Abate,, J., & Whitt,, W. (1999). Computing Laplace transforms for numerical inversion via continued fractions. INFORMS Journal on Computing, 11(4), 394–405.

Andersson,, H., & Britton,, T. (2000). Stochastic epidemic models and their statistical analysis. New York, NY: Springer.

Anscombe,, F. J. (1953). Sequential estimation. Journal of the Royal Statistical Society: Series B, 15(1), 1–29.

Athreya,, K. B., & Ney,, P. E. (2004). Branching processes. New York: Dover Publications.

Bailey,, N. T. (1975). %22The mathematical theory of infectious diseases and its applications%22 New York: Hafner Press.

Bailey,, N. T. J. (1964). The elements of stochastic processes with applications to the natural sciences. New York, NY: Wiley.

Ball,, F. (1986). A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Advances in Applied Probability, 18, 289–310.

Bankier,, J. D., & Leighton,, W. (1942). Numerical continued fractions. American Journal of Mathematics, 64(1), 653–668.

Bladt,, M., & Sorensen,, M. (2005). Statistical inference for discretely observed Markov jump processes. Journal of the Royal Statistical Society: Series B, 67(3), 395–410.

Blanch,, G. (1964). Numerical evaluation of continued fractions. SIAM Review, 6(4), 383–421.

Bordes,, G., & Roehner,, B. (1983). Application of Stieltjes theory for S‐fractions to birth and death processes. Advances in Applied Probability, 15(3), 507–530.

Cai,, H., & Luo,, X. (1994). Stochastic control of an epidemic process. International Journal of Systems Science, 25(4), 821–828.

Clancy,, D. (1999). Optimal intervention for epidemic models with general infection and removal rate functions. Journal of Mathematical Biology, 39(4), 309–331.

Craviotto,, C., Jones,, W. B., & Thron,, W. J. (1993). A survey of truncation error analysis for Padé and continued fraction approximants. Acta Appl. Math., 33, 211–272.

Crawford,, F. W., Minin,, V. N., & Suchard,, M. A. (2014). Estimation for general birth‐death processes. Journal of the American Statistical Association, 109(506), 730–747.

Crawford,, F. W., Stutz,, T. C., & Lange,, K. (2016). Coupling bounds for approximating birth–death processes by truncation. Statistics %26 Probability Letters, 109, 30–38.

Crawford,, F. W., & Suchard,, M. A. (2012). Transition probabilities for general birth‐death processes with applications in ecology, genetics, and evolution. Journal of Mathematical Biology, 65, 553–580.

Crawford,, F. W., Weiss,, R. E., & Suchard,, M. A. (2015). Sex, lies, and self‐reported counts: Bayesian mixture models for longitudinal heaped count data via birth‐death processes. Annals of Applied Statistics, 9(2), 572–596.

Crawford,, F. W., & Zelterman,, D. (2015). Markov counting models for correlated binary responses. Biostatistics, 16(3), 427–440.

Cuyt,, A., Petersen,, V., Verdonk,, B., Waadeland,, H., & Jones,, W. (2008). Handbook of continued fractions for special functions. Berlin, Germany: Springer.

Darwin,, J. H. (1956). The behaviour of an estimator for a simple birth and death process. Biometrika, 43(1), 23–31.

Dauxois,, J.‐Y. (2004). Bayesian inference for linear growth birth and death processes. Journal of Statistical Planning and Inference, 121(1), 1–19.

Dempster,, A. P., Laird,, N. M., & Rubin,, D. B. (1977). Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society: Series B, 39(1), 1–38.

Demuth,, J. P., De Bie,, T., Stajich,, J. E., Cristianini,, N., & Hahn,, M. W. (2006). The evolution of mammalian gene families. PLoS ONE, 1(1), e85.

Dobson,, A. J. (2001). An introduction to generalized linear models. Boca Raton, FL: CRC Press.

Doss,, C. R., Suchard,, M. A., Holmes,, I., Kato‐Maeda,, M., & Minin,, V. N. (2013). Fitting birth‐death processes to panel data with applications to bacterial DNA fingerprinting. Annals of Applied Statistics, 7(4), 2315–2335.

Faddy,, M., & Bosch,, R. (2001). Likelihood‐based modeling and analysis of data underdispersed relative to the Poisson distribution. Biometrics, 57(2), 620–624.

Faddy,, M. J. (1997). Extended poisson process modelling and analysis of count data. Biometrical Journal, 39(4), 431–440.

Feller,, W. (1971). An introduction to probability theory and its applications. New York, NY: Wiley.

Flajolet,, P., & Guillemin,, F. (2000). The formal theory of birth‐and‐death processes, lattice path combinatorics and continued fractions. Advances in Applied Probability, 32(3), 750–778.

Fudenberg,, D., Imhof,, L., Nowak,, M. A., & Taylor,, C. (2004). *Stochastic evolution as a generalized Moran process*. Unpublished manuscript.

Gani,, J., & Jerwood,, D. (1972). The cost of a general stochastic epidemic. Journal of Applied Probability, 9(2), 257–269.

Gani,, J., & McNeil,, D. R. (1971). Joint distributions of random variables and their integrals for certain birth‐death and diffusion processes. Advances in Applied Probability, 3(2), 339–352.

Gani,, J., & Swift,, R. (2008). A simple approach to the integrals under three stochastic processes. Journal of Statistical Theory and Practice, 2(4), 559–568.

Gaver,, D. (1969). Highway delays resulting from flow‐stopping incidents. Journal of Applied Probability, 6(1), 137–153.

Guillemin,, F., & Pinchon,, D. (1998). Continued fraction analysis of the duration of an excursion in an *M/M/∞* system. Journal of Applied Probability, 35(1), 165–183.

Guillemin,, F., & Pinchon,, D. (1999). Excursions of birth and death processes, orthogonal polynomials, and continued fractions. Journal of Applied Probability, 36(3), 752–770.

Guo,, X., & Hernández‐Lerma,, O. (2009). Continuous‐time Markov decision processes. Berlin: Springer.

Guttorp,, P. (1991). Statistical inference for branching processes. New York: Wiley.

Hernández‐Suárez,, C., & Castillo‐Chavez,, C. (1999). A basic result on the integral for birth‐death Markov processes. Mathematical Biosciences, 161(1), 95–104.

Ho,, L. S. T., Xu,, J., Crawford,, F. W., Minin,, V. N., & Suchard,, M. A. (in press). Birth/birth‐death processes and their computable transition probabilities with biological applications. Journal of Mathematical Biology.

Hobolth,, A., & Jensen,, J. L. (2005). Statistical inference in evolutionary models of DNA sequences via the EM algorithm. Statistical Applications in Genetics and Molecular Biology, 4(1).

Hobolth,, A., & Stone,, E. A. (2009). Simulation from endpoint‐conditioned, continuous‐time Markov chains on a finite state space, with applications to molecular evolution. Annals of Applied Statistics, 3(3), 1024–1231.

Holmes,, I., & Bruno,, W. J. (2001, September). Evolutionary HMMs: A Bayesian approach to multiple alignment. Bioinformatics, 17(9), 803–820.

Holmes,, I., & Rubin,, G. (2002). An expectation maximization algorithm for training hidden substitution models. Journal of Molecular Biology, 317(5), 753–764.

Ismail,, M. E. H., Letessier,, J., & Valent,, G. (1988). Linear birth and death models and associated Laguerre and Meixner polynomials. Journal of Approximation Theory, 55(3), 337–348.

Jahnke,, T., & Huisinga,, W. (2007). Solving the chemical master equation for monomolecular reaction systems analytically. Journal of Mathematical Biology, 54(1), 1–26.

Jerwood,, D. (1970). A note on the cost of the simple epidemic. Journal of Applied Probability, 7(2), 440–443.

Kaplan,, N. (1974). Limit theorems for the integral of a population process with immigration. Stochastic Processes and their Applications, 2(3), 281–294.

Karev,, G. P., Wolf,, Y. I., Rzhetsky,, A. Y., Berezovskaya,, F. S., & Koonin,, E. V. (2002). Birth and death of protein domains: A simple model of evolution explains power law behavior. BMC Evolutionary Biology, 2(1), 18.

Karlin,, S., & McGregor,, J. (1957a). The classification of birth and death processes. Transactions of the American Mathematical Society, 86(2), 366–400.

Karlin,, S., & McGregor,, J. (1958a). Linear growth, birth and death processes. Journal of Mathematics and Mechanics, 7(4), 643–662.

Karlin,, S., & McGregor,, J. L. (1958b). Many server queueing processes with poisson input and exponential service times. Pacific Journal of Mathematics, 8(1), 87–118.

Karlin,, S., & McGregor,, J. L. (1957b). The differential equations of birth‐and‐death processes, and the Stieltjes moment problem. Transactions of the American Mathematical Society, 85(2), 489–546.

Karlin,, S., & Taylor,, H. M. (1975). A first course in stochastic processes. New York: Academic Press.

Keiding,, N. (1975). Maximum likelihood estimation in the birth‐and‐death process. Annals of Statistics, 3(2), 363–372.

Kendall,, D. G. (1948). On the generalized “birth‐and‐death” process. The Annals of Mathematical Statistics, 19, 1–15.

Kendall,, D. G. (1949). Stochastic processes and population growth. Journal of the Royal Statistical Society: Series B: Methodological, 11(2), 230–282.

Kimmel,, M., & Axelrod,, D. E. (2016). Branching processes in biology. New York: Springer.

Kingman,, J. F. (1982). On the genealogy of large populations. Journal of Applied Probability, 19(A), 27–43.

Klar,, B., Parthasarathy,, P., & Henze,, N. (2010). Zipf and lerch limit of birth and death processes. Probability in the Engineering and Informational Sciences, 24(1), 129–144.

Krone,, S. M., & Neuhauser,, C. (1997). Ancestral processes with selection. Theoretical Population Biology, 51, 210–237.

Lange,, K. (1995). A gradient algorithm locally equivalent to the EM algorithm. Journal of the Royal Statistical Society: Series B, 57(2), 425–437.

Lange,, K. (2010a). Applied probability (2nd ed.). New York, NY: Springer.

Lange,, K. (2010b). Numerical analysis for statisticians (2nd ed.). New York, NY: Springer Hardcover.

Lee,, J., Weiss,, R. E., & Suchard,, M. A. (2011). *Using a birth‐death process to account for reporting errors in longitudinal self‐reported counts of behavior*. Unpublished UCLA Biostatistics Technical Report.

Lefévre,, C. (1981). Optimal control of a birth and death epidemic process. Operations Research, 29(5), 971–982.

Lenin,, R., & Parthasarathy,, P. (2000). A birth‐death process suggested by a chain sequence. Computers %26 Mathematics with Applications, 40(2–3), 239–247.

Lorentzen,, L., & Waadeland,, H. (1992). Continued fractions with applications. Amsterdam, the Netherlands: North‐Holland.

Mayrose,, I., Barker,, M. S., & Otto,, S. P. (2010). Probabilistic models of chromosome number evolution and the inference of polyploidy. Systematic Biology, 59(2), 132–144.

McFarland,, C. D., Mirny,, L. A., & Korolev,, K. S. (2014). Tug‐of‐war between driver and passenger mutations in cancer and other adaptive processes. Proceedings of the National Academy of Sciences, 111(42), 15138–15143.

McNeil,, D. (1970). Integral functionals of birth and death processes and related limiting distributions. Annals of Mathematical Statistics, 41(2), 480–485.

Metzner,, P., Dittmer,, E., Jahnke,, T., & Schütte,, C. (2007). Generator estimation of Markov jump processes. Journal of Computational Physics, 227(1), 353–375.

Moran,, P. (1951). Estimation methods for evolutive processes. Journal of the Royal Statistical Society: Series B: Methodological, 13, 141–146.

Moran,, P. (1953). The estimation of the parameters of a birth and death process. Journal of the Royal Statistical Society: Series B: Methodological, 15, 241–245.

Moran,, P. A. P. (1958). Random processes in genetics. Mathematical Proceedings of the Cambridge Philosophical Society, 54, 60–71.

Murphy,, J., & O`donohoe,, M. (1975). Some properties of continued fractions with applications in markov processes. IMA Journal of Applied Mathematics, 16(1), 57–71.

Nee,, S. (2006). Birth‐death models in macroevolution. Annual Review of Ecology, Evolution, and Systematics, 37, 1–17.

Nee,, S., May,, R. M., & Harvey,, P. H. (1994). The reconstructed evolutionary process. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 344(1309), 305–311.

Neuts,, M. F. (1995). Algorithmic probability: A collection of problems (stochastic modeling series). London: Chapman and Hall/CRC.

Norris,, J. R. (1998). *Markov chains* (No. 2008). Cambridge University Press.

Novozhilov,, A. S., Karev,, G. P., & Koonin,, E. V. (2006). Biological applications of the theory of birth‐and‐death processes. Briefings in Bioinformatics, 7(1), 70–85.

Nowak,, M. A., Michor,, F., & Iwasa,, Y. (2003). The linear process of somatic evolution. Proceedings of the National Academy of Sciences, 100(25), 14966–14969.

Nowak,, M. A., Sasaki,, A., Taylor,, C., & Fudenberg,, D. (2004). Emergence of cooperation and evolutionary stability in finite populations. Nature, 428(6983), 646.

Ohkubo,, J. (2014). Karlin–McGregor‐like formula in a simple time‐inhomogeneous birth–death process. Journal of Physics A: Mathematical and Theoretical, 47(40), 405001.

Parthasarathy,, P., Lenin,, R., Schoutens,, W., & Van Assche,, W. (1998). A birth and death process related to the Rogers–Ramanujan continued fraction. Journal of Mathematical Analysis and Applications, 224(2), 297–315.

Pollett,, P. (2003). Integrals for continuous‐time markov chains. Mathematical Biosciences, 182(2), 213–225.

Pollett,, P., & Stefanov,, V. (2003). %22A method for evaluating the distribution of the total cost of a random process over its lifetime%22. In International Congress on Modelling and Simulation (Vol. 4, pp. 1863–1867). Townsville, Australia: Modelling and Simulation Society of Australia and New Zealand.

Puri,, P. S. (1966). On the homogeneous birth‐and‐death process and its integral. Biometrika, 53(1–2), 61–71.

Puri,, P. S. (1968). %22Some further results on the birth‐and‐death process and its integral%22. In Mathematical proceedings of the Cambridge Philosophical Society (Vol. 64, pp. 141–154). Cambridge, UK: Cambridge University Press.

Puri,, P. S. (1971). A method for studying the integral functionals of stochastic processes with applications: I. Markov chain case. Journal of Applied Probability, 8(2), 331–343.

Puri,, P. S. (1972a). A method for studying the integral functional of stochastic processes with applications. II. Sojourn time distributions for Markov chains. Probability Theory and Related Fields, 23(2), 85–96.

Puri,, P. S. (1972b). %22A method for studying the integral functionals of stochastic processes with applications III%22. In Proceedings of the sixth Berkeley symposium on mathematical statistics and probility (Vol. 3, pp. 481–500). Berkeley, CA: University of California Press.

Reed,, W. J., & Hughes,, B. D. (2004). A model explaining the size distribution of gene and protein families. Mathematical Biosciences, 189(1), 97–102.

Renshaw,, E. (1993). Modelling biological populations in space and time. Cambridge, UK: Cambridge University Press.

Renshaw,, E. (2011). Stochastic population processes: Analysis, approximations, simulations. Oxford, UK: Oxford University Press.

Reynolds,, J. F. (1973). On estimating the parameters of a birth‐death process. Australian %26 New Zealand Journal of Statistics, 15(1), 35–43.

Rosenberg,, N. A., Tsolaki,, A. G., & Tanaka,, M. M. (2003). Estimating change rates of genetic markers using serial samples: Applications to the transposon IS6110 in *Mycobacterium tuberculosis*. Theoretical Population Biology, 63(4), 347–363.

Ross,, S. M. (1995). Stochastic processes (2nd ed.). New York: Wiley.

Sagitov,, S. (2013). Linear‐fractional branching processes with countably many types. Stochastic Processes and their Applications, 123(8), 2940–2956.

Stefanov,, V., & Wang,, S. (2000). A note on integrals for birth‐death processes. Mathematical Biosciences, 168(2), 161–165.

Tan,, W. Y., & Piantadosi,, S. (1991). On stochastic growth processes with application to stochastic logistic growth. Statistica Sinica, 1, 527–540.

Thorne,, J., Kishino,, H., & Felsenstein,, J. (1991). An evolutionary model for maximum likelihood alignment of DNA sequences. Journal of Molecular Evolution, 33(2), 114–124.

Wall,, H. S. (1948). Analytic theory of continued fractions. New York, NY: D. Van Nostrand.

Wallis,, J. (1972). *Opera mathematica. Volume 1. oxoniae e theatro shedoniano*. Hildesheim, Germany: Georg Olms Verlag (Reprinted by Georg Olms Verlag).

Wickwire,, K. (1977). Mathematical models for the control of pests and infectious diseases: A survey. Theoretical Population Biology, 11(2), 182–238.

Wolff,, R. W. (1965). Problems of statistical inference for birth and death queuing models. Operations Research, 13(3), 343–357.

Xu,, J., Guttorp,, P., Kato‐Maeda,, M., & Minin,, V. N. (2015). Likelihood‐based inference for discretely observed birth–death‐shift processes, with applications to evolution of mobile genetic elements. Biometrics, 71(4), 1009–1021.