LOWER BOUNDS FOR THE NORMAL APPROXIMATION OF A SUM OF INDEPENDENT RANDOM VARIABLES

The rates of convergence to the normal distribution are investigated for a sum of independent random variables. Using Stein's method, we derive a lower bound of the uniform distance between two distributions of independent sum and normal.     

A Weighted Central Limit Theorem Under Sublinear Expectations

In this article, we investigate a central limit theorem for weighted sum of independent random variables under sublinear expectations. It is turned out that our results are natural extensions of the results obtained by Peng (2008 Peng , S. (2008). A New Central Limit Theorem under Sublinear Expectations. arXiv:0803. 2656vl [math.PR]. [Google Scholar]) and Li and Shi (2010 Li , M. , Shi , Y. ( 2010 ). A general central limit theorem under sublinear expectations . Sci. China Ser. A 53 : 1989 – 1994 .[Crossref] , [Google Scholar]).     

A CENTRAL LIMIT THEOREM FOR TESTING EXPONENTIALITY*

To test exponentiality of a population, we modify the test statistic based on the sample entropy in the literature, and establish its limiting distribution theorem under weaker conditions, which improves the existing results.

     

Diagrams for Set Theory & Probability Problems of Four or More Variables

It is pointed out that in many one-sided testing situations for a real-valued parameter θ, the monotonicity of the power function hinges on the stochastic order of the underlying family of distributions [F_θ] rather than on the stronger property of monotone likelihood ratio of the family. An elementary proof, accessible to students of introductory probability and statistics, is presented.     

Correction

In many probability and mathematical statistics courses the probability generating function (PGF) is typically overlooked in favor of the more utilized moment generating function. However, for certain types of random variables, the PGF may be more appealing. For example, sums of independent, non-negative, integer-valued random variables with finite support are easily studied via the PGF. In particular, the exact distribution of the sum can easily be calculated. Several illustrative classroom examples, with varying degrees of difficulty, are presented. All of the examples have been implemented using the R statistical software package.     

ON QUANTILES OF SUMS

In this paper we investigate the relationship between the quantiles of a sum of independent continuous random variables and those of its components. Results concerning this relationship are given for the special cases of symmetric distributions, gamma distributions, and for the difference of identically distributed random variables.     

NEW LARGE SAMPLE AND BOOTSTRAP METHODS ON SHAPE SPACES IN HIGH LEVEL ANALYSIS OF NATURAL IMAGES

This paper, dedicated to the 80th birthday of Professor C. R. Rao, deals with asymptotic distributions of Fréchet sample means and Fréchet total sample variance that are used in particular for data on projective shape spaces or on 3D shape spaces. One considers the intrinsic means associated with Riemannian metrics that are locally flat in a geodesically convex neighborhood around the support of a probability measure on a shape space or on a projective shape space. Such methods are needed to derive tests concerning variability of planar projective shapes in natural images or large sample and bootstrap confidence intervals for 3D mean shape coordinates of an ordered set of landmarks from laser images.     

Simulation and Analytical Approach to the Identification of Significant Factors

We develop our previous works concerning the identification of the collection of significant factors determining some, in general, nonbinary random response variable. Such identification is important, e.g., in biological and medical studies. Our approach is to examine the quality of response variable prediction by functions in (certain part of) the factors. The prediction error estimation requires some cross-validation procedure, certain prediction algorithm, and estimation of the penalty function. Using simulated data, we demonstrate the efficiency of our method. We prove a new central limit theorem for introduced regularized estimates under some natural conditions for arrays of exchangeable random variables.     

Asymptotic Properties of Random Weighted Empirical Distribution Function

This article studies the asymptotic properties of the random weighted empirical distribution function of independent random variables. Suppose X₁, X₂, ???, X_n is a sequence of independent random variables, and this sequence is not required to be identically distributed. Denote the empirical distribution function of the sequence by F_n(x). Based on the random weighting method and F_n(x), the random weighted empirical distribution function H_n(x) is constructed and the asymptotic properties of H_n are discussed. Under weak conditions, the Glivenko–Cantelli theorem and the central limit theorem for the random weighted empirical distribution function are obtained. The obtained results have also been applied to study the distribution functions of random errors of multiple sensors.     

Some Estimates for the Rate of Convergence in the Multidimensional Central Limit Theorem for Homogeneous Markov Chains

This paper obtains some estimates for the rate of convergence in the multi-dimensional central limit theorem for vector-valued functions of a homogeneous Markov chain without assuming the finiteness of their absolute third moment. These estimates have a universal character and generalize the results that hold when the third moments are finite.     

Editorial collaborators

Confidence intervals obtained by bootstrap methods and normal approximation are compared, based on output data from terminating and steady-state simulations. Bootstrap intervals are equal or better than normal approximation intervals in actual probability coverages. Furthermore, bootstrap methods capture the skewness in the distribution of outputs and, therefore, are more desirable than normal approximation.     

Large Deviations for Sums of Independent Non Identically Distributed Random Variables

This article considers large deviation results for sums of independent non identically distributed random variables, generalizing the result of Petrov (1968 Petrov , V. V. ( 1968 ). Asymptotic behavior of probabilities of large deviations . Theor. Probab. Appl. 13 : 408 – 420 . [Google Scholar]) by using a weaker and more natural condition on bounds of the cumulant generating functions of the sequence of random variables.     

On Large Deviations of Sums of Independent Random Variables

Extensions of some limit theorems are proved for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramér's condition. The large deviation x-region under consideration is broader than in the classical Cramér's theorem, and the estimate of the remainder is uniform with respect to x. The corresponding asymptotic expansion with arbitrarily many summands is also obtained.     

On rates of convergence to infinitely divisible laws for dependent random variables

Large O and small o approximations of the expected value of a class of functions (modified K-functional and Lipschitz class) of the normalized partial sums of dependent random variables by the expectation of the corresponding functions of infinitely divisible random variables have been established. As a special case, we have obtained rates of convergence to the Stable Limit Laws and to the Weak Laws of Large Numbers. The technique used is the conditional version of the operator method of Trotter and the Taylor expansion.     

A note on asymptotic distributions in directed exponential random graph models with bi-degree sequences

The asymptotic normality of a fixed number of the maximum likelihood estimators (MLEs) in the directed exponential random graph models with an increasing bi-degree sequence has been established recently. In this article, we further derive a central limit theorem for a linear combination of all the MLEs with an increasing dimension. Simulation studies are provided to illustrate the asymptotic results.     

Constants on a Uniform Berry-Esseen Bound on Some Borel Sets in ℝ k via Stein's Method

For each n, k ∈ ?, let Y _i  = (Y _i1, Y _i2,…, Y _ik ), 1 ≤ i ≤ n be independent random vectors in ? ^k with finite third moments and Y _ij are independent for all j = 1, 2,…, k. In this article, we use the Stein's technique to find constants in uniform bounds for multidimensional Berry-Esseen inequality on a closed sphere, a half plane and a rectangular set.     

A Criterion for Stepwise Regression

In “stepwise” regression analysis, the usual procedure enters or removes variables at each “step” on the basis of testing whether certain partial correlation coefficients are zero. An alternative method suggested in this paper involves testing the hypothesis that the mean square error of prediction does not decrease from one step to the next. This is equivalent to testing that the partial correlation coefficient is equal to a certain nonzero constant. For sample sizes sufficiently large, Fisher's z transformation can be used to obtain an asymptotically UMP unbiased test. The two methods are contrasted with an example involving actual data.