Consequently, our results hold substantial generality, enabling the extension of Rice’s formula to address specific scenarios within these subclasses.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:In this paper, we apply Rice’s formula, typically employed to calculate the mean number of upcrossings for stationary Gaussian processes, and extend it to the broader framework of generalized mixtures of Gaussian processes.
📊
Key Findings & Data:The class of generalized mixtures of Gaussian distributions, recently introduced by [3], is highly comprehensive and includes significant subclasses such as mean mixtures of Gaussian, variance mixtures of Gaussian, meanvariance mixtures of Gaussian, and even scale mixtures of skew-Gaussian distributions.
💡
Conclusion & Impact:Consequently, our results hold substantial generality, enabling the extension of Rice’s formula to address specific scenarios within these subclasses.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Crossings · generalized mixture of Gaussian distribution · Rice’s formula
Abstract
In this paper, we apply Rice’s formula, typically employed to calculate the mean number of upcrossings for stationary Gaussian processes, and extend it to the broader framework of generalized mixtures of Gaussian processes. The class of generalized mixtures of Gaussian distributions, recently introduced by [3], is highly comprehensive and includes significant subclasses such as mean mixtures of Gaussian, variance mixtures of Gaussian, meanvariance mixtures of Gaussian, and even scale mixtures of skew-Gaussian distributions. Consequently, our results hold substantial generality, enabling the extension of Rice’s formula to address specific scenarios within these subclasses.
Desmond, A., Mardani-fard, H. A., Jamalizadeh, A., Roozegar, R. (2025). Generalized mixtures of Gaussian processes with an application to Bitcoin daily price analysis. Hacettepe Journal of Mathematics and Statistics, Volume: 54 Issue: 1. https://doi.org/10.15672/hujms.1472824
K. Aas and I.H. Haff, The generalized hyperbolic skew Student’s t-distribution, J. Financ. Econ. 4, 275-309, 2006.
M. Abdi, M. Madadi, N. Balakrishnan and A. Jamalizadeh, Family of mean-mixtures of multivariate normal distributions: Properties, inference and assessment of multivariate skewness, J. Multivar. Anal. 181, 104679, 2021.
R.B. Arellano-Valle and A. Azzalini, A formulation for continuous mixtures of multivariate Gaussian distributions, J. Multivar. Anal. 185, 104780, 2021.
A. Azzalini, A class of distributions which includes the Gaussian ones, Scand. J. Stat. 12, 171–178, 1985.
M.D. Branco and D.K. Dey, A general class of multivariate skew-elliptical distributions, J. Multivar. Anal. 79(1), 99–113, 2001.
O. Barndorff-Nielsen, J. Kent and M. Sorensen, Normal variance-mean mixtures and Z distributions, Int. Stat. Rev. 50, 145–159, 1982.
M.L. Bianchi and G.L. Tassinari, Estimation for multivariate normal rapidly decreasing tempered stable distributions, J. Stat. Comput. Simul. 94(1), 103–125, 2024.
A.F. Desmond and B.T. Guy, Crossing theory for non-Gaussian stochastic processes with an application to hydrology, Water Resour. Res. 27(10), 2791–2797, 1991.
P. Embrechts, C. Kluppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, New York, 1997.
C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, 4th Edition, Springer, New York, 2009.
C. Klüppelberg and M.G. Rasmussen, Outcrossings of safe regions by generalized hyperbolic processes, Stat. Probab. Lett. 83, 2197–2204, 2013.
G. Lindgren, Values and crossings for the 2-process and other functions of multidimensional Gaussian processes, with reliability applications, Adv. Appl. Probab. 12(3), 746–774, 1980.
G. Lindgren, Stationary Stochastic Processes: Theory and Applications, In: Texts in Statistical Science, Chapman & Hall/CRC, 2012.
G. Lindgren, Gaussian Integrals and Rice Series in Crossing Distributions–to Compute the Distribution of Maxima and Other Features of Gaussian Processes, Stat. Sci. 34, 100–128, 2019.
G. Lindgren and I. Rychlik, Slepian Models and Regression Approximations in Crossing and Extreme Value Theory, Int. Stat. Rev. 59, 195-225, 1991.
M. Mahfoud and G.P. Patil, On weighted distributions, In G. Kallianpur et al. (eds.) Statistics and Probability Essays in Honor of C. R. Rao, Amsterdam: North Holland, 479-492, 1982.
J. Masoliver and M. Palassini, Counting of level crossings for inertial random processes: Generalization of the Rice formula, Phys. Rev. E. 107, 024-111, 2023.
M. Naderi, W.L. Hung, T.I. Lin and A. Jamalizadeh, A novel mixture model using the multivariate normal mean–variance mixture of Birnbaum Saunders distributions and its application to extrasolar planets, J. Multivar. Anal. 171, 126-138, 2019.
H. Negarestani, A. Jamalizadeh, S. Shafiei and N. Balakrishnan,Mean mixtures of Gaussian distributions: properties, inference and application, Metrika. 82, 501-528, 2019.
C. F. Nordin and D. M. Rosbjerg, Applicationsof crossingtheory in hydrology, Bull. Int. Assoc. Sci. Hydrol. 15, 27-43, 1970.
C. R. Rao, On discrete distributions arising out of methods of ascertainment, Sankhya A. 27, 311-324, 1965.
S.O. Rice, Mathematical analysis of random noise, Bell Syst. Tech. J. 23, 282–332, 1944.
S.O. Rice, Mathematical analysis of random noise, Bell Syst. Tech. J. 24, 46-156, 1945.
I. Rychlik, Note on Cycle Counts in Irregular Loads, Fatigue Fract. Eng. Mater. Struct. 16, 377-390, 1993.
I. Rychlik, On Some Reliability Applications of Rice Formula for Intensity of Level Crossings, Extremes. 3, 331-348, 2000.
F. Vilca, N. Balakrishnan and C. Zeller, Multivariate Skew-Normal Generalized Hyperbolic distribution and its properties, J. Multivar. Anal. 128, 73-85, 2014.