Expressions for Marginal Mean Excess and Marginal Expected Shortfall Measures under Bivariate Scale Mixture of Normal Distribution

Roohollah Roozegar (), Narayanaswamy Balakrishnan, Heydar Ali Mardani-Fard, Anthony F. Desmond and Ahad Jamalizadeh
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Roohollah Roozegar: Yasouj University
Narayanaswamy Balakrishnan: McMaster University
Heydar Ali Mardani-Fard: Yasouj University
Anthony F. Desmond: University of Guelph
Ahad Jamalizadeh: Shahid Bahonar University

Methodology and Computing in Applied Probability, 2025, vol. 27, issue 2, 1-18

Abstract: Abstract Here two important risk measures–marginal expected shortfall (MES) and marginal mean excess (MME)–for bivariate risk vectors $$(Y_{1},Y_{2})$$ ( Y 1 , Y 2 ) are studied. Usually, deriving explicitly these measures is challenging and is done through asymptotic methods. In this paper, we derive explicit expressions for these measures when the joint risk factor $$(Y_{1},Y_{2})$$ ( Y 1 , Y 2 ) follows a bivariate normal distribution. As risk factors commonly exhibit heavy-tailed behavior, we extend our findings to attain exact expressions for MES and MME, under scale mixture of normal (SMN) risk factors. This class include important distributions, such as symmetric generalized hyperbolic (SGH) and Student- $$t$$ t distributions, and the established results are extended to include these subclasses.

Keywords: Extended skew-normal distribution; Extended skew-scale mixture of normal distribution; Marginal expected shortfall; Marginal mean excess; Systemic risk; 62E15; 91G70 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s11009-025-10156-8

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