}_{j_1'}}{\hat{N}_k(x-u)}\,du}\\&\quad =\int _0^x {\varvec{\alpha }e^{\mathbf{{T}}u}\varvec{t}_{j_1'}}\,\big \{\varvec{e}_{j_1}^\top (\mathbf{{I}}-e^{\mathbf{{T}}(x-u)})\varvec{r}_{1,k}-\varvec{e}_{j_1}^\top \sum _{i=2}^k\hat{{\mathcal {B}}}_{1i}(x-u)\varvec{r}_{i,k}\big \}\, du\\&\quad =\varvec{\alpha }(\mathbf{{I}}-e^{\mathbf{{T}}x})(-\mathbf{{T}})^{-1}\varvec{t}_{j_1'} \varvec{e}_{j_1}^\top \varvec{r}_{1,k} -\varvec{\alpha }{\mathcal {B}}_{12}(x) \varvec{r}_{1,k}\\&\quad \quad -\varvec{\alpha }\sum _{i=2}^k\int _0^x e^{\mathbf{{T}}u} \varvec{t}_{j_1'} \varvec{e}_{j_1}^\top \hat{{\mathcal {B}}}_{1,i}(x-u) \,du \,\varvec{r}_{i,k}. \end{aligned}$$

Using Lemma 4.4 and (11), we can show that

$$\begin{aligned} \int _0^x e^{\mathbf{{T}}u} \varvec{t}_{j_1'} \varvec{e}_{j_1}^\top \hat{{\mathcal {B}}}_{1,i}(x-u) \,du = {\mathcal {B}}_{1,i+1}(x), \end{aligned}$$

so that by properly redefining the indices we finally obtain what we need.\(\square \)

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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

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