Abstract
The goal of this chapter is to recover the distribution function of a counting random variable representing a countable event defined on a set of multidimensional variables, whose dependence structure is known. After a review of the existing main contributions in counting statistics, we will introduce our new approach to the problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This family is a set of probability distributions that represents a generalization of the natural exponential family. A lot of very common probability distributions belong to the class of exponential dispersion family, among them are normal distribution, binomial distribution, Poisson distribution, negative-binomial distribution, gamma distribution, inverse Gaussian distribution, and Tweedie distribution.
- 2.
We refer to \(\hat {D}(i,k)\) as the number of the ways in which one can distribute the integer i into k groups without taking into account the order of the groups (c.ds for short). If we take into account the groups’ cardinalities, then we refer to \(\hat {D}^c(i,k)\) as the number of the compatible (for cardinalities’ groups) combinatorial distributions.
- 3.
Following this calling order, the c.c.d. considered here is the second one, i.e. j = 2.
- 4.
The concept of compatibility allows to restrict our interest to the scenarios distributing a number of events that doesn’t exceed the cardinality of the group itself.
- 5.
The rule which explains how the coordinates are generated is presented in the following Corollary.
- 6.
In the case of empty set, we delete the correspondent vector element. For example, if the set of elements equal to one into the s-th group and for the j-th c.c.d. is empty, we correspond only \(u_{s,2}^j\) to \(u_{s,w}^j,\forall w\) and \(v_{s,2}^j\) to \(v_{s,w}^j,\forall w\).
- 7.
We assume that this c.c.ds are compatibles with the equivalent cardinalities of groups determined by the clustering method and then compensated for the homogeneous assumption.
- 8.
The representation has been done smoothing and interpolating the points process, for graphical convenience.
References
Bernardi, E., & Romagnoli, S. (2011). Computing the volume of an high-dimensional semi-unsupervised hierarchical copula. International Journal of Computer Mathematics, 88(12), 2591–2607.
Bernardi, E., & Romagnoli, S. (2013). A clusterized copula-based probability distribution of a counting variable for high-dimensional problems. The Journal of Credit Risk, 9(2), 3–26.
Bortkiewicz, L. J. (1898). The Law of Small Numbers. Leipzig: B.G. Teubner.
Cameron, A. C., & Trivedi, P. K. (1998). Regression Analysis of Count Data. (Econometric Society Monograph, Vol. 30). Cambridge: Cambridge University Press.
Consul, P. C., & Jain, G. C. (1973). A generalization of the Poisson distribution. Technometrics, 15(4), 791–799.
Davis, M., & Lo, V. (2001). Infectious defaults. Quantitative Finance, 1, 382–386.
Driessen, J. (2005). Is default event risk priced in corporate bonds? Review of Financial Studies, 18, 165–195.
Duffee, G. R. (1999). Estimating the price of default risk. The Review of Financial Studies, 12(1), 197–226.
Duffie, D., & Singleton, K. (1999). Simulating correlated defaults. Stanford University, working paper.
Elizalde, A. (2005). Credit Risk Models I: Default Correlation in Intensity Models. working paper.
Greenwood, M., & Yule, G. U. (1920). An inquiry in to the nature of frequency distributions of multiple happenings, with particular reference to the occurrence of multiple attacks of disease or repeated accidents. Journal of the Royal Statistical Society, 83, 255–279.
Jarrow, R., & Yu, F. (2001). Counterparty risk and the pricing of defaultable securities. Journal of Finance, 56, 1765–1799.
Joe, H., & Zhu, R. (2005). Generalized Poisson distribution: The property of mixture of Poisson and comparison with negative binomial distribution. Biometrical Journal, 47, 219–229.
Jørgensen, B., & De Souza, M. C. P. (1994). Fitting Tweedie’s compound Poisson model to insurance claims data. Scandinavian Actuarial Journal, 1994, 69–93.
Kemp, C. D. (1967). Stuttering-Poisson distributions. Journal of the Statistical Enquiry of Ireland, 21(5), 151–157.
Lomnitz, C. (1994). Fundamentals of Earthquake Prediction. New York: Wiley.
Mack, T. (1997). Measuring the variability of chain ladder reserve estimates. CCasualty Actuarial Society Forum, Spring 1994.
McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models. London: Chapman and Hall.
Poisson, S. D. (1837). Probabilité des jugements en matière criminelle et en matière civile, précédées des règle générales du calcul des probabilitiés. Paris: Bachelier.
Renshaw, A. E., & Verrall, R. J. (1998). A stochastic model underlying the chain-ladder technique. British Actuarial Journal, 4, 903–923.
Revfeim, K. J. A. (1984). An initial model of the relationship between rainfall events and daily rainfalls. Journal of Hydrology, 75, 357–364.
Thompson, C. S. (1984). Homogeneity analysis of a rainfall series: An application of the use of a realistic rainfall model. Journal of Climatology, 4, 609–619.
Tweedie, M. C. K. (1984). An index which distinguishes between some important exponential families. In J. K. Ghosh & J. Roy (Eds.), Applications and New Directions. Proceeding of the Indian Statistical Golden Jubilee International Conference (pp. 579–604). Calcutta: Indian Statistical Institute.
Wüthrich, M. V. (2003). Claims reserving using Tweedie’s compound Poisson model. Astin Bulletin, 33(2), 331–346.
Zang, F. X. (2003). What did the credit market expect of Argentina default? Evidence from Default Swap Data. working paper, Federal Reserve Board, Division of Research and Statistics, Washington.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bernardi, E., Romagnoli, S. (2021). Counting a Random Event: Traditional Approach and New Perspectives. In: Counting Statistics for Dependent Random Events. Springer, Cham. https://doi.org/10.1007/978-3-030-64250-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-64250-1_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-64249-5
Online ISBN: 978-3-030-64250-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)