Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T08:08:40.719Z Has data issue: false hasContentIssue false

Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

Published online by Cambridge University Press:  17 March 2021

Krzysztof Bartoszek*
Affiliation:
Linköping University
Torkel Erhardsson*
Affiliation:
Linköping University
*
*Postal address: Department of Computer and Information Science, Linköping University, SE-581 83 Linköping, Sweden.
**Postal address: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden. Email address: [email protected]

Abstract

Explicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

Type
Original Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adamczak, R. and Miłoś, P. (2015). CLT for Ornstein–Uhlenbeck branching particle system. Electron. J. Prob. 20, 135.CrossRefGoogle Scholar
Ané, C., Ho, L. S. T. and Roch, S. (2017). Phase transition on the convergence rate of parameter estimation under an Ornstein–Uhlenbeck diffusion on a tree. J. Math. Biol. 74, 355385.CrossRefGoogle Scholar
Barbour, A. D. and Chen, L. H. Y. (eds) (2005). An Introduction to Stein’s Method (Lecture Notes Ser., Vol. 4, Inst. Math. Sci., National University of Singapore). World Scientific Publishing, Singapore.CrossRefGoogle Scholar
Barbour, A. D. and Chen, L. H. Y. (eds) (2005). Stein’s Method and Applications (Lecture Notes Ser., Vol. 5, Inst. Math. Sci., National University of Singapore). World Scientific Publishing, Singapore.CrossRefGoogle Scholar
Barbour, A. D. and Hall, P. (1984). Stein’s method and the Berry–Esseen theorem. Austral. J. Statist. 26, 815.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Science Publications.Google Scholar
Bartoszek, K. (2014). Quantifying the effects of anagenetic and cladogenetic evolution. Math. Biosci. 254, 4257.CrossRefGoogle ScholarPubMed
Bartoszek, K. (2020). A central limit theorem for punctuated equilibrium. Stoch. Models 36, 473517.CrossRefGoogle Scholar
Bartoszek, K., Pienaar, J., Mostad, P., Andersson, S. and Hansen, T. F. (2012). A phylogenetic comparative method for studying multivariate adaptation. J. Theoret. Biol. 314, 204215.CrossRefGoogle ScholarPubMed
Bartoszek, K. and Sagitov, S. (2015). Phylogenetic confidence intervals for the optimal trait value. J. Appl. Prob. 52, 11151132.CrossRefGoogle Scholar
Bokma, F. (2002). Detection of punctuated equilibrium from molecular phylogenies. J. Evol. Biol. 15, 10481056.CrossRefGoogle Scholar
Chen, L. H. Y. and Shao, Q.-M. (2005). Stein’s method for normal approximation. In An Introduction to Stein’s Method, eds. Barbour, A. D. and Chen, L. H. Y., World Scientific Publishing, Singapore, pp. 159.Google Scholar
Gernhard, T. (2008). The conditioned reconstructed process. J. Theoret. Biol. 253, 769778.CrossRefGoogle ScholarPubMed
Hansen, T. F. (1997). Stabilizing selection and the comparative analysis of adaptation. Evolution 51, 13411351.CrossRefGoogle ScholarPubMed
Müller, A. (1997). Integral probability metrics and their generating classes of functions. Adv. Appl. Prob. 29, 429443.CrossRefGoogle Scholar
Petersen, K. (1983). Ergodic Theory. Cambridge University Press.CrossRefGoogle Scholar
R Core Team (2017). R: a Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna. Available at https://www.R-project.org.Google Scholar
Ren, Y. X., Song, R. and Zhang, R. (2014). Central limit theorems for supercritical branching Markov processes. J. Funct. Anal. 266, 17161756.CrossRefGoogle Scholar
Stadler, T. (2011). Simulating trees with a fixed number of extant species. Syst. Biol. 60, 676684.CrossRefGoogle ScholarPubMed
Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. 6th Berkeley Symp. Math. Statist. Prob., Vol. 2, University of California Press, Berkeley, pp. 583602.Google Scholar