A mixed bivariate distribution with exponential and geometric marginals. (English) Zbl 1072.62003
Let \(X_1, X_2, \dots\) be \(i.i.d.\) (independent and identically distributed) exponential random variables with parameter \(\beta > 0\) and the probability density function (\(p.d.f.\)) defined as \(f(x) = \beta\cdot e^{-\beta x}, x > 0.\) Let \(N\) be a geometric random variable with p.d.f. \(h(n) = P(N = n) = p(1 - p)^{n - 1}, n = 1, 2,\dots,\) independent of every \(X_i\) random variable and, since the exponential distribution is closed under geometric compound, the random sum \(X =\sum^{N}_{i=1} X_i\) has an exponential distribution with the parameter \(p\cdot\beta\).
The aim of this paper is to study the joint distribution of \(X\) and \(N\), and to show that the resulting class \((X, N)\) of mixed bivariate distributions, referred hereafter as BEG (Bivariate distributions with Exponential and Geometric marginals), carries remarkable properties and supports useful applications in stochastic modelling. Section 2 of the paper defines the mixed BEG class of distributions and derives their basic properties: relationship between BEG distribution and the Laplace distribution, survival function computation, convergence in distribution. Section 3 presents certain conditional distributions that are useful in the practical implementation of goodness of fit analyses connected with the BEG model. Section 4 gives three representations of BEG distributions involving random and deterministic sums of i.i.d. random vectors.
The following results generalize similar properties of univariate exponential and geometric distributions, and justify the applicability of the BEG distributions as stochastic models: infinite divisibility, stability with respect to geometric summation, and compound Poisson representation. Section 5 considers maximum likelihood estimation of the BEG parameters, showing that the obtained maximum likelihood estimators are consistent, asymptotically normal, and efficient ones. Section 6 illustrates the modelling potential of the proposed BEG distributions on a currency exchange data set, \(N\) representing the number of consecutive positive daily \(log\)-returns of currency exchange rates.
The numerical findings prove a very good fit in all five studied cases, entailing that the BEG model captures not only the marginal distributions of the data but also the conditional distributions very well, concluding the remarkable fit of the BEG model to the considered currency data. The modelling potential of the investigated laws may be also successfully applied in various fields such as water resources, climate research, and finance.
The aim of this paper is to study the joint distribution of \(X\) and \(N\), and to show that the resulting class \((X, N)\) of mixed bivariate distributions, referred hereafter as BEG (Bivariate distributions with Exponential and Geometric marginals), carries remarkable properties and supports useful applications in stochastic modelling. Section 2 of the paper defines the mixed BEG class of distributions and derives their basic properties: relationship between BEG distribution and the Laplace distribution, survival function computation, convergence in distribution. Section 3 presents certain conditional distributions that are useful in the practical implementation of goodness of fit analyses connected with the BEG model. Section 4 gives three representations of BEG distributions involving random and deterministic sums of i.i.d. random vectors.
The following results generalize similar properties of univariate exponential and geometric distributions, and justify the applicability of the BEG distributions as stochastic models: infinite divisibility, stability with respect to geometric summation, and compound Poisson representation. Section 5 considers maximum likelihood estimation of the BEG parameters, showing that the obtained maximum likelihood estimators are consistent, asymptotically normal, and efficient ones. Section 6 illustrates the modelling potential of the proposed BEG distributions on a currency exchange data set, \(N\) representing the number of consecutive positive daily \(log\)-returns of currency exchange rates.
The numerical findings prove a very good fit in all five studied cases, entailing that the BEG model captures not only the marginal distributions of the data but also the conditional distributions very well, concluding the remarkable fit of the BEG model to the considered currency data. The modelling potential of the investigated laws may be also successfully applied in various fields such as water resources, climate research, and finance.
Reviewer: Neculai Curteanu (Iaşi)
MSC:
| 62E10 | Characterization and structure theory of statistical distributions |
| 62F10 | Point estimation |
| 60E05 | Probability distributions: general theory |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 91B30 | Risk theory, insurance (MSC2010) |
| 91B32 | Resource and cost allocation (including fair division, apportionment, etc.) |
Keywords:
double exponential distribution; mixed bivariate distribution; infinite divisibility; Laplace distribution; maximum likelihood estimation; random summation; currency exchange rates modelingReferences:
| [1] | Amaral, L. A.N.; Buldyrev, S. V.; Havlin, S.; Salinger, M. A.; Stanley, H. E., Power law scaling for a system of interacting units with complex internal structure, Phys. Rev. Lett., 80, 7, 1385-1388 (1998) |
| [2] | Arnold, B. C., Some characterizations of the exponential distribution by geometric compounding, SIAM J. Appl. Math., 24, 242-244 (1973) · Zbl 0251.60010 |
| [3] | Ayebo, A.; Kozubowski, T. J., An asymmetric generalization of Gaussian and Laplace laws, J. Probab. Statist. Sci., 1, 2, 187-210 (2003) |
| [4] | Biondi, F.; Panorska, A. K.; Kozubowski, T. J., Stochastic modeling of regime shifts, Climate Res., 23, 23-30 (2002) |
| [5] | Boothe, P.; Glassman, D., The statistical distribution of exchange rates, J. Int. Econ., 22, 297-319 (1987) |
| [6] | Buldyrev, S. V.; Amaral, L. A.N.; Havlin, S.; Leschhorn, H.; Maass, P.; Salinger, M. A.; Stanley, H. E.; Stanley, M. H.R., Scaling behavior in economics, II. modeling of company growth, J. Phys. I (France), 7, 4, 635-650 (1997) |
| [7] | Chenyao, D.; Mittnik, S.; Rachev, S. T., Distribution of exchange ratesa geometric summation stable model, (Proceedings of the Seminar on Data Analysis, September 12-17 (1996), Sozopol: Sozopol Bulgaria) · Zbl 1098.62520 |
| [8] | Cook, E.R., Krusic, P.J., 2003. The North American Drought Atlas. EOS Transactions of the American Geophysical Union 84, Abstract GC52A-01.; Cook, E.R., Krusic, P.J., 2003. The North American Drought Atlas. EOS Transactions of the American Geophysical Union 84, Abstract GC52A-01. |
| [9] | Feller, W., 1960. An Introduction to Probability Theory and its Applications, Vol. I, 2nd edition. Wiley, New York.; Feller, W., 1960. An Introduction to Probability Theory and its Applications, Vol. I, 2nd edition. Wiley, New York. · Zbl 0155.23101 |
| [10] | González, J.; Valdés, J. B., Bivariate drought recurrence analysis using tree-ring reconstructions, J. Hydrol. Eng., 8, 247-258 (2003) |
| [11] | Johnson, N. L.; Kotz, S.; Kemp, A. W., Univariate Discrete Distributions (1993), Wiley: Wiley New York |
| [12] | Kalashnikov, V., Geometric SumsBounds for Rare Events with Applications (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0881.60043 |
| [13] | Kim, T.-W.; Valdés, J. B.; Yoo, C., Nonparametric approach for estimating return periods of droughts in arid regions, J. Hydrol. Eng., 8, 237-246 (2003) |
| [14] | Klebanov, L. B.; Maniya, G. M.; Melamed, I. A., A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables, Theory Probab. Appl., 29, 791-794 (1984) · Zbl 0579.60016 |
| [15] | Koedijk, K. G.; Schafgans, M. M.; de Vries, C. G., The tail index of exchange rates returns, J. Int. Econ., 29, 93-108 (1990) |
| [16] | Kotz, S.; Kozubowski, T. J.; Podgórski, K., The Laplace Distribution and GeneralizationsA Revisit with Applications to Communications, Economics, Engineering, and Finance (2001), Birkhäuser: Birkhäuser Boston · Zbl 0977.62003 |
| [17] | Kozubowski, T. J.; Podgórski, K., Asymmetric Laplace distributions, Math. Sci., 25, 37-46 (2000) · Zbl 0961.60026 |
| [18] | Kozubowski, T. J.; Podgórski, K., Asymmetric Laplace laws and modeling financial data, Math. Comput. Modelling, 34, 1003-1021 (2001) · Zbl 1002.60012 |
| [19] | Kozubowski, T. J.; Podgórski, K., A log-Laplace growth-rate model, Math. Sci., 28, 49-60 (2003) · Zbl 1051.60015 |
| [20] | Lee, Y.; Amaral, L. A.N.; Canning, D.; Meyer, M.; Stanley, H. E., Universal features in the growth dynamics of complex organizations, Physical Rev. Lett., 81, 15, 3275-3278 (1998) · Zbl 0910.90038 |
| [21] | Madan, D. B.; Carr, P. P.; Chang, E. C., The variance gamma process and option pricing, Eur. Finance Rev., 2, 79-105 (1998) · Zbl 0937.91052 |
| [22] | Mathier, L.; Perreault, L.; Bobée, B.; Ashkar, F., The use of geometric and gamma-related distributions for frequency analysis of water deficit, Stochastic Hydrol. Hydraul., 6, 239-254 (1992) |
| [23] | McFarland, J. W.; Pettit, R. R.; Sung, S. K., The distribution of foreign exchange price changestrading day effects and risk measurement, J. Finance, 37, 3, 693-715 (1982) |
| [24] | McFarland, J. W.; Pettit, R. R.; Sung, S. K., The distribution of foreign exchange price changestrading day effects and risk measurement—a reply, J. Finance, 42, 1, 189-194 (1987) |
| [25] | Nolan, J. P., Maximum likelihood estimation and diagnostics for stable distributions, (Barndorff-Nielsen, O. E.; Mikosh, T.; Resnick, S., Lévy Processes (2001), Birkhäuser: Birkhäuser Boston), 379-400 · Zbl 0971.62008 |
| [26] | Rao, C. R., Linear Statistical Inference and its Applications (1973), Wiley: Wiley New York · Zbl 0169.21302 |
| [27] | Reed, W. J., The Pareto, Zipf and other power laws, Econ. Lett., 74, 15-19 (2001) · Zbl 1007.91046 |
| [28] | Shiau, J.-T.; Shen, H. W., Recurrence analysis of hydrologic droughts of differing severity, J. Water Resour. Plann. Manage., 127, 30-40 (2001) |
| [29] | So, J. C., The distribution of foreign exchange price changestrading day effects and risk measurement—a comment, J. Finance, 42, 1, 181-188 (1987) |
| [30] | Stanley, M. H.R.; Amaral, L. A.N.; Buldyrev, S. V.; Havlin, S.; Leschorn, H.; Maass, P.; Salinger, M. A.; Stanley, H. E., Scaling behavior in the growth of companies, Nature, 379, 804 (1996) |
| [31] | Takayasu, H.; Okuyama, K., Country dependence on company size distributions and a numerical model based on competition and cooperation, Fractals, 6, 1, 67-69 (1998) |
| [32] | Todorovic, P.; Woolhiser, D. A., Stochastic structure of the local pattern of precipitation, (Shen, H. W., Stochastic Approaches to Water Resources, Vol. 2 (1976), Colorado State University: Colorado State University Fort Collins, Colorado) |
| [33] | Tucker, A. L.; Pond, L., Probability distribution of exchange rate changes, Rev. Econ. Stud., 70, 638-647 (1988) |
| [34] | Westerfield, J. M., An examination of foreign exchange risk under fixed and floating rate regimes, J. Int. Econ., 7, 181-200 (1977) |
| [35] | Zelenhasić, E.; Salvai, A., A method of streamflow drought analysis, Water Resour. Res., 23, 1, 156-168 (1987) |
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