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.     

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.     

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.     

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.     

Generalized definition of the geometric mean of a non negative random variable

The first probabilistic definition of the geometric mean of a non negative random variable under certain assumptions was given in Feng et al. (2013 Feng, C., Wang, H., Tu, X. (2013). Geometric mean of nonnegative random variable. Commun. Stat.—Theory Methods 42:2714–2717.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). In this paper, we generalize the definition to a larger class of random variables. We also show the basic properties of the geometric mean and point out its discontinuity and instability. Some convergence properties are studied as well, for which we emphasize its link to the positive moments of the random variable. A discussion of potential applications of the new definition in biomedical research and open questions to complete the theory of geometric mean is highlighted.     

A note on the Edgeworth approximation to the distribution of Spearman's Rho with a correction to Pearson's approximation

The approximation of Edgeworth's form of the Gram-Charlier series to the exact cumulative distribution function (c.d.f.) of Spearman's Rank Correlation Coefficient is examined. For both the uncorrected and continuity correction cases, the maximum differences between the Edgeworth approximation and the exact c.d.f. is presented for 9 ≤ n ≤ 18. A correction to a similar, earlier table for the Pearson Type II approximation is also incorporated and comparisons made.     

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.     

A bivariate generalized geometric distribution with applications

This paper proposes a bivariate version of the univariate discrete generalized geometric distribution considered by Gómez–Déniz (2010 Gómez–Déniz, E. (2010). Another generalization of the geometric distribution. Test 19:399–415.[Crossref], [Web of Science ®] , [Google Scholar]). The proposed bivariate distribution can have a positive or negative correlation coefficient which can be used for modeling bivariate-dependent count data. After discussing some of its properties, maximum likelihood estimation is discussed. Two illustrative examples are given for fitting and demonstrating the usefulness of the new bivariate distribution proposed here.     

On uniform approximation of Poisson variates by normal ones

In this paper the accuracy of the normal approximation to the Poisson is treated from the viewpoint of direct approximation of Poisson variables by normal ones. The conclusions that are derived on the accuracy of this approximation lead (among others) to very useful results on confidence limits for the mean of a linear combination of independent Poisson variables; these latter are employed in precise determination of the composition of a mixture of radioactive isotopes by means of a scintillation counter.     

A geometric time-series model with an alternative dependent Bernoulli counting series

In this article, we first introduce an alternative way for construction of the generalized binomial thinning operator with dependent counting series. Some properties of this thinning operator are derived and discussed. Then, by using this thinning operator, we introduce an integer-valued time-series model with geometric marginals. Some conditional and unconditional properties of this model are derived and discussed. Some estimation methods are considered and for some of them, asymptotic properties of the obtained estimates are derived. Performances of the estimates are discussed through some simulations. Finally, a real data example is considered and the goodness-of-fit of this model is compared with the models based on the binomial, negative binomial, and dependent binomial thinning operators.     

Some characteristics on the selection of spline smoothing parameter

The smoothing spline method is used to fit a curve to a noisy data set, where selection of the smoothing parameter is essential. An adaptive C_p criterion (Chen and Huang 2011 Chen, C. S., and H. C. Huang. 2011. An improved C_p criterion for spline smoothing. Journal of Statistical Planning and Inference 141:445–52.[Crossref], [Web of Science ®] , [Google Scholar]) based on the Stein’s unbiased risk estimate has been proposed to select the smoothing parameter, which not only considers the usual effective degrees of freedom but also takes into account the selection variability. The resulting fitted curve has been shown to be superior and more stable than commonly used selection criteria and possesses the same asymptotic optimality as C_p. In this paper, we further discuss some characteristics on the selection of smoothing parameter, especially for the selection variability.     

A geometric time series model with a new dependent Bernoulli counting series

ABSTRACT

New generalized binomial thinning operator with dependent counting series is introduced. An integer valued time series model with geometric marginals based on this thinning operator is constructed. Main features of the process are analyzed and determined. Estimation of the parameters are presented and some asymptotic properties of the obtained estimators are discussed. Behavior of the estimators is described through the numerical results. Also, model is applied on the real data set and compared to some relevant INAR(1) models.     

Some theoretical results on approximation methods 1

In this paper we give an approximation procedure to surfaces which are defined on a _p-dimensional region and are observable (disturbed with some noice) according to an experimental design. In this procedure we combine clustering methods, discriminant analysis and smoothing techniques.

In the second part of the paper is considered some investigations on statistical properties of linear smoothing procedures. We assume linear models and for a broad class of models we prove the consistence of the estimation of the expectation of observations after smoothing.

In the last section we give some results on efficiency.     

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.     

A note on an approximation to neyman type a distributions

This note outlines a steepest descent approximation to the tail probability of the Neyman type A distribution.