A product autoregressive model with log-Laplace marginal distribution
DOI:
https://doi.org/10.6092/issn.1973-2201/3650Abstract
The log-Laplace distribution and its properties are considered. Some important properties like multiplicative infinite divisibility, geometric multiplicative infinite divisibility and self-decomposability are discussed. A first order product autoregressive model with log-Laplace marginal distribution is developed. Simulation studies are conducted as well as sample path properties and estimation of parameters of the process are discussed. Further multivariate extensions are also considered.
References
M. ABRAMOWITZ, I.A. STEGUN (1964). Handbook of Mathematical Functions, U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series 55.
R.A. BAGNOLD, O. BARNDORFF-NIELSEN (1980). The pattern of natural size distribution. “Sedimentology” 27, pp. 199-207.
O. BARNDORFF-NIELSEN (1977). Exponentially decreasing distributions for the logarithm of particle size. ‘Proceedings of Royal Society London A” 353, pp. 401-419.
L. BONDESSON (1979). A general result on infinite-divisibility. “Annals of Probability” 7, pp. 965-979.
L. BONDESSON (1981). Discussion of the paper Cox “Statistical analysis of time series: Some recent developments”. “Scandinavian Journal of Statistics” 8, pp. 93-115.
D.R. COX (1981). Statistical Analysis of Time Series: Some Recent Developments. “Scandinavian Journal of Statistics” 8, pp. 93-115.
E. DAMSLETH, A.H. EL-SHAARAWI (1989). ARMA models with double-exponentially distributed noise. “Jornal of Royal Statistical Society B” 51(1), pp. 61-69.
L.S. DEWALD, P.A.W. LEWIS (1985). A new Laplace second-order autoregressive time-series model-NLAR(2). “IEEE Transactions on Information Theory” IT 31(5), pp. 645-651.
M. FRÉCHET (1939). Sur les formules de répartition des revenus, “Revue l’Institut International de Statistique” 7(1), pp. 32-38.
Y. FUJITHA (1993). A generalization of the results of Pillai. “Annals of Institute of Statistical Mathematics” 45, pp. 361-365.
D.P. GAVER, P.A.W. LEWIS (1980). First order autoregressive gamma sequences and point processes. “Advances in Applied Probability” 12, pp. 727-745.
J.D. GIBSON (1986). Data compression of a first order intermittently excited AR process. In Statistical Image Processing and Graphics. pp. 115-126 (Edited by E.J. Wegman and D.J. Depriest, Marcel Deckrer Inc., New York).
M.J. HARTLEY, N.S. REVANKAR (1974). On the estimation of the Pareto law from underreported data, “Journal of Econometrics” 2, pp. 327-341.
D.V. HINKLEY, N.S. REVANKAR (1977). Estimation of the Pareto law from under reported data, “Journal of Econometrics” 5, pp. 1-11.
T. INOUE (1978). On Income Distribution: The Welfare Implications of the General Equilibrium Model, and the Stochastic Processes of Income Distribution Formation, Ph.D. Thesis, University of Minnesota.
K. JAYAKUMAR (1997). First order autoregressive semi-alpha-Laplace processes, “Statistica”, LVII, pp. 455-463.
K.K. JOSE, T. LISHAMOL, J. SREEKUMAR (2008). Autoregressive processes with normal Laplace marginals. “Statistics and Probability Letters” 78, pp. 2456-2462.
K.K. JOSE, S.R. NAIK (2008). A class of asymmetric pathway distributions and an entropy interpretation. “Physica A: Statistical Mechanics and its Applications” 387(28), pp. 6943-6951.
K. K. JOSE, B. ABRAHAM (2011). A Count Model based on with Mittag-Leffler interarrival times, “Statistica”, anno LXXI, n. 4, 501-514.
O. JULIÀ, J. VIVES-REGO (2005). Skew-Laplace distribution in Gram-negative bacterial axenic cultures: new insights into intrinsic cellular heterogeneity. “Microbiology” 151, pp. 749-755.
L.B. KLEBANOV, G.M. MANIYA, I.A. MELAMED (1984). A problem of Zolotarev and analogs of infinitely divisible and stable distribution in a scheme for summing a random number of random variables. “Theory of Probability and its Applications” 29, pp. 791-794.
S. KOTZ, T.J. KOZUBOWSKI, K. PODGORSKI (2001). The Laplace Distribution and Generalizations – A Revisit with Applications to Communications, Economics, Engineering and Finance. Birkhäuser, Boston.
T.J. KOZUBOWSKI, K. PODGORSKI (2003). Log-Laplace distributions. “International Journal of Mathematics” 3 (4), pp. 467-495.
T.J. KOZUBOWSKI, K. PODGORSKI (2010). Random self-decomposability and autoregressive processes. “Statistics and Probability Letters” 80, pp. 1606-1611.
E. KRISHNA, K. K. JOSE (2011). Marshall-Olkin assymmetric Laplace distribution and processes, “Statistica”, anno LXXI, n. 4, 453-467.
A.J. LAWRANCE (1978). Some autoregressive models for point processes. “In Proceedings of Bolyai Mathematical Society Colloquiuon point processes and queuing problems” 24, Hungary, pp. 257-275.
A.J. LAWRANCE (1982). The innovation distribution of a gamma distributed autoregressive process. “Scandinavian Journal of Statistics” 9, pp. 234-236.
M. MAEJIMA, Y. NAITO (1998). Semi-self decomposable distributions and a new class of limit theorems. “Probability Theory and Related Fields” 112, pp. 13-31.
E.D. MCKENZIE (1982). Product autoregression: a time-series characterization of the gamma distribution. “ Journal of Applied Probability” 19, pp. 463-468.
R.N. PILLAI (1990). Harmonic mixtures and geometric infinite divisibility. “Jornal of Indian Statistical Association” 28, pp. 87-98.
R.N. PILLAI, E. SANDHYA (1990). Distributions with complete monotone derivative and geometric infinite divisibility. “Advances in Applied Probability” 22, pp. 751-754.
B. PUNATHUMPARAMBATH (2011). A new family of skewed slash distributions generated by the normal kernel, “Statistica”, anno LXXI, n. 3, 345-353.
W.J. REED (2007). Brownian Laplace motion and its use in financial modelling. “Communications in Statistics-Theory and Methods” 36, pp. 473-484.
W.J. REED, M. JORGENSEN (2004). The double Pareto lognormal distribution-a new parametric model for size distributions. “Communications in Statistics-Theory and Methods” 33(8), pp. 1733-1753.
V. SEETHALEKSHMI, K.K. JOSE (2004). An autoregressive process with geometric -Laplace marginals. “Statistical Papers” 45, pp. 337-350.
V. SEETHALEKSHMI, K.K. JOSE (2006). Autoregressive processes with Pakes and geometric Pakes generalized Linnik marginals. “Statistics and Probability Letters” 76(2), 318-326.
D.N. SHANBHAG, M. SREEHARI (1977). On certain self-decomposable distributions. “Z. Warscheinlichkeitstch” 38, pp. 217-222.
C.H. SIM (1994). Modelling non-normal first order autoregressive time series. “Journal of Forecasting” 13, pp. 369-381.
V.R.R. UPPULURI (1981). Some properties of log-Laplace distribution. “Statistical Distributions in Scientific Work” 4, (eds., G.P. Patil, C. Taillie and B. Baldessari), Dordrecht: Reidel, 105-110.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2012 Statistica
This journal is licensed under a Creative Commons Attribution 3.0 Unported License (full legal code).
Authors accept to transfer their copyrights to the journal.
See also our Open Access Policy.