3,596 research outputs found
On Stein's method for products of normal random variables and zero bias couplings
In this paper we extend Stein's method to the distribution of the product of
independent mean zero normal random variables. A Stein equation is obtained
for this class of distributions, which reduces to the classical normal Stein
equation in the case . This Stein equation motivates a generalisation of
the zero bias transformation. We establish properties of this new
transformation, and illustrate how they may be used together with the Stein
equation to assess distributional distances for statistics that are
asymptotically distributed as the product of independent central normal random
variables. We end by proving some product normal approximation theorems.Comment: 34 pages. To appear in Bernoulli, 2016
Approximation for absolutely continuous functions by Stancu Beta operators
In this paper, we obtain an exact estimate for the first-order absolute moment of Stancu Beta operators by means of the Stirling formula and integral operations. Then we use this estimate for establishing a theorem on approximation of absolutely continuous functions by Stancu Beta operators.Отримано точну оцiнку для абсолютного моменту бета-операторiв Станку першого порядку iз використанням формули Стiрлiнга та iнтегральних операцiй. Цю оцiнку використано для встановлення теореми про наближення абсолютно неперервних функцiй бета-операторами Станку
Solving Dynamic Discrete Choice Models Using Smoothing and Sieve Methods
We propose to combine smoothing, simulations and sieve approximations to
solve for either the integrated or expected value function in a general class
of dynamic discrete choice (DDC) models. We use importance sampling to
approximate the Bellman operators defining the two functions. The random
Bellman operators, and therefore also the corresponding solutions, are
generally non-smooth which is undesirable. To circumvent this issue, we
introduce a smoothed version of the random Bellman operator and solve for the
corresponding smoothed value function using sieve methods. We show that one can
avoid using sieves by generalizing and adapting the `self-approximating' method
of Rust (1997) to our setting. We provide an asymptotic theory for the
approximate solutions and show that they converge with root-N-rate, where
is number of Monte Carlo draws, towards Gaussian processes. We examine their
performance in practice through a set of numerical experiments and find that
both methods perform well with the sieve method being particularly attractive
in terms of computational speed and accuracy
A Stein characterisation of the generalized hyperbolic distribution
The generalized hyperbolic (GH) distributions form a five parameter family of
probability distributions that includes many standard distributions as special
or limiting cases, such as the generalized inverse Gaussian distribution,
Student's -distribution and the variance-gamma distribution, and thus the
normal, gamma and Laplace distributions. In this paper, we consider the GH
distribution in the context of Stein's method. In particular, we obtain a Stein
characterisation of the GH distribution that leads to a Stein equation for the
GH distribution. This Stein equation reduces to the Stein equations from the
current literature for the aforementioned distributions that arise as limiting
cases of the GH superclass.Comment: 19 pages, to appear in ESAIM: Probability and Statistics, 2017
Stein's method for comparison of univariate distributions
We propose a new general version of Stein's method for univariate
distributions. In particular we propose a canonical definition of the Stein
operator of a probability distribution {which is based on a linear difference
or differential-type operator}. The resulting Stein identity highlights the
unifying theme behind the literature on Stein's method (both for continuous and
discrete distributions). Viewing the Stein operator as an operator acting on
pairs of functions, we provide an extensive toolkit for distributional
comparisons. Several abstract approximation theorems are provided. Our approach
is illustrated for comparison of several pairs of distributions : normal vs
normal, sums of independent Rademacher vs normal, normal vs Student, and
maximum of random variables vs exponential, Frechet and Gumbel.Comment: 41 page
Stein's method on the second Wiener chaos : 2-Wasserstein distance
In the first part of the paper we use a new Fourier technique to obtain a
Stein characterizations for random variables in the second Wiener chaos. We
provide the connection between this result and similar conclusions that can be
derived using Malliavin calculus. We also introduce a new form of discrepancy
which we use, in the second part of the paper, to provide bounds on the
2-Wasserstein distance between linear combinations of independent centered
random variables. Our method of proof is entirely original. In particular it
does not rely on estimation of bounds on solutions of the so-called Stein
equations at the heart of Stein's method. We provide several applications, and
discuss comparison with recent similar results on the same topic
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