925 research outputs found
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
An accurate approximation formula for gamma function
In this paper, we present a very accurate approximation for gamma function:
\begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi x}\left(
\dfrac{x}{e}\right) ^{x}\left( x\sinh \frac{1}{x}\right) ^{x/2}\exp \left(
\frac{7}{324}\frac{1}{ x^{3}\left( 35x^{2}+33\right) }\right) =W_{2}\left(
x\right) \end{equation*} as , and prove that the function
is strictly
decreasing and convex from onto , where \begin{equation*} \beta =\frac{22\,025}{22\,032}-\ln
\sqrt{2\pi \sinh 1}\approx 0.00002407. \end{equation*}Comment: 9 page
Two asymptotic expansions for gamma function developed by Windschitl's formula
In this paper, we develop Windschitl's approximation formula for the gamma
function to two asymptotic expansions by using a little known power series. In
particular, for with , we have \begin{equation*}
\Gamma \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left(
x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1}\tfrac{\left(
2k\left( 2k-2\right) !-2^{2k-1}\right) B_{2k}}{2k\left( 2k\right) !x^{2k-1}}
+R_{n}\left( x\right) \right) \end{equation*} with \begin{equation*} \left|
R_{n}\left( x\right) \right| \leq \frac{\left| B_{2n}\right| }{2n\left(
2n-1\right) }\frac{1}{x^{2n-1}} \end{equation*} for all , where
is the Bernoulli number. Moreover, we present some approximation formulas for
gamma function related to Windschitl's approximation one, which have higher
accuracy.Comment: 14 page
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
Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion
The Negative Binomial distribution becomes highly skewed under extreme
dispersion. Even at moderately large sample sizes, the sample mean exhibits a
heavy right tail. The standard Normal approximation often does not provide
adequate inferences about the data's mean in this setting. In previous work, we
have examined alternative methods of generating confidence intervals for the
expected value. These methods were based upon Gamma and Chi Square
approximations or tail probability bounds such as Bernstein's Inequality. We
now propose growth estimators of the Negative Binomial mean. Under high
dispersion, zero values are likely to be overrepresented in the data. A growth
estimator constructs a Normal-style confidence interval by effectively removing
a small, pre--determined number of zeros from the data. We propose growth
estimators based upon multiplicative adjustments of the sample mean and direct
removal of zeros from the sample. These methods do not require estimating the
nuisance dispersion parameter. We will demonstrate that the growth estimators'
confidence intervals provide improved coverage over a wide range of parameter
values and asymptotically converge to the sample mean. Interestingly, the
proposed methods succeed despite adding both bias and variance to the Normal
approximation
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