Non Uniform Bounds on Geometric Approximation Via Stein's Method and w-Functions

In this article, we use Stein's method and w-functions to give uniform and non uniform bounds in the geometric approximation of a non negative integer-valued random variable. We give some applications of the results of this approximation concerning the beta-geometric, Pólya, and Poisson distributions.     

Exponential bounds for normal approximation of the number of descents and inversions

In this article, we study two well-known statistics, called descent and inversion. The normal approximation by Stein’s method has been considered. Polynomial non uniform bounds of such approximation were obtained by Chuntee and Neammanee in 2013 Chuntee, W., Neammanee, K. (2013). Bounds on normal approximations for the number of descents and inversion. Commun. Stat. A - Theor.to appear. [Google Scholar]. In this article, we sharpen the bound to the exponential growth by using Stein’s method and techniques from the work of Chen, Shao, and Fang in the same year.     

Bounds on Normal Approximations for the number of Descents and Inversions

In this article, we use Stein’s method to give non uniform bounds for normal approximation for the number of descents and inversions. Moreover, we also improve a constant on uniform bounds for normal approximation of those random variables in Fulman’s work in 2004. Our constants are 13.42 and 14.24, respectively.     

Non uniform exponential bounds on normal approximation by Stein’s method and monotone size bias couplings

It is known that the normal approximation is applicable for sums of non negative random variables, W, with the commonly employed couplings. In this work, we use the Stein’s method to obtain a general theorem of non uniform exponential bound on normal approximation base on monotone size bias couplings of W. Applications of the main result to give the bound on normal approximation for binomial random variable, the number of bulbs on at the terminal time in the lightbulb process, and the number of m runs are also provided.     

Normal approximation for call function via Stein’s method

A Collateralized dept Obligation (CDO) is a cause of the Hamburger crisis in the USA. We use call function for pricing the CDO. In this paper, we first give uniform and non uniform bounds on normal approximation for the call function without correction term. In this part, we assume that the third and fourth moments of random variables exist, respectively. Second, we present uniform and non uniform bounds on normal approximation for the call function with a correction term under the assumption that the sixth moments of random variables is finite. Our techniques are Stein’s method and the zero bias transformation.     

Stein’s method and the distribution of the product of zero mean correlated normal random variables

Abstract

Over the last 80?years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables. Motivated by this historical interest, we use a recent technique from the Stein’s method literature to obtain a simple new proof, which also serves as an exposition of a general method that may be useful in related problems.     

Non uniform bound on a combinatorial central limit theorem

ABSTRACT

In this paper, we give a non uniform bound on combinatorial central limit theorem by using Stein’s method. This improves the result of Neammanee and Rattanawong (2009 Neammanee, K., Rattanawong, P. (2009). Non-uniform bound on normal approximation of Latin hypercube sampling. JMR 1:28–42. [Google Scholar]) from the finiteness of the sixth moment to the finiteness of the third moment.     

Half-normal approximation for statistics of symmetric simple random walk

In 2013, Döbler used Stein’s method to obtain the uniform bounds in half-normal approximation for three statistics of a symmetric simple random walk; the maximum value, the number of returns to the origin and the number of sign changes up to a given time n. In this paper, we give the non-uniform bounds for these statistics by using Stein’s method and the concentration inequality approach.     

Multiparameter estimation in truncated power series distributions under the stein's loss

LetX ₁,…,X _p be p(≥2)independent random variables, where each X.has a distribution belonging to a one parameter truncated power series

distribution. The problem is to estimate simultaneously the unknown parameters under asymmetric loss developed by James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob. 1, 361-380). Several new classes of dominating estimators are obtained by solving a certain difference inequality.     

Statistical analysis of the number of self-overlapping leftmost repeats in an homogeneous stationary Markov chain on finite states

ABSTRACT

This article addresses the problem of repeats detection used in the comparison of significant repeats in sequences. The case of self-overlapping leftmost repeats for large sequences generated by an homogeneous stationary Markov chain has not been treated in the literature. In this work, we are interested by the approximation of the number of self-overlapping leftmost long enough repeats distribution in an homogeneous stationary Markov chain. Using the Chen–Stein method, we show that the number of self-overlapping leftmost long enough repeats distribution is approximated by the Poisson distribution. Moreover, we show that this approximation can be extended to the case where the sequences are generated by a m-order Markov chain.     

The normal tempered stable regression model

Abstract

This paper deals with the statistical studies of the normal tempered stable model defined by Barndorff-Nielsen and Shephard. It represents the natural extension of the normal inverse Gaussian one introduced by Barndorff-Nielsen. We basically use the Monte-Carlo’s approximation in order to simulate this distribution. We introduce a linear regression model with normal tempered stable error. We apply this model for the analyzing of the daily logarithm returns data on CAC40 index. The parameters estimation results show that this model better deals with long tailed distribution which is the case for the CAC40 logarithm returns.     

Exact and approximate runs distributions

Let R = R_n denote the total (and unconditional) number of runs of successes or failures in a sequence of n Bernoulll (p) trials, where p is assumed to be known throughout. The exact distribution of R is related to a convolution of two negative binomial random variables with parameters p and q (=1-p). Using the representation of R as the sum of 1 - dependent indicators, a Berry - Esséen theorem is derived; the obtained rate of sup norm convergence is O(n^-½). This yields an unconditional version of the classical result of Wald and Wolfowitz (1940). The Stein - Chen method for m - dependent random variables is used, together with a suitable coupling, to prove a Poisson limit theorem for R. but with the limiting support set being the set of odd integers, Total variation error bounds (of order O(p) are found for the last result. Applications are indicated.     

On the property of multivariate generalized hyperbolic distribution and the Stein-type inequality

This note consists of two parts . In the first part, we provide a pedagogic review on the multivariate generalized hyperbolic (MGH) distribution. We show that this probability family is close under margining, conditioning, and linear transforms; however, such property does not hold for its subclasses. In the second part, we obtain the Stein-type inequality in the context of MGH distribution. Moreover, we apply the Stein-type inequality to prove a lower bound for Var[h(X)]. Particularly, we present examples when X belongs to some well-known subclasses in MGH family.     

Approximating moments of continuous functions of random variables using Bernstein polynomials

Bernstein polynomials have many interesting properties. In statistics, they were mainly used to estimate density functions and regression relationships. The main objective of this paper is to promote further use of Bernstein polynomials in statistics. This includes (1) providing a high-level approximation of the moments of a continuous function $g\left(X\right)$ of a random variable $X$, and (2) proving Jensen’s inequality concerning a convex function without requiring second differentiability of the function. The approximation in (1) is demonstrated to be quite superior to the delta method, which is used to approximate the variance of $g\left(X\right)$ with the added assumption of differentiability of the function. Two numerical examples are given to illustrate the application of the proposed methodology in (1).     

Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments

Approximate t-tests of single degree of freedom hypotheses in generalized least squares analyses (GLS) of mixed linear models using restricted maximum likelihood (REML) estimates of variance components have been previously developed by Giesbrecht and Burns (GB), and by Jeske and Harville (JH), using method of moment approximations for the degrees of freedom (df) for the tstatistics. This paper proposes approximate Fstatistics for tests of multiple df hypotheses using one-moment and two-moment approximations which may be viewed as extensions of the GB and JH methods. The paper focuses specifically on tests of hypotheses concerning the main-plot treatment factor in split-plot experiments with missing data. Simulation results indicate usually satisfactory control of Type I error rates.