Abstract
Methods for simulating dependent sequences of continuous positivevalued random variables with exponential, uniform, Gamma, and mixed exponential marginal distributions are given. In most cases the sequences are first-order, linear autoregressive, Markovian processes. A very broad two-parameter family of this type, GNEAR(l), with exponential marginals and both positive and negative correlation is defined and its transformation to a similar multiplicative process with uniform marginals is given. It is shown that for a subclass of this two-parameter family extension to mixed exponential marginals is possible, giving a model of broad applicability for analyzing data and modelling stochastic systems, although negative correlation is more difficult to obtain than in the exponential case. Finally, two schemes for autoregressive sequences with Gamma distributed marginals are outlined. Efficient simulation of some of these schemes is discussed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Gaver, D. P. and Lewis, P. A. W., “First Order Autoregressive Gamma Sequences and Point Processes,” Adv. in Appl. Probab., 12, 1980, 727–745.
Lawrance, A. J. and Lewis, P. A. W., “A Moving Average Exponential Point Process (EMA1),” J. Appl. Probab., 14, 1977, 98–113.
Jacobs, P. A. and Lewis, P. A. W., “A Mixed Autoregressive-Moving Average Exponential sequence and Point Process (EARMA 1,1),” Adv. in Appl. Probab., 9, 1977, 87–104.
Process J. Roy. Statist. Soc. Ser. B, 42, 1980, 150–161.
Lawrance, A. J. and Lewis, P. A. W., “A New Autoregressive Time-Series Model in Exponential Variables (NEAR(l)),” Adv. In Appl. Probab., 13, 1981, 826–845.
Lawrance, A. J., “The Mixed Exponential Solution to the First-Order Autoregressive Model,” J. Appl. Probab., 17, 1980, 546–552.
Lawrance, A. J., “Some Autoregressive Models for Point Processes,” Point Process and Queuing Problems, Colloquia Mathematica Societates Janos Bolyai, 24, P. Bartfai and J. Tomko eds., North Holland, 1981, Amsterdam, 257–275.
Lewis, P. A. W., “Simple Models for Positive-Valued and Discrete-Valued Time Series with ARMA Correlation Structure,” Proc. Fifth International Symposium Mult. Anal., Multivariate Analysis-V, P. R. Krishnaiah, ed. North-Ho Hand, Amsterdam, 1980, 151–166.
Lawrance, A. J. and Lewis, P. A. W., “Simulation of some Autoregressive sequences of positive random variables.” Proc. 1979 Winter Simulation Conference, Highland, H.J., Spiegel, M. G. and Shannon, R. eds., 1979. IEEE Press, N.Y., 301–308.
Cox, D. R. and Lewis, P. A. W., Statistical Analysis of Series of Events, Methuen, London, 1966.
Lewis, P. A. W. and Shedler, G. S., “Analysis and Modelling of Point Processes in Computer Systems,” Bull. ISI, XLVII (2), 1978, 193–219.
Jacobs, P. A., “A Closed Cyclic Queueing Network with Dependent Exponential Service Times,” J. A.pl. Probab., 15, 1978, 573–589.
Jacobs, P. A., “Heavy traffic results for single-server queues with dependent (EARMA) service and interarrival times,” Adv. in Appl. Probab., 12, 1980, 517–529.
Jacobs, P. A. and Lewis, P. A. W., “Discrete Time Series Generated by Mixtures, I: Correlational and Runs Properties,” J. Roy. Statist. Soc. Ser. B., 40, 1978, 94–105.
Jacobs, P. A. and Lewis, P. A. W., “Discrete Time Series Generated by Mixtures II: Asymptotic Properties,” J. Roy. Statist. Soc. Ser. B, 40, 1978, 222–228.
Vervaat, W., “On a stochastic difference equation and a representation of non-negative infinitely divisible random variables,” Adv. in Appl. Probab., 11, 1979, 750–783.
G. S. Fishman, personal communication.
Lawrance, A. J. and Lewis, P. A. W., “A mixed exponential time series model, NMEAR(p,q),” Naval Postgraduate School Technical Report NPS55-80-012, 1980. To appear in Management Sci.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1982 Birkhäuser Boston, Inc.
About this chapter
Cite this chapter
Lawrance, A.J., Lewis, P.A.W. (1982). Generation of some First-Order Autoregressive Markovian Sequences of Positive Random Variables with Given Marginal Distributions. In: Disney, R.L., Ott, T.J. (eds) Applied Probability-Computer Science: The Interface Volume 1. Progress in Computer Science, vol 2. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-5791-2_15
Download citation
DOI: https://doi.org/10.1007/978-1-4612-5791-2_15
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-5793-6
Online ISBN: 978-1-4612-5791-2
eBook Packages: Springer Book Archive