Tail conditional moments for elliptical and log-elliptical distributions. (English) Zbl 1371.60041
Summary: In this paper we provide the tail conditional moments for the class of elliptical distributions, which was introduced in [D. Kelker, Sankhyā, Ser. A 32, 419–430 (1970; Zbl 0223.60008)] and was widely discussed in [A. K. Gupta et al., Elliptically contoured models in statistics and portfolio theory. 2nd ed. New York, NY: Springer (2013; Zbl 1306.62028)] and for the class of log-elliptical distributions. These families of distributions include some important members such as the normal, Student-t, logistic, Laplace, and log-normal distributions. We give analytic formulae for the \(n\)th higher order unconditional moments of elliptical distributions, which has not been provided before. We also propose novel risk measures, the tail conditional skewness and the tail conditional kurtosis, for examining the skewness and the kurtosis of the tail of loss distributions, respectively.
MSC:
| 60E05 | Probability distributions: general theory |
| 91B30 | Risk theory, insurance (MSC2010) |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
Keywords:
elliptical distributions; log-elliptical distributions; tail conditional expectation; tail conditional moments; tail varianceReferences:
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