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.

     

Reasonable sample sizes for convergence to normality

The central limit theorem says that, provided an estimator fulfills certain weak conditions, then, for reasonable sample sizes, the sampling distribution of the estimator converges to normality. We propose a procedure to find out what a “reasonably large sample size” is. The procedure is based on the properties of Gini's mean difference decomposition. We show the results of implementations of the procedure from simulated datasets and data from the German Socio-economic Panel.     

From Student's t 2 Distribution to Student's t 3 Distribution Through Characterizations

Some simple regressional properties of order statistics, similar to those obtained for t ₂ distribution in the recent statistical literature, are shown to characterize Student's t ₃ distribution.     

Sample Sizes and the Central Limit Theorem: The Poisson Distribution as an Illustration

Poisson distributions are often used to show that the central limit theorem is valid even for discrete and for highly skewed distributions. It is not so commonly appreciated that they can also be used to demonstrate that, in some cases, very large sample sizes may not be enough to invoke the theorem. Binomial distributions can be used in a similar manner.     

On Statistical Properties of Palindromes in DNA

The main aim of this article is to determine the distribution of the number of palindromes of a fixed length in a DNA sequence. Palindrome in DNA is a part of the DNA sequence which is equal to its complementary sequence read backwards (C ～ G, A ～ T). We examine the general case where the distribution of the nucleobases is arbitrary. Further, we determine conditions in which Normal approximation can be used. Special attention is given to coding and non-coding regions in DNA and to the distribution of bases in those parts of DNA.     

L'Hospital's Rule and the Central Limit Theorem

Beginning probability students are often confused by the use of Taylor polynomials in the proof of the central limit theorem. This article provides a proof of the central limit theorem based on L'Hospital's rule rather than on Taylor polynomials.     

Limit theorems for dependent Bernoulli variables with statistical inference

We consider a class of dependent Bernoulli variables where the conditional success probability is a linear combination of the last few trials and the original success probability. We obtain its limit theorems including the strong law of large numbers, weak invariance principle, and law of the iterated logarithm. We also derive some statistical inference results which make the model applicable. Simulation results are exhibited as well to show that with small sample size the convergence rate is satisfying and the proposed estimators behave well.     

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 Note on Functional Forms of Order Statistics

Functional forms of order statistics, as the solution of a system of equations, are studied. The case of the smaller and the larger of two random variables is discussed in detail. Some applications for normal and binomial distributions are presented.     

Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

Undergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.     

More on progressively Type‐II censored order statistics for multivariate observations

In this paper, we consider some results on distribution theory of multivariate progressively Type‐II censored order statistics. We also establish some characterizations of Freund's bivariate exponential distribution based on the lack of memory property.     

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.     

Limitations of P-Values and R-squared for Stepwise Regression Building: A Fairness Demonstration in Health Policy Risk Adjustment

ABSTRACT

Stepwise regression building procedures are commonly used applied statistical tools, despite their well-known drawbacks. While many of their limitations have been widely discussed in the literature, other aspects of the use of individual statistical fit measures, especially in high-dimensional stepwise regression settings, have not. Giving primacy to individual fit, as is done with p-values and R², when group fit may be the larger concern, can lead to misguided decision making. One of the most consequential uses of stepwise regression is in health care, where these tools allocate hundreds of billions of dollars to health plans enrolling individuals with different predicted health care costs. The main goal of this “risk adjustment” system is to convey incentives to health plans such that they provide health care services fairly, a component of which is not to discriminate in access or care for persons or groups likely to be expensive. We address some specific limitations of p-values and R² for high-dimensional stepwise regression in this policy problem through an illustrated example by additionally considering a group-level fairness metric.     

A note on the almost sure central limit theorem for the product of partial sums of m-dependent random variables

We present an almost sure central limit theorem for the product of the partial sums of m-dependent random variables. In order to obtain the main result, we prove a corresponding almost sure central limit theorem for a triangular array.     

Comments on zacks' recent paper on one-sided cusum procedures

The goal of this paper is to propose approximations for the cdf and the inverse cdf of the normal sample median. The presented methodology, which seems to not have been investigated before, suggests to fit the normal sample median distribution with a symmetrical Johnson S_U: distribution having ap-proximatively the same second and fourth moments. The results obtained with this approach, compared with the normal approximation, are very impressive, especially for the inverse cdf. One important application of the inverse cdf approximation of the normal sample median is the computation of accurate α‐level median/range control limits for any value of α (and not only for the popular value α = 0.0027). This paper can be also viewed as an homage to Professor N.L. Johnson's works by making a link between two of his major papers.     

Regression in a copula model for bivariate count data

In many cases of modeling bivariate count data, the interest lies on studying the association rather than the marginal properties. We form a flexible regression copula-based model where covariates are used not only for the marginal but also for the copula parameters. Since copula measures the association, the use of covariates in its parameters allow for direct modeling of association. A real-data application related to transaction market basket data is used. Our goal is to refine and understand whether the association between the number of purchases of certain product categories depends on particular demographic customers’ characteristics. Such information is important for decision making for marketing purposes.     

A family of bivariate distributions with non-elliptical contours

In this paper, a family of copulas with two parameters is proposed and its dependence analysis is performed. The corresponding family of bivariate distributions with specified marginals is constructed. For normal marginals, the new distributions are non-elliptical and can be applied in data analysis. They provide various alternative hypotheses for testing normality. Finally, an example is given.     

An Elementary Introduction to Maximum Likelihood Estimation for Multinomial Models: Birch's Theorem and the Delta Method

A fairly complete introduction to the large sample theory of parametric multinomial models, suitable for a second-year graduate course in categorical data analysis, can be based on Birch's theorem (1964) and the delta method (Bishop, Fienberg, and Holland 1975). I present an elementary derivation of a version of Birch's theorem using the implicit function theorem from advanced calculus, which allows the presentation to be relatively self-contained. The use of the delta method in deriving asymptotic distributions is illustrated by Rao's (1973) result on the distribution of standardized residuals, which complements the presentation in Bishop, Fienberg, and Holland. The asymptotic theory is illustrated by two examples.