Non-Gaussian conditional linear AR(1) models

Australian and New Zealand Journal of Statistics

Volumn 42, Issue 4, 2000, Pages 479-495

  Grunwald, Gary K ^a,b   Hyndman, Rob J ^a   Tedesco, Leanna ^a,c   Tweedie, Richard L ^a,d  

^a MONASH UNIVERSITY   (Australia)

^b UNIVERSITY OF COLORADO   (United States)

^c Tillinghast Towers Perrin   (Australia)

^d UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Autoregression; Data analysis; Exponential time series; Gamma time series; Non Gaussian time series; Poisson time series

Indexed keywords

EID: 0034358979     PISSN: 13691473     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-842X.00143     Document Type: Article

References (70)

1. AL-OSH, M.A. & ALY, E-E.A.A. (1992). First order autoregressive time series with negative binomial and geometric marginals. Comm. Statist. Theory Methods 21, 2483-2492.
2. AL-OSH, M.A. & ALZAID, A.A. (1987). First order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal 8, 261-275.
3. ALTMAN, N.S. (1990). Kernel smoothing of data with correlated errors. J. Amer. Statist. Assoc. 85, 749-759.
4. ALZAID, A.A. & AL-OSH, M.A. (1988). First order integer-valued autoregressive process: distributional and regression properties. Statist. Neerlandica 42, 53-61.
5. ANDEL, J. (1988). On AR(1) processes with exponential white noise. Comm. Statist. Theory Methods 17, 1481-1495.
6. BARTLETT, M.S. (1946). On the theoretical specification of sampling properties of autocorrelated time series. J. R. Stat. Soc. Ser. B Stat. Methodol. 8, 27-41.
7. BELL, C.B. & SMITH, E.P. (1986). Inference for non-negative autoregressive schemes. Comm. Statist. Theory Methods 15, 2267-2293.
8. BESAG, J.E. (1974). Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B Stat. Methodol. 36, 192-225.
9. BICKEL, P.J. & DOKSUM, K.A. (1977). Mathematical Statistics: Basic Ideas and Selected Topics. San Francisco: Holden-Day.
10. BRECKLING, J. (1989). The Analysis of Directional Time Series: Applications to Wind Speed and Direction. Berlin: Springer-Verlag. Lecture Notes in Statistics, 61.
11. BROCKWELL, P.J. & DAVIS, R.A. (1991). Time Series: Theory and Methods, 2nd edn. New York: Springer-Verlag.
12. CHANDRASEKHAR, S. (1954). Selected papers on noise and stochastic processes. In Stochastic Problems in Physics and Astronomy, ed. N. Wax. New York: Dover.
13. DEWALD, L.S. & LEWIS, P.A.W. (1985). A new Laplace second-order autoregressive time series model - NLAR(2). IEEE Trans. Inf. Theory 31, 645-651.
14. DIGGLE, P.J., LIANG, K.Y. & ZEGER, S.L. (1994). Analysis of Longitudinal Data. Oxford: Clarendon Press.
15. FAHRMEIR, L. (1992). State space modeling and conditional mode estimation for categorical time series. In New Directions in Time Series Analysis, Part 1, eds D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt & M.S. Taqqu. New York: Springer.
16. FEIGIN, P. & TWEEDIE, R.L. (1985). Random coefficient autoregressive processes: a Markov chain analysis of stationarity and finiteness of moments. J. Time Ser. Anal. 6, 1-14.
17. FRANKE, J. & SELIGMANN, T. (1993). Conditional maximum likelihood estimates for INAR(1) processes and their application to modelling epileptic seizure counts. In Developments in Time Series Analysis: in Honour of Maurice B. Priestley, ed. T. Subba Rao. New York: Chapman and Hall.
18. GAVER D.P. & LEWIS, P.A.W. (1980). First-order autoregressive gamma sequences and point processes. Adv. in Appl. Probab. 12, 726-745.
19. GRUNWALD, G.K. & FEIGIN, P. (1996). First Order Product Autoregressive Processes. Research Report 2. Department of Statistics, University of Melbourne.
20. GRUNWALD, G.K. & HYNDMAN, R.J. (1998). Smoothing non-Gaussian time series with autoregressive structure. Comput. Statist. Data Anal. 28, 171-191.
21. GRUNWALD, G.K. & JONES, R.J. (1999). Markov models for time series with mixed distribution. Environmetrics 10 (in press).
22. GRUNWALD, G.K., HAMZA, K. & HYNDMAN, R.J. (1997). Some properties and generalizations of Bayesian time series models. J. R. Stat. Soc. Ser. B Stat. Methodol. 59 , 615-626.
23. GRUWVALD, G.K., HYNDMAN, R.J. & TEDESCO, L. (1995). A Unified View of Linear AR(1) Models. Research Report. Department of Statistics, University of Melbourne. Available at http://www-personal.buseco.monash.edu.au/~hyndman/papers/arl.htm
24. GUTTORP, P. (1991). Statistical Inference for Branching Processes. New York: Wiley.
25. HART, J.D. (1991). Kernel regression estimation with time series errors. J. R. Stat. Soc. Ser. B Stat. Methodol. 53, 173-187.
26. HARVEY, A.C. & FERNANDES, C. (1989). Time series models for count or qualitative observations (with discussion). J. Bus. Econom. Statist. 7, 407-422.
27. HASTIE, T.J. & TIBSHIRANI, R.J. (1990). Generalized Additive Models. London: Chapman and Hall.
28. HEYDE, C.C. & SENETA, E. (1972). Estimation theory for growth and immigration rates in a multiplicative process. J. Appl. Probab. 9, 235-256.
29. HUTTON, J.L. (1990). Non-negative time series models for dry riverflow. J. Appl. Probab. 27, 171-182.
30. HYNDMAN, R.J. (1999). Nonparametric additive regression models for binary time series', In Proceedings, 1999 Australasian Meeting of the Econometric Society, 7-9 July 1999. University of Technology, Sydney.
31. HYNDMAN, R.J. & GRUNWALD, G.K. (2000). Generalized additive modelling of mixed distribution Markov models with application to Melbourne's rainfall, Aust. N. Z. J. Stat. 42, 145-158.
32. JACOBS, P.A. & LEWIS, P.A.W. (1978a). Discrete time series generated by mixtures I: correlational and runs properties. J. R. Stat. Soc. Ser. B Stat. Methodol. 40, 94-105.
33. JACOBS, P.A. & LEWIS, P.A.W. (1978b). Discrete time series generated by mixtures II: asymptotic properties. J. R. Stat. Soc. Ser. B Stat. Methodol. 40, 222-228.
34. JUDGE, G.G., HILL, R.C., GRIFHTHS, W.E., LÜTKEPOHL, H. & LEE, T-C. (1985). The Theory and Practice of Econometrics, 2nd edn. New York: John Wiley & Sons.
35. KANTER, M. (1975). Autoregression for discrete processes mod 2. J. Appl. Probab. 12, 371-375.
36. KASHIWAGI, N. & YANAGIMOTO, T. (1992). Smoothing serial count data through a state-space model. Biometrics 48, 1187-1194.
37. KITAGAWA, G. (1987). Non-Gaussian state-space modeling of nonstationary time series. J. Amer. Statist. Assoc. 82, 1032-1063.
38. LAWRANCE, A.J. (1982). The innovation distribution of a gamma distributed autoregressive process. Scand. J. Statist. 9, 234-236.
39. LAWRANCE, A.J. & LEWIS, P.A.W. (1981). A new autoregressive time series model in exponential variables (NEAR(1)). Adv. in Appl. Probab. 13, 826-845.
40. LAWRANCE, A.J. & LEWIS, P.A.W. (1985). Modelling and residual analysis of nonlinear autoregressive time series in exponential variables (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 47, 165-202.
41. LEWIS, P.A.W. (1985). Some simple models for continuous variate time series. Water Resources Bulletin 21, 635-644.
42. LEWIS, P.A.W. & MCKENZIE, E. (1991). Minification processes and their transformations. J. Appl. Probab. 28, 45-57.
43. LEWIS, P.A.W., MCKENZIE, E. & HUGUS, D.K. (1989). Gamma processes. Comm. Statist. Stochastic Models 5, 1-30.
44. LI, W.K. (1994). Time series models based on generalized linear models: some further results. Biometrics 50, 506-511.
45. LYE, J.N. & MARTIN, V.L. (1994). Non-linear time series modelling and distributional flexibility. J. Time Ser. Anal. 15, 65-84.
46. MCCULLAGH, P. & NELDER, J.A. (1989). Generalized Linear Models, 2nd edn. London: Chapman and Hall.
47. MCKENZIE, E. (1982). Product autoregression: a time series characterization of the gamma distribution. J. Appl. Probab. 19, 463-468.
48. MCKENZIE, E. (1985a). Some simple models for discrete variate time series. Water Resources Bulletin 21, 645-650.
49. MCKENZIE, E. (1985b). An autoregressive process for beta random variables. Management Sci. 31, 988-997.
50. MCKENZIE, E. (1988). Some ARMA models for dependent sequences of Poisson counts. Adv. in Appl. Probab. 20, 822-835.
51. MEYN, S.P. & TWEEDIE, R.L. (1993). Markov Chains and Stochastic Stability. New York: Springer-Verlag.
52. MORRIS, C.N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10, 65-80.
53. MORRIS, C.N. (1983). Natural exponential families with quadratic variance functions: statistical theory. Ann. Statist. 11, 515-529.
54. NICHOLLS, D.F. & QUINN, B.G. (1982). Random Coefficient Autoregressive Models: An Introduction. Berlin: Springer-Verlag. Lecture Notes in Statistics, 11.
55. PARZEN, E. (1962). Stochastic Processes. San Francisco: Holden-Day.
56. RAO, P.S. & JOHNSON, D.H. (1988). A first-order AR model for non-Gaussian time series. In Proceedings of IEEE International Conference on ASSP. Vol. 3, pp. 1534-1537.
57. SHEPHARD, N. (1995). Generalized Linear Autoregressions. Economics Working Paper 8. Oxford: Nuffield College.
58. SIM, C.H. (1986). Simulation of Weibull and gamma autoregressive stationary processes. Comm. Statist. Simulation Comput. 15, 1141-1146.
59. SIM, C.H. (1990). First-order autoregressive models for gamma and exponential processes. J. Appl. Probab. 27, 325-332.
60. SIM, C.H. (1993). First-order autoregressive logistic processes. J. Appl. Probab. 30, 467-470.
61. SIM, C.H. (1994). Modelling non-normal first-order autoregressive time series. J. Forecasting 13, 369-381.
62. STERN, R.D. & COE, R. (1984). A model fitting analysis of daily rainfall data (with discussion). J. Roy. Statist. Soc. Ser. A 147, 1-34.
63. SWIFT, A.L. & JANACEK, G.J. (1991). Forecasting non-normal time series. J. Forecasting 10, 501-520.
64. TAVARES, L. V. (1977). The exact distribution of extremes in a non-Gaussian process. Stochastic Process. Appl. 5, 151-156.
65. TAVARES, L.V. (1980a). A non-Gaussian Markovian model to simulate hydrologie processes. J. Hydrology 46, 281-287.
66. TAVARES, L.V. (1980b). An exponential Markovian stationary process. J. Appl. Probab. 17, 1117-1120.
67. TSAY, R.S. (1992). Model checking via parametric bootstraps in time series analysis. Applied Statistics 41, 1-15.
68. WEISS, G. (1985). Half-hourly precipitation and stream flow, River Hirnant, Wales, U.K., November and December, 1972. In Data: a Collection of Problems from many Fields for the Student and Research Worker, eds D.F. Andrews & A.M. Herzberg. New York: Springer-Verlag.
69. WEST, M., HARRISON, P.J. & MIGON, H.S. (1985). Dynamic generalized linear models and Bayesian forecasting (with discussion). J. Amer. Statist. Assoc. 80, 73-96.
70. ZEGER, S.L. & QAQISH, B. (1988). Markov regression models for time series: a quasi-likelihood approach. Biometrics 44, 1019-1031.