Parameter estimation for fractional birth and fractional death processes

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Abstract

The fractional birth and the fractional death processes are more desirable in practice than their classical counterparts as they naturally provide greater flexibility in modeling growing and decreasing systems. In this paper, we propose formal parameter estimation procedures for the fractional Yule, the fractional linear death, and the fractional sublinear death processes. The methods use all available data possible, are computationally simple and asymptotically unbiased. The procedures exploited the natural structure of the random inter-birth and inter-death times that are known to be independent but are not identically distributed. We also showed how these methods can be applied to certain models with more general birth and death rates. The computational tests showed favorable results for our proposed methods even with relatively small sample sizes. The proposed methods are also illustrated using the branching times of the plethodontid salamanders data of \cite{hal79}.

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Extinction rates can be estimated from molecular phylogenies.

Richard Harvey, S Nee, Bennett May … (1994)

Molecular phylogenies can be used to reject null models of the way we think evolution occurred, including patterns of lineage extinction. They can also be used to provide maximum likelihood estimates of parameters associated with lineage birth and death rates. We illustrate: (i) how molecular phylogenies provide information about the extent to which particular clades are likely to be under threat from extinction; (ii) how cursory analyses of molecular phylogenies can lead to incorrect conclusions about the evolutionary processes that have been at work; and (iii) how different evolutionary processes leave distinctive marks on the structure of reconstructed phylogenies.

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Fractional pure birth processes

Enzo Orsingher, Federico Polito (2010)

We consider a fractional version of the classical nonlinear birth process of which the Yule--Furry model is a particular case. Fractionality is obtained by replacing the first order time derivative in the difference-differential equations which govern the probability law of the process with the Dzherbashyan--Caputo fractional derivative. We derive the probability distribution of the number \(\mathcal{N}_{\nu}(t)\) of individuals at an arbitrary time \(t\). We also present an interesting representation for the number of individuals at time \(t\), in the form of the subordination relation \(\mathcal{N}_{\nu}(t)=\mathcal{N}(T_{2\nu}(t))\), where \(\mathcal{N}(t)\) is the classical generalized birth process and \(T_{2\nu}(t)\) is a random time whose distribution is related to the fractional diffusion equation. The fractional linear birth process is examined in detail in Section 3 and various forms of its distribution are given and discussed.