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.     

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.     

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.     

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.     

A comparison of some approximations to the Wilcoxon-Mann-Whitney distribution

The following approximations to the exact distribution of the Wilcoxon rank sum test (Mann-Whitney U-test) are compared at or near significance levels .05, .025, .01 and .005: the normal approximation with continuity correction, the simple and complete version of the Edgeworth series approximation proposed by Fix and Hodges (1955), Buckle, Kraft and van Eeden’s (1969a) uniform approximation and Iman’s (1976) recently proposed approximation. The comparison takes into account simplicity of application as well as closeness, and some preference rules are suggested for different user objectives.     

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.     

已知Gini相关系数对Copula函数上界的改进

如果已知Copula函数的某些信息,通常其Frechet-Hoeffding上下界则可以进一步收窄。R.B.Nelson分别在已知Kendall相关系数tau、Spearman相关系数rho、Blomqvist相关系数beta的条件下,给出了Copula函数的上下确界。鉴此,在给定Gini相关系数gamma的条件下,证明Copula函数的上界也可以进一步变窄,并给出上确界的表达式。     

Closed-Form Approximation for the AOQL of Attributes Acceptance Sample Plans

We derive upper and lower bounds at the point at which the average outgoing quality limit (AOQL) of an attributes acceptance sampling plan is achieved. Using a simple average of these bounds to approximate the ordinate of the AOQL, we develop an accurate, closed-form approximation to the AOQL. The bounds and approximation show how the parameters of a sampling plan affect the AOQL and can be used to study the behavior of the AOQL and other measures of the plan's performance.     

Asymptotically optimal Berry-Esseen-type bounds for distributions with an absolutely continuous part

Recursive and closed form upper bounds are offered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The method employed is a modification of the method of compositions along with Zolotarev's ideal metric. The approximation error in the CLT obtained vanishes at a rate O(n^−k/2+1), provided that the common distribution of the summands possesses an absolutely continuous part, and shares the same k−1 (k?3) first moments with the standard normal distribution. Moreover, for the first time, these new uniform Berry-Esseen-type bounds are asymptotically optimal, that is, the ratio of the true distance to the respective bound converges to unity for a large class of distributions of the summands. Thus, apart from the correct rate, the proposed error estimates incorporate an optimal asymptotic constant (factor). Finally, three illustrative examples are presented along with numerical comparisons revealing that the new bounds are sharp enough even to be used in practical statistical applications.     

Tables of the cumulative non-central chi-square distribution Part 51

Some bounds and approximations for the mean of second largest order sta¬tistic from an arbitrary continuous distribution symmetric about zero are obtained. For the standard normal case, FOURIER coefficients involved in the approximation procedure are tabulated and the bounds obtained are compared with other similar methods     

Sharp bounds on the expectations of second record values from symmetric populations

Combining the greatest convex minorant approximation (Moriguti, S. (1953). A modification of Schwarz's inequality with applications to distributions. Ann. Math. Statist., 24, 107–113.) with the Hölder inequality, we establish sharp bounds on the expectations of the second record statistics from symmetric populations. We also determine the distributions for which the bounds are attained. The optimal bounds are numerically evaluated and compared with other classical rough ones.     

Critical review and comparison of continuity correction methods: The normal approximation to the binomial distribution

We provide a comprehensive and critical review of Yates’ continuity correction for the normal approximation to the binomial distribution, in particular emphasizing its poor ability to approximate extreme tail probabilities. As an alternative method, we also review Cressie's finely tuned continuity correction. In addition, we demonstrate how Yates’ continuity correction is used to improve the coverage probability of binomial confidence limits, and propose new confidence limits by applying Cressie's continuity correction. These continuity correction methods are numerically compared and illustrated by data examples arising from industry and medicine.     

Tail Probability Ratios of Normal and Student’s t Distributions

The ratio of normal tail probabilities and the ratio of Student’s t tail probabilities have gained an increased attention in statistics and related areas. However, they are not well studied in the literature. In this paper, we systematically study the functional behaviors of these two ratios. Meanwhile, we explore their difference as well as their relationship. It is surprising that the two ratios behave very different to each other. Finally, we conclude the paper by conducting some lower and upper bounds for the two ratios.     

Approximations for a fixed sample size selection procedure for negative binomial populations

A large sample approximation of the least favorable configuration for a fixed sample size selection procedure for negative binomial populations is proposed. A normal approximation of the selection procedure is also presented. Optimal sample sizes required to be drawn from each population and the bounds for the sample sizes are tabulated. Sample sizes obtained using the approximate least favorable configuration are compared with those obtained using the exact least favorable configuration. Alternate form of the normal approximation to the probability of correct selection is also presented. The relation between the required sample size and the number of populations involved is studied.     

A non uniform bound on geometric approximation with w-functions

The aim of this paper is a use of Stein’s method and w-functions to determine a non uniform bound on the geometric approximation for a non negative integer-valued random variable. Some applications of the obtained results are provided to approximate the negative hypergeometric, Pólya and negative Pólya distributions.     

An improvement of a non-uniform bound for combinatorial central limit theorem

In this article, we develop a new Rosenthal Inequality for uniform random permutation sums of random variables with finite third moments and apply it to obtain a sharp non-uniform bound for the combinatorial central limit theorem using the Stein's method and the exchangeable pair techniques. The obtained bound is shown to be sharper than other existing bounds.     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.     

Error bounds for the perturbation solution of the transition density under a multi-factor CIR term structure model with weak mean-reversion effect

Abstract

We consider a multi-factor Cox-Ingersoll-Ross (CIR) model of the term structure of interest rates with weak mean-reversion effect. We use perturbation theory to analyze its conditional characteristic function illustrated by a system of Riccati equations and derive the error bounds for the perturbation approximations. Using the Fourier inversion theorem, we clarify that the perturbation approximation of the conditional characteristic function can be applied to estimate the transition density and likelihood function. We provide their error bounds and accuracy orders. Finally, we discuss the performance of the perturbation approximation in estimating the transition density via simulation.     

A matching prior for the product of normal means based on the modified profile likelihood

In this paper, we develop a matching prior for the product of means in several normal distributions with unrestricted means and unknown variances. For this problem, properly assigning priors for the product of normal means has been issued because of the presence of nuisance parameters. Matching priors, which are priors matching the posterior probabilities of certain regions with their frequentist coverage probabilities, are commonly used but difficult to derive in this problem. We developed the first order probability matching priors for this problem; however, the developed matching priors are unproper. Thus, we apply an alternative method and derive a matching prior based on a modification of the profile likelihood. Simulation studies show that the derived matching prior performs better than the uniform prior and Jeffreys’ prior in meeting the target coverage probabilities, and meets well the target coverage probabilities even for the small sample sizes. In addition, to evaluate the validity of the proposed matching prior, Bayesian credible interval for the product of normal means using the matching prior is compared to Bayesian credible intervals using the uniform prior and Jeffrey’s prior, and the confidence interval using the method of Yfantis and Flatman.     

MONOTONE BOUNDS ON THE CHI-SQUARED APPROXIMATION TO THE DISTRIBUTION OF PEARSON'S XSTATISTICS12

Monotone bounds, depending on the underlying multinomial probabilities and the sample size, are given for the chi-squared approximation to the distribution of Pearson's statistic for goodness of fit for simple hypotheses. These bounds apply to the distribution of a single statistic and to the joint distribution of two statistics associated with the margins of a two-way table, in both the central and non-central cases.