Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T11:11:04.699Z Has data issue: false hasContentIssue false
  • English
  • Français

Reward distributions associated with some block tridiagonal transition matrices with applications to identity by descent

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Valeri T. Stefanov  and
Frank Ball
Valeri T. Stefanov*
Affiliation:
The University of Western Australia
Frank Ball*
Affiliation:
University of Nottingham
*
∗ Postal address: School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia. Email address: stefanov@maths.uwa.edu.au
∗∗ Postal address: School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK. Email address: fgb@maths.nott.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Markov and semi-Markov processes with block tridiagonal transition matrices for their embedded discrete-time Markov chains are underlying stochastic models in many applied probability problems. In particular, identity-by-descent (IBD) problems for uncle-type and cousin-type relationships fall into this class. More specifically, the exact distributions of relevant IBD statistics for two individuals in either an uncle-type or cousin-type relationship are of interest. Such statistics are the amount of genome shared IBD by the two related individuals on a chromosomal segment and the number of IBD pieces on such a segment. These lead to special reward distributions associated with block tridiagonal transition matrices for continuous-time Markov chains. A method is provided for calculating explicit, closed-form expressions for Laplace transforms of general reward functions for such Markov chains. Some calculation results on the cumulative probabilities of relevant IBD statistics via a numerical inversion of the Laplace transforms are also provided for uncle/nephew and first-cousin relationships.

Keywords

Semi-Markov processstopping timereward functionLaplace transformgenomic continuum model

MSC classification

Primary: 60E10: Characteristic functions; other transforms
Secondary: 92D10: Genetics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 523 - 545
Copyright
Copyright © Applied Probability Trust 2009 

References

Abate, J. and Whitt, W. (2006). A unified framework for numerically inverting Laplace transforms. INFORMS J. Computing 18, 408–421.Google Scholar
Abate, J., Choudhury, G. L. and Whitt, W. (1998). Numerical inversion of multidimensional Laplace transforms by the Laguerre method. Performance Evaluation 31, 229–243.Google Scholar
Asmussen, S. and Pihlsgård, M. (2004). Transient properties of many-server queues and related QBDs. Queueing Systems 4, 249–270.Google Scholar
Ball, F. and Stefanov, V. T. (2005). Evaluation of identity-by-descent probabilities for half-sibs on continuous genome. Math. Biosci. 196, 215–225.Google Scholar
Ball, F., Milne, R. K. and Yeo, G. F. (1994). Continuous-time Markov chains in a random environment, with applications to ion channel modelling. Adv. Appl. Prob. 26, 919–946.CrossRefGoogle Scholar
Bickeboller, H. and Thompson, E. A. (1996a). Distribution of genome shared IBD by half-sibs: approximation by the Poisson clumping heuristic. Theoret. Pop. Biol. 50, 66–90.Google Scholar
Bickeboller, H. and Thompson, E. A. (1996b). The probability distribution of the amount of an individual's genome surviving to the following generation. Genetics 143, 1043–1049.Google Scholar
Browning, S. (2000). A Monte Carlo approach to calculating probabilities for continuous identity by descent data. J. Appl. Prob. 37, 850–864.Google Scholar
Cannings, C. (2003). The identity by descent process along the chromosome. Human Heredity 56, 126–130.Google Scholar
Choudhury, G. L., Lucantoni, D. M., and Whitt, W. (1994). Multidimensional transform inversion with applications to the transient M/G/1 queue. Ann. Appl. Prob. 4, 719–740.CrossRefGoogle Scholar
Donnelly, K. P. (1983). The probability that related individuals share some section of genome identical by descent. Theoret. Pop. Biol. 23, 34–63.Google Scholar
Gaver, D. P., Jacobs, P. A. and Latouche, G. (1984). Finite birth-and-death models in randomly changing environments. Adv. Appl. Prob. 16, 715–731.Google Scholar
Iosifescu, M. (1980). Finite Markov Processes and Their Applications. John Wiley, Chichester.Google Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Lange, K. (2002). Mathematical and Statistical Methods for Genetic Analysis, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Li, Q.-L. and Cao, J. (2004). Two types of RG-factorizations of quasi-birth-and-death processes and their applications to stochastic integral functionals. Stoch. Models 20, 299–340.Google Scholar
Lucantoni, D. M., Choudhury, G. L. and Whitt, W. (1994). The transient BMAP/G/1 Queue. Commun. Statist. Stoch. Models 10, 145–182.Google Scholar
Stefanov, V. T. (2000a). Distribution of genome shared IBD by two individuals in grandparent-type relationship. Genetics 156, 1403–1410.Google Scholar
Stefanov, V. T. (2000b). On some waiting time problems. J. Appl. Prob. 37, 756–764.Google Scholar
Stefanov, V. T. (2002). Statistics on continuous IBD data: exact distribution evaluation for a pair of full(half)-sibs and a pair of a (great-)grandchild with a (great-)grandparent. BMC Genetics 3, No. 7.Google Scholar
Stefanov, V. T. (2004). Distribution of the amount of genetic material from a chromosomal segment surviving to the following generation. J. Appl. Prob. 41, 345–354. (Correction: 41 (2004), 1248–1249)CrossRefGoogle Scholar
Stefanov, V. T. (2006). Exact distributions for reward functions on semi-Markov and Markov additive processes. J. Appl. Prob. 43, 1053–1065.Google Scholar
Thompson, E. A. (2000). Statistical Inference From Genetic Data on Pedigrees (NSF-CBMS Regional Conf. Ser. Prob. Statist. 6). American Statistical Association, Alexandria, VA.Google Scholar
Walters, K. and Cannings, C. (2005). The probability density of the total IBD length over a single autosome in unilineal relationships. Theoret. Pop. Biol. 68, 55–63.Google Scholar