218 research outputs found
Variance-Gamma approximation via Stein's method
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
, where the and 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
for smooth test functions. We end with a simple application to binary sequence
comparison.Comment: 39 pages. Published Versio
Wasserstein and Kolmogorov error bounds for variance-gamma approximation via Stein's method I
The variance-gamma (VG) distributions form a four parameter family that
includes as special and limiting cases the normal, gamma and Laplace
distributions. Some of the numerous applications include financial modelling
and approximation on Wiener space. Recently, Stein's method has been extended
to the VG distribution. However, technical difficulties have meant that bounds
for distributional approximations have only been given for smooth test
functions (typically requiring at least two derivatives for the test function).
In this paper, which deals with symmetric variance-gamma (SVG) distributions,
and a companion paper \cite{gaunt vgii}, which deals with the whole family of
VG distributions, we address this issue. In this paper, we obtain new bounds
for the derivatives of the solution of the SVG Stein equation, which allow for
approximations to be made in the Kolmogorov and Wasserstein metrics, and also
introduce a distributional transformation that is natural in the context of SVG
approximation. We apply this theory to obtain Wasserstein or Kolmogorov error
bounds for SVG approximation in four settings: comparison of VG and SVG
distributions, SVG approximation of functionals of isonormal Gaussian
processes, SVG approximation of a statistic for binary sequence comparison, and
Laplace approximation of a random sum of independent mean zero random
variables.Comment: 37 pages, to appear in Journal of Theoretical Probability, 2018
Rates of convergence in normal approximation under moment conditions via new bounds on solutions of the Stein equation
New bounds for the -th order derivatives of the solutions of the normal
and multivariate normal Stein equations are obtained. Our general order bounds
involve fewer derivatives of the test function than those in the existing
literature. We apply these bounds and local approach couplings to obtain an
order bound, for smooth test functions, for the distance between
the distribution of a standardised sum of independent and identically
distributed random variables and the standard normal distribution when the
first moments of these distributions agree. We also obtain a bound on the
convergence rate of a sequence of distributions to the normal distribution when
the moment sequence converges to normal moments.Comment: 16 pages. Final version. To appear in Journal of Theoretical
Probabilit
Derivative formulas for Bessel, Struve and Anger-Weber functions
We derive formulas for the derivatives of general order for the functions
and , where is a Bessel,
Struve or Anger--Weber function.Comment: 9 page
Inequalities for modified Bessel functions and their integrals
Simple inequalities for some integrals involving the modified Bessel
functions and are established. We also obtain a
monotonicity result for and a new lower bound, that involves gamma
functions, for .Comment: 13 pages. Final version. To appear in Journal of Mathematical
Analysis and Application
Inequalities for integrals of the modified Struve function of the first kind
Simple inequalities for some integrals involving the modified Struve function
of the first kind are established. In most cases, these
inequalities have best possible constant. We also deduce a tight double
inequality, involving the modified Struve function , for a
generalized hypergeometric function.Comment: 9 pages. To appear in Results in Mathematics, 2018
Inequalities for integrals of modified Bessel functions and expressions involving them
Simple inequalities are established for some integrals involving the modified
Bessel functions of the first and second kind. In most cases, we show that we
obtain the best possible constant or that our bounds are tight in certain
limits. We apply these inequalities to obtain uniform bounds for several
expressions involving integrals of modified Bessel functions. Such expressions
occur in Stein's method for variance-gamma approximation, and the results
obtained in this paper allow for technical advances in the method. We also
present some open problems that arise from this research.Comment: 20 pages. Final version. To appear in Journal of Mathematical
Analysis and Application
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