Variance-Gamma approximation via Stein's method

By Robert E. Gaunt. 2013

107 citations recorded by the source corpus.

Variance-Gamma distributions are widely used in financial modelling and contain as special cases the normal, Gamma and Laplace distributions. In this paper we extend Stein's method to this class of distributions. In particular, we obtain a Stein equation and smoothness estimates for its solution. This Stein equation has the attractive property of reducing to the known normal and Gamma Stein equations for certain parameter values. We apply these results and local couplings to bound the distance between sums of the form $\sum_{i,j,k=1}^{m,n,r}X_{ik}Y_{jk}$, where the $X_{ik}$ and $Y_{jk}$ are independent and identically distributed random variables with zero mean, by their limiting Variance-Gamma distribution. Through the use of novel symmetry arguments, we obtain a bound on the distance that is of order $m^{-1}+n^{-1}$ for smooth test functions. We end with a simple application to binary sequence comparison.

Subfields: Algebra and Number Theory, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8fea74c05cc5dfea76a08d0f2e977feaac4e31a8

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Citations in the corpus

Chi-square approximation by Stein's method with application to Pearson's statistic (2015, methodology): “Over the years, Stein’s method has been extended to many other distributions, such as the Poisson [5], multinomial [20], exponential [4], [27], Laplace [29], variance-gamma [14], and the gamma distribution [21], [25], which we develop further in this paper.”
Stein's method for comparison of univariate distributions (2014, methodology): “Outside of the Gaussian and Poisson frameworks, for univariate distributions the method has now also been shown to be effective for: exponential approximation [19, 79], Gamma approximation [68, 81, 70], binomial approximation [31], Beta approximation [43, 28], the asymptotics of rank distributions [”
On Stein's method for products of normal random variables and zero bias couplings (2013, methodology): “It should also be noted that the PN(2, σ(2)) distribution is a member of the class of variance-gamma distributions, and that (9) is indeed a special case of the variance-gamma Stein equation, recently introduced by Gaunt [16].”
The Variance-Gamma Distribution: A Review (2023, unknown): “VG approximations for double Wiener-Itô integrals have been studied by Azmoodeh, Eichelsbacher and Thäle [7], Eichelsbacher and Thäle [28] and Gaunt [41] via Stein’s method for VG approximation, which was first developed by Gaunt [36], for which the starting point is the Stein characterisation (2.”