# zbMATH — the first resource for mathematics

Generalized skew-elliptical distributions and their quadratic forms. (English) Zbl 1083.62043
Summary: This paper introduces generalized skew-elliptical distributions (GSE), which include the multivariate skew-normal, skew-$$t$$, skew-Cauchy, and skew-elliptical distributions as special cases. GSE are weighted elliptical distributions but the distribution of any even function in GSE random vectors does not depend on the weight function. In particular, this holds for quadratic forms in GSE random vectors. This property is beneficial for inference from non-random samples. We illustrate the latter point on a data set of Australian athletes.

##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H12 Estimation in multivariate analysis
Full Text:
##### References:
 [1] Anscombe, F. J. (1950). Sampling theory of the negative binomial and logarithmic series distributions,Biometrika,37, 358–382. · Zbl 0039.14202 [2] Breiman, L. (1968).Probability, Addison-Wesley, Reading, Massachusetts. · Zbl 0174.48801 [3] Bunge, J. and Fitzpatrick, M. (1993). Estimating the number of species: A review,Journal of the American Statistical Association,88, 364–373. [4] Charalambides, C. A. and Singh, J. (1988). A review of the Stirling numbers, their generalizations and statistical applications,Communications in Statistics, Theory and Methods,17, 2533–2595. · Zbl 0696.62025 [5] Engen, S. (1974). On species frequency models,Biometrika,61, 263–270. · Zbl 0281.62062 [6] Engen, S. (1977). Comments on two different approaches to the analysis of species frequency data,Biometrics,33, 205–213. · Zbl 0351.92007 [7] Engen, S. (1978).Stochastic Abundance Models, Chapman and Hall, London. · Zbl 0429.62075 [8] Ewens, W. J. (1972). The sampling theory of selectively neutral alleles,Theoretical Population Biology,3, 87–112. · Zbl 0245.92009 [9] Feller, W. (1957).An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed., Wiley, New York. · Zbl 0077.12201 [10] Fisher, R. A., Corbet, A. S. and Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population,Journal of Animal Ecology,12, 42–58. [11] Good, I. J. (1953). The population frequencies of species and the estimation of population parameters,Biometrika,40, 237–264. · Zbl 0051.37103 [12] Gupta, R. C. (1974). Modified power series distributions and some of its applications,Sankhyā B,36, 288–298. · Zbl 0318.62009 [13] Hoshino, N. (2001). Applying Pitman’s sampling formula to microdata disclosure risk assessment,Journal of Official Statistics,17, 499–520. [14] Hoshino, N. (2002). On limiting random partition structure derived from the conditional inverse Gaussian-Poisson distribution, Technical Report, CMU-CALD-02-100, School of Computer Science, Carnegie Mellon University. [15] Hoshino, N. (2003). Random clustering based on the conditional inverse Gaussian-Poisson distribution,Journal of the Japan Statistical Society,33, 105–117. · Zbl 1023.62014 [16] Hoshino, N. and Takemura, A. (1998). Relationship between logarithmic series model and other superpopulation models useful for microdata disclosure risk assessment,Journal of the Japan Statistical Society,28(2), 125–134. · Zbl 1008.62704 [17] Johnson, N. L., Kotz, S. and Kemp, A. W. (1993).Univariate Discrete Distributions, 2nd ed., Wiley, New York. · Zbl 1092.62010 [18] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1997).Discrete Multivariate Distributions, Wiley, New York. · Zbl 0868.62048 [19] Jørgensen, B. (1982).Statistical Properties of the Generalized Inverse Gaussian Distribution, Lecture Notes in Statistics, No. 9, Springer, New York. · Zbl 0486.62022 [20] Kemp, A. W. (1978). Cluster size probabilities for generalized Poisson distributions,Communications in Statistics, Theory and Methods,7, 1433–1438. · Zbl 0394.62045 [21] Khatri, C. G. and Patel, I. R. (1961). Three classes of univariate discrete distributions,Biometrics,17, 567–575. · Zbl 0111.15904 [22] Mandelbrot, B. B. (1983).The Fractal Geometry of Nature, W. H. Freeman and Company, New York. · Zbl 1194.30028 [23] Mehninick, E. F. (1964). A comparison of some species individuals diversity indices applied to samples of field insects.Ecology,45, 859–861. [24] Mosimann, J. E. (1962). On the compound multinomial distribution, the multivariate {$$\beta$$}-distribution and correlations among proportions,Biometrika,49, 65–82. · Zbl 0105.12502 [25] Noack, A. (1950). A class of random variables with discrete distributions,Annals of Mathematical Statistics,21, 127–132. · Zbl 0036.08601 [26] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions,Probability Theory and Related Fields,102, 145–158. · Zbl 0821.60047 [27] Seshadri, V. (1999).The Inverse Gaussian Distribution, Springer, New York. · Zbl 0942.62011 [28] Sibuya, M. (1979). Generalized hypergeometric, digamma and trigamma distribution,Annals of the Institute of Statistical Mathematics,31, 373–390. · Zbl 0448.62008 [29] Sibuya, M. (1993). A random clustering process,Annals of the Institute of Statistical Mathematics,45, 459–465. · Zbl 0802.60010 [30] Sibuya, M., Yoshimura, M. and Shimizu, R. (1964). Negative multinomial distribution,Annals of the Institute of Statistical Mathematics,16, 409–426. · Zbl 0146.39303 [31] Sichel, H. S. (1971). On a family of discrete distributions particularly suited to represent long-tailed frequency data,Proceedings of the Third Symposium on Mathematical Statistics (ed. N. F. Laubscher),S.A. C.S.I.R., Pretoria, 51–97. · Zbl 0274.60012 [32] Sichel, H. S. (1974). On a distribution representing sentence-length in written prose,Journal of the Royal Statistical Society, Ser. A,137, 25–34. [33] Sichel, H. S. (1992). Anatomy of the generalized inverse Gaussian-Poisson distribution with special applications to bibliometric studies,Information Processing and Management,28, 5–17. [34] Sichel, H. S. (1997). Modelling species-abundance frequencies and species-individual functions with the generalized inverse Gaussian-Poisson distribution,South African Statistical Journal,31, 13–37. · Zbl 0888.62115 [35] Steutel, F. W. and van Harn, K. (1979). Discrete analogues of self-decomposability and stability,Annals of Probability,7, 893–899. · Zbl 0418.60020 [36] Watterson, G. A. (1974). Models for the logarithmic species abundance distributions,Theoretical Population Biology,6, 217–250. · Zbl 0292.92003 [37] Willmot, G. E. (1986). Mixed compound Poisson distributions,ASTIN Bulletin,16, S59-S79. [38] Willmot, G. E. (1989). A remark on the Poisson-Pascal and some other contagious distributions,Statistics and Probability Letters,7, 217–220. · Zbl 0662.62011 [39] Yamato, H., Sibuya, M. and Nomachi, T. (2001). Ordered sample from two-parameter GEM distribution,Statistics and Probability Letters,55, 19–27. · Zbl 1003.62008 [40] Zipf, G. K. (1949).Human Behavior and the Principle of Least Effort, Addison-Wesley, Cambridge, Massachusetts.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.