The Behaviour of an Estimator for a Simple Birth and Death Process

Darwin, J. H.

doi:10.1093/biomet/43.1-2.23

Cited by 27 publications

(17 citation statements)

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“…Speciation events give birth to new species, and extinction events result in the death of species. This representation is practical for modelling the diversification of lineages since, with some further assumptions, this agrees with the birth-and-death processes which have been extensively studied in the past (Kendall 1948a(Kendall , b, 1949Darwin, 1956;Keiding, 1975). Nee et al (1994b) proposed a method to estimate both speciation and extinction rates of a lineage using the reconstructed phylogenetic relationships of the living species, for instance using molecular data.…”

Section: Introductionmentioning

confidence: 52%

Can extinction rates be estimated without fossils?

Paradis

2004

Journal of Theoretical Biology

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There is considerable interest in the possibility of using molecular phylogenies to estimate extinction rates. The present study aims at assessing the statistical performance of the birth-death model fitting approach to estimate speciation and extinction rates by comparison to the approach considering fossil data. A simulation-based approach was used. The diversification of a large number of lineages was simulated under a wide range of speciation and extinction rate values. The estimators obtained with fossils performed better than those without fossils. In the absence of fossils (e.g. with a molecular phylogeny), the speciation rate was correctly estimated in a wide range of situations; the bias of the corresponding estimator was close to zero for the largest trees. However, this estimator was substantially biased when the simulated extinction rate was high. On the other hand the estimator of extinction rate was biased in a wide range of situations. Surprisingly, this bias was lesser with medium-sized trees. Some recommendations for interpreting results from a diversification analysis are given.

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Section: Introductionmentioning

confidence: 52%

Can extinction rates be estimated without fossils?

Paradis

2004

Journal of Theoretical Biology

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“…More realistically, if there is an intrinsic death rate (or, for species and higher taxa, extinction rate) μ , then one has a time-homogeneous birth and death process defined by the differential equation, e.g., Kendall (1948), Darwin (1954), and Keiding (1975),

\begin{matrix} right \frac{d p_{n}}{d t} & left = λ (n - 1) p_{n - 1} - (λ + μ) n p_{n} + μ (n + 1) p_{n + 1} for n > 0, \\ right \frac{d p_{0}}{d t} & left = μ p_{1} . \end{matrix}

The solutions for λ ≠ μ given p 1 (0)=1 are given by

\begin{matrix} right p_{n} (t) & left = \frac{{(λ - μ)}^{2} \exp [- (λ - μ) t]}{{(λ - μ \exp [- (λ - μ) t])}^{2}} {true(\frac{λ - λ \exp [- (λ - μ) t]}{λ - μ \exp [- (λ - μ) t]} true)}^{n - 1} for n > 0, \\ right p_{0} (t) & left = \frac{μ - μ \exp [- (λ - μ) t]}{λ - μ \exp [- (λ - μ) t]} . \end{matrix}

In the special case where λ = μ , the solutions simplify to:

\begin{matrix} right p_{n} (t) & left = \frac{{(λ t)}^{n - 1}}{{(λ t + 1)}^{n + 1}} for n > 0, \\ right p_{0} (t) & left = \frac{λ t}{λ t + 1} . \end{matrix}

...…”

Section: Introductionmentioning

confidence: 99%

The Distribution of Family Sizes Under a Time-Homogeneous Birth and Death Process

Moschopoulos¹,

Shpak²

2010

Communications in Statistics - Theory and Methods

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The number of extant individuals within a lineage, as exemplified by counts of species numbers across genera in a higher taxonomic category, is known to be a highly skewed distribution. Because the sublineages (such as genera in a clade) themselves follow a random birth process, deriving the distribution of lineage sizes involves averaging the solutions to a birth and death process over the distribution of time intervals separating the origin of the lineages. In this article, we show that the resulting distributions can be represented by hypergeometric functions of the second kind. We also provide approximations of these distributions up to the second order, and compare these results to the asymptotic distributions and numerical approximations used in previous studies. For two limiting cases, one with a relatively high rate of lineage origin, one with a low rate, the cumulative probability densities and percentiles are compared to show that the approximations are robust over a wide range of parameters. It is proposed that the probability distributions of lineage size may have a number of relevant applications to biological problems such as the coalescence of genetic lineages and in predicting the number of species in living and extinct higher taxa, as these systems are special instances of the underlying process analyzed in this article.

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“…The maximum likelihood estimator can be constructed using the probability distribution as given by Darwin (1956). Although the likelihood function takes an explicit form, we found that the calculation of the maximum likelihood estimator became numerically challenging as the size of the process got large, thereby warranting alternative approaches to be investigated.…”

Section: Examples and Simulationsmentioning

confidence: 99%

“…Kendall (1949), Darwin (1956), and Keiding (1974, 1975) investigated maximum likelihood estimation for the pure birth and the linear birth-and-death processes. Oh, Severo and Slivka (1991) proposed approximate maximum likelihood estimators for a class of pure birth processes.…”

Section: Introductionmentioning

confidence: 99%

Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

Chen

Hyrien

2011

Journal of Statistical Planning and Inference

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This article deals with quasi- and pseudo-likelihood estimation in a class of continuous-time multi-type Markov branching processes observed at discrete points in time. “Conventional” and conditional estimation are discussed for both approaches. We compare their properties and identify situations where they lead to asymptotically equivalent estimators. Both approaches possess robustness properties, and coincide with maximum likelihood estimation in some cases. Quasi-likelihood functions involving only linear combinations of the data may be unable to estimate all model parameters. Remedial measures exist, including the resort either to non-linear functions of the data or to conditioning the moments on appropriate sigma-algebras. The method of pseudo-likelihood may also resolve this issue. We investigate the properties of these approaches in three examples: the pure birth process, the linear birth-and-death process, and a two-type process that generalizes the previous two examples. Simulations studies are conducted to evaluate performance in finite samples.

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The Behaviour of an Estimator for a Simple Birth and Death Process

Cited by 27 publications

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Can extinction rates be estimated without fossils?

Can extinction rates be estimated without fossils?

The Distribution of Family Sizes Under a Time-Homogeneous Birth and Death Process

Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

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