Abstract
Statistical modeling of the dependence within applied stochastic models has become an important goal in many fields of science, since Pearson’s correlation does not provide a complete description of the dependence structure of the random variables, being strongly affected from extreme endpoints, and correlation zero does not imply independence, except in the case of multivariate normal distributions. The construction of bounds for the variability of the distributions of some applied stochastic models when there is only partial information on the dependence structure of the models is the main purpose of this paper. We consider stochastic models in engineering, hydrology, and biosciences, that are defined by mixtures with stochastic environmental parameters. We study stochastic monotonicity and directional convexity properties of some functionals of random variables that are used to define these models. Variability comparisons between the mixture models in terms of the dependence between the stochastic environments are obtained. Stochastic bounds and examples are derived from modeling the dependence structure by some known notions.
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References
Begon M, Harper JL, Townsend CR (1996) Ecology: individuals, populations, and communities, 3a edn. Blackwell, Oxford
Belzunce F, Ortega EM, Ruiz JM (2009) Ageing properties of a discrete-time failure and repair model. IEEE Trans Reliab 58:161–171
Blischke WR, Murthy DNP (2000) Reliability. Modeling, prediction and optimization. Wiley, New York
Chang C-S, Shanthikumar JG, Yao DD (1994) Stochastic convexity and stochastic majorization. In: Yao DD (ed) Stochastic modeling and analysis of manufacturing systems. Ser Oper Res
Cigizoglu HK, Bayazit M (2000) A generalized seasonal model for flow duration curve. Hydrol Process 14:1053–1067
Dracup JA, Lee KS, Paulson EG (1980) On the statistical characteristics of drought events. Water Resour Res 16:289–296
Escudero LF, Ortega EM (2008) Actuarial comparisons of aggregate claims with randomly right truncated claims. Insur Math Econ 43:255–262
Fang Y (2005) Modeling and performance analysis for wireless mobile networks: a new analytical approach. IEEE/ACM Trans Netw 13:989–1002
Fernández-Ponce JM, Ortega EM, Pellerey F (2008) Convex comparisons for random sums in random environments and applications. Probab Eng Inform Sci 22:389–413
Fisher RA, Corbet AS, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. J Anim Ecol 12:42–58
Foerster J (2003) UWB channel modeling sub-committee report final. IEEE P802.15 working group for wireless personal area networks (February 2003)
Genest C, Marceau E, Mesfioui M (2002) Upper stop-loss bounds for sums of possibly dependent risks with given means and variances. Stat Probab Lett 57:33–41
Goovaerts M, Dhaene J, De Schepper A (2000) Stochastic upper bounds for present value functions. J Risk Insur 67:1–14
Harris TE (1963) The theory of branching processes. Springer, Berlin
Jagers P, Klebaner FC (2000) Population-size-dependent and age-dependent branching processes. Stoch Process Appl 87:235–254
Kaas R, Dhaene J, Goovaerts M (2000) Upper and lower bounds for sums of random variables. Insur Math Econ 27:151–168
Kalashnikov V (1997) Geometric sums: bounds for rare events with applications. Kluwer, Boston
Lánský P (1996) A stochastic model for circulatory transport in pharmacokinetics. Math Biosci 132:141–167
Magurran AE (2004) Measuring biological diversity. Blackwell, Oxford
Marinacci M, Montrucchio L (2005) Ultramodular functions. Math Oper Res 30:311–332
Meester LE, Shanthikumar JG (1993) Regularity of stochastic processes. A theory based on directional convexity. Probab Eng Inform Sci 7:343–360
Meester LE, Shanthikumar JG (1999) Stochastic convexity on general space. Math Oper Res 24:472–494
Nadarajah S (2008) A review of results on sums of random variables. Acta Appl Math 103:131–140
Nakagami M (1960) The m-distribution general formula of intensity distribution of rapid fading. In: Statistical methods in radio wave propagation. Pergamon, Oxford, pp 3–36
Oloffson P (1996) Branching processes with local dependencies. Ann Appl Probab 6:238–268
Ortega EM, Escudero LF (2008) On expected utility for financial insurance portfolios with stochastic dependencies. Eur J Oper Res (online first)
Podolski H (1972) The distribution of a product of n independent random variables with generalized gamma distribution. Demonstratio Mathematica 4:119–123
Romeo M, Da Costa V, Bardou F (2003) Broad distribution effects in sums of lognormal random variables. Eur Phys J B32:513–525
Rüschendorf L (2004) Comparison of multivariate risks and positive dependence. Adv Appl Probab 41:391–406
Sakamoto H (1943) On the distributions of the product and the quotient of the independent and uniformly distributed random variables. Tohoku Math J 49:243–260
Scaillet O (2006) A Kolmogorov-Smirnov type test for positive quadrant dependence. Can J Stat 33:415–427
Shaked M, Shanthikumar JG (1990) Parametric stochastic convexity and concavity of stochastic processes. Ann Inst Stat Math 42:509–531
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York
Stoyan D (1983) Comparisons methods for queues and other stochastic models. Wiley, New York
Stuart A (1962) Gamma-distributed products of independent random variables. Biometrika 49:564–565
Weiss M (1984) A note on the role of generalized Inverse Gaussian distribution of circulatory transit times in pharmacokinetics. J Math Biol 20:95–102
Willmot G, Lin X (2001) Lundberg approximations for compounds distributions with insurance applications. Springer, New York
Wright EM (1954) An inequality for convex functions. Am Math Mon 61:620–622
Yaari ME (1987) The dual theory of choice under risk. Econometrica 55:95–115
Yang DW, Nadarajah S (2006) Drought modeling and products of random variables with exponential kernel. Stoch Environ Res Risk Assess 21:123–129
Zhang Y, Soong BH (2004) Channel holding time in hierarchical cellular systems. IEEE Commun Lett 8:614–616
Acknowledgments
The authors are sincerely grateful to the Editor-in-Chief and the referees for insightful comments that lead to improvements in the paper. Thanks are due to Prof. Concepción Paredes, Dr. Federico Dicenta, Dr. Encarnación Ortega, for generous assessment or useful references.
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Escudero, L.F., Ortega, E.M. & Alonso, J. Variability comparisons for some mixture models with stochastic environments in biosciences and engineering. Stoch Environ Res Risk Assess 24, 199–209 (2010). https://doi.org/10.1007/s00477-009-0310-6
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DOI: https://doi.org/10.1007/s00477-009-0310-6