Parameter Dependence in Stochastic Modeling—Multivariate Distributions
Affiliation(s)
Department of Mathematics and Computer Science, Oakton Community College, Des Plaines, USA. Department of Mathematics, Northeastern Illinois University, Chicago, USA.
Department of Mathematics and Computer Science, Oakton Community College, Des Plaines, USA. Department of Mathematics, Northeastern Illinois University, Chicago, USA.
ABSTRACT
We start with analyzing stochastic dependence in a classic bivariate normal density framework. We focus on the way the conditional density of one of the random variables depends on realizations of the other. In the bivariate normal case this dependence takes the form of a parameter (here the “expected value”) of one probability density depending continuously (here linearly) on realizations of the other random variable. The point is, that such a pattern does not need to be restricted to that classical case of the bivariate normal. We show that this paradigm can be generalized and viewed in ways that allows one to extend it far beyond the bivariate or multivariate normal probability distributions class.
KEYWORDS
Multivariate Probability Distributions, Stochastic Dependence Paradigms, Multivariate Gaussian Distributions, Parameter Dependence Method of Construction, Conditioning, Stress, Biomedical Applications
Multivariate Probability Distributions, Stochastic Dependence Paradigms, Multivariate Gaussian Distributions, Parameter Dependence Method of Construction, Conditioning, Stress, Biomedical Applications
Cite this paper
Filus, J. and Filus, L. (2014) Parameter Dependence in Stochastic Modeling—Multivariate Distributions. Applied Mathematics, 5, 928-940. doi: 10.4236/am.2014.56088.
Filus, J. and Filus, L. (2014) Parameter Dependence in Stochastic Modeling—Multivariate Distributions. Applied Mathematics, 5, 928-940. doi: 10.4236/am.2014.56088.
References
[1] Filus, J.K. and Filus, L.Z. (2013) A Method for Multivariate Probability Distributions Construction via Parameter Dependence. Communications in Statistics: Theory and Methods, 42, 716-721.
http://dx.doi.org/10.1080/03610926.2012.731549 [2] Filus, J.K. and Filus, L.Z. (2000) A Class of Generalized Multivariate Normal Densities. Pakistan Journal of Statistics, 16, 11-32. [3] Filus, J.K. and Filus, L.Z. (2007) On New Multivariate Probability Distributions and Stochastic Processes with Systems Reliability and Maintenance Applications. Methodology and Computing in Applied Probability, 9, 426-446, [4] Filus, J.K. and Filus, L.Z. (2006) On Some New Classes of Multivariate Probability Distributions. Pakistan Journal of Statistics, 1, 21-42. [5] Filus, J.K. and Filus, L.Z. (2008) On Multicomponent System Reliability with Microshocks-Microdamages Type of Components’ Interaction. Proceedings of the International Multiconference of Engineers and Computer Scientists, Lecture Notes in Engineering and Computer Science, Hong Kong, 19-21 March 2008, 1945-1951. [6] Filus, J.K. and Filus, L.Z. (2010) Weak Stochastic Dependence in Biomedical Applications. American Institute of Physics Conference Proceedings 1281, Numerical Analysis and Applied Mathematics, III, 1873-1876. [7] Kotz, S., Balakrishnan, N. and Johnson, N.L. (2000) Continuous Multivariate Distributions (Vol. 1). 2nd Edition, J. Wiley & Sons, Inc, New York, 217-218.
http://dx.doi.org/10.1002/0471722065 [8] Barlow, R.E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York. [9] Arnold, B.C., Castillo, E. and Sarabia, J.M. (1999) Conditional Specification of Statistical Models. Springer Series in Statistics, Springer Verlag, New York. [10] Castillo, E. and Galambos, J. (1990) Bivariate Distributions with Weibull Conditionals. Analysis Mathematica, 16, 3-9.
http://dx.doi.org/10.1007/BF01906769 [11] Filus, J.K., Filus, L.Z. (2008) Construction of New Continuous Stochastic Processes. Pakistan Journal of Statistics, 24, 227-251. [12] Filus, J.K., Filus, L.Z. and Arnold, B.C. (2010) Families of Multivariate Distributions Involving “Triangular” Transformations. Communications in Statistics—Theory and Methods, 39, 107-116. [13] Meeker, W.Q. and Escobar, L.A. (1998) Statistical Methods for Reliability Data. John Wiley & Sons, Inc., New York. [14] Nelson, W. (1990) Accelerated Testing: Statistical Models, Test Plans and Data Analysis. Wiley, New York.
http://dx.doi.org/10.1002/9780470316795 [15] (2013) www.ReliaWiki.org/index.php/Accelerated_Life_Testing_Data_Analysis_Reference [16] Cox, D.R. (1972) Regression Models and Life Tables (with Discussion). Journal of the Royal Statistical Society, B74, 187-220. [17] Collett, D. (2003) Modeling Survival Data in Medical Research. 2nd Edition, Chapman @ Hall/CRS A CRC Press Company, London, New York, Washington, D.C. [18] Filus, J.K., Filus, L.Z. and Krysiak, Z. (2013) Analytical Statistical and Simulation Models Utilized in Modeling the Risk in Finance. 5th International Conference on Risk Analysis, Tomar, Portugal, 30th May-1st June. [19] Filus, J.K. (1986) A Problem in Reliability Optimization. Journal of the Operational Research Society, 37, 407-412. [20] Filus, J.K. (1987) The Load Optimization of a Repairable System with Gamma Distributed Time-to-Failure. Reliability Engineering, 18, 275-284. http://dx.doi.org/10.1016/0143-8174(87)90032-1 [21] Levitin, G. and Amari, S.V. (2009) Optimal Load Distribution in Series-Parallel Systems. Reliability Engineering and System Safety, 94, 254-260.
http://dx.doi.org/10.1016/j.ress.2008.03.001 [22] Nourelfath, M. and Yalaoui, F. (2012) Integrated Load Distribution and Production Planning in Series-Parallel Multistate Systems. Reliability Engineering and System Safety, 106, 138-145. http://dx.doi.org/10.1016/j.ress.2012.06.006 [23] Freund, J.E. (1961) A Bivariate Extension of the Exponential Distribution. Journal of the American Statistical Association, 56, 971-977.
http://dx.doi.org/10.1080/01621459.1961.10482138 [24] Lu, J. (1989) Weibull Extensions of the Freund and Marshal-Olkins Bivariate Exponential Models. IEEE Transactions on Reliability, 38, 615-619.
http://dx.doi.org/10.1109/24.46492 [25] Filus, J.K. (1991) On a Type of Dependencies between Weibull Life times of System Components. Reliability Engineering and System Safety, 31s, 267-280.
http://dx.doi.org/10.1016/0951-8320(91)90071-E
[1] Filus, J.K. and Filus, L.Z. (2013) A Method for Multivariate Probability Distributions Construction via Parameter Dependence. Communications in Statistics: Theory and Methods, 42, 716-721.
http://dx.doi.org/10.1080/03610926.2012.731549 [2] Filus, J.K. and Filus, L.Z. (2000) A Class of Generalized Multivariate Normal Densities. Pakistan Journal of Statistics, 16, 11-32. [3] Filus, J.K. and Filus, L.Z. (2007) On New Multivariate Probability Distributions and Stochastic Processes with Systems Reliability and Maintenance Applications. Methodology and Computing in Applied Probability, 9, 426-446, [4] Filus, J.K. and Filus, L.Z. (2006) On Some New Classes of Multivariate Probability Distributions. Pakistan Journal of Statistics, 1, 21-42. [5] Filus, J.K. and Filus, L.Z. (2008) On Multicomponent System Reliability with Microshocks-Microdamages Type of Components’ Interaction. Proceedings of the International Multiconference of Engineers and Computer Scientists, Lecture Notes in Engineering and Computer Science, Hong Kong, 19-21 March 2008, 1945-1951. [6] Filus, J.K. and Filus, L.Z. (2010) Weak Stochastic Dependence in Biomedical Applications. American Institute of Physics Conference Proceedings 1281, Numerical Analysis and Applied Mathematics, III, 1873-1876. [7] Kotz, S., Balakrishnan, N. and Johnson, N.L. (2000) Continuous Multivariate Distributions (Vol. 1). 2nd Edition, J. Wiley & Sons, Inc, New York, 217-218.
http://dx.doi.org/10.1002/0471722065 [8] Barlow, R.E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York. [9] Arnold, B.C., Castillo, E. and Sarabia, J.M. (1999) Conditional Specification of Statistical Models. Springer Series in Statistics, Springer Verlag, New York. [10] Castillo, E. and Galambos, J. (1990) Bivariate Distributions with Weibull Conditionals. Analysis Mathematica, 16, 3-9.
http://dx.doi.org/10.1007/BF01906769 [11] Filus, J.K., Filus, L.Z. (2008) Construction of New Continuous Stochastic Processes. Pakistan Journal of Statistics, 24, 227-251. [12] Filus, J.K., Filus, L.Z. and Arnold, B.C. (2010) Families of Multivariate Distributions Involving “Triangular” Transformations. Communications in Statistics—Theory and Methods, 39, 107-116. [13] Meeker, W.Q. and Escobar, L.A. (1998) Statistical Methods for Reliability Data. John Wiley & Sons, Inc., New York. [14] Nelson, W. (1990) Accelerated Testing: Statistical Models, Test Plans and Data Analysis. Wiley, New York.
http://dx.doi.org/10.1002/9780470316795 [15] (2013) www.ReliaWiki.org/index.php/Accelerated_Life_Testing_Data_Analysis_Reference [16] Cox, D.R. (1972) Regression Models and Life Tables (with Discussion). Journal of the Royal Statistical Society, B74, 187-220. [17] Collett, D. (2003) Modeling Survival Data in Medical Research. 2nd Edition, Chapman @ Hall/CRS A CRC Press Company, London, New York, Washington, D.C. [18] Filus, J.K., Filus, L.Z. and Krysiak, Z. (2013) Analytical Statistical and Simulation Models Utilized in Modeling the Risk in Finance. 5th International Conference on Risk Analysis, Tomar, Portugal, 30th May-1st June. [19] Filus, J.K. (1986) A Problem in Reliability Optimization. Journal of the Operational Research Society, 37, 407-412. [20] Filus, J.K. (1987) The Load Optimization of a Repairable System with Gamma Distributed Time-to-Failure. Reliability Engineering, 18, 275-284. http://dx.doi.org/10.1016/0143-8174(87)90032-1 [21] Levitin, G. and Amari, S.V. (2009) Optimal Load Distribution in Series-Parallel Systems. Reliability Engineering and System Safety, 94, 254-260.
http://dx.doi.org/10.1016/j.ress.2008.03.001 [22] Nourelfath, M. and Yalaoui, F. (2012) Integrated Load Distribution and Production Planning in Series-Parallel Multistate Systems. Reliability Engineering and System Safety, 106, 138-145. http://dx.doi.org/10.1016/j.ress.2012.06.006 [23] Freund, J.E. (1961) A Bivariate Extension of the Exponential Distribution. Journal of the American Statistical Association, 56, 971-977.
http://dx.doi.org/10.1080/01621459.1961.10482138 [24] Lu, J. (1989) Weibull Extensions of the Freund and Marshal-Olkins Bivariate Exponential Models. IEEE Transactions on Reliability, 38, 615-619.
http://dx.doi.org/10.1109/24.46492 [25] Filus, J.K. (1991) On a Type of Dependencies between Weibull Life times of System Components. Reliability Engineering and System Safety, 31s, 267-280.
http://dx.doi.org/10.1016/0951-8320(91)90071-E