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has been cited by the following article:

• TITLE: Value at Risk and Expected Shortfall for Normal Variance Mean Mixtures of Finite Weighted Inverse Gaussian Distributions

  AUTHORS: Calvin B. Maina, Patrick G. O. Weke, Carolyne A. Ogutu, Joseph A. M. Ottieno

  KEYWORDS: Risk Measures, Backtesting, Weighted Distribution, Normal Mixture, EM-Algorithm

  JOURNAL NAME: Journal of Mathematical Finance, Vol.12 No.1, February 16, 2022

  ABSTRACT: The Normal Inverse Gaussian (NIG) distribution, a special case of the Generalized Hyperbolic Distribution (GHD) has been frequently used for financial modelling and risk measures. In this work, we consider other normal Variance mean mixtures based on finite mixtures of special cases for Generalised Inverse Gaussian as mixing distributions. The Expectation-Maximization (EM) algorithm has been used to obtain the Maximum Likelihood (ML) estimates of the proposed models for some financial data. We estimate Value at risk (VaR) and Expected Shortfall (ES) for the fitted models. The Kupiec likelihood ratio (LR) has been applied for backtesting of VaR. Akaike Information Creterion (AIC), Bayesian Information Creterion (BIC) and Log-likelihood have been used for model selection. The results clearly show that the proposed models are good alternatives to NIG for determining VaR and ES.