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.     

Quadratic reflected BSDEs and related obstacle problems for PDEs

Abstract

In this paper, we study a kind of reflected backward stochastic differential equations (BSDEs) whose generators are of quadratic growth in z and linear growth in y. We first give an estimate of solutions to such reflected BSDEs. Then under the condition that the generators are convex with respect to z, we can obtain a comparison theorem, which implies the uniqueness of solutions for this kind of reflected BSDEs. Besides, the assumption of convexity also leads to a stability property in the spirit of above estimate. We further establish the nonlinear Feynman-Kac formula of the related obstacle problems for partial differential equations (PDEs) in our framework. At last, a numerical example is given to illustrate the applications of our theoretical results, as well as its connection with an optimal stopping time problem.     

A Note on the Comparison between Simple Mean and Mean Based on Distinct Units in Sampling with Replacement

Basu (1958) and Raj and Khamis (1958) have shown that in sampling with replacement the average over distinct units possesses lower variance than the average over the entire sample, including repetitions. While Basu proves the result by the use of the Rao-Blackwell theorem, Raj and Khamis prove it by making use of the distribution of the number of distinct units u in the sample, and the proof is quite involved. Here, an elementary algebraic proof is given.     

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.     

《公司法》研究的新视角——法律经济学分析方法的提炼

我国的企业公司法立法从1956年开始,根据经济发展的需要和实践一直在进行修订,其目的在于在提高公司的经营绩效,增强市场竞争力,但目前来说仍未达到立法者的预期目标。主要从法律经济学的基点入手研究、总结法律经济学研究方法在公司法制度改革中的应用态势,并考虑到法律经济学分析方法将在公司法的改革中继续深化和进一步升华的问题,使公司法在制度设计中达到协调交易成本最小化而使经济收益最大化,使公司在经营活动中更加具有活力和市场竞争力。     

Empirical Study of Six Tests for Equality of Populations with Zero-Inflated Continuous Distributions

We evaluated the properties of six statistical methods for testing equality among populations with zero-inflated continuous distributions. These tests are based on likelihood ratio (LR), Wald, central limit theorem (CLT), modified CLT (MCLT), parametric jackknife (PJ), and nonparametric jackknife (NPJ) statistics. We investigated their statistical properties using simulated data from mixed distributions with an unknown portion of non zero observations that have an underlying gamma, exponential, or log-normal density function and the remaining portion that are excessive zeros. The 6 statistical tests are compared in terms of their empirical Type I errors and powers estimated through 10,000 repeated simulated samples for carefully selected configurations of parameters. The LR, Wald, and PJ tests are preferred tests since their empirical Type I errors were close to the preset nominal 0.05 level and each demonstrated good power for rejecting null hypotheses when the sample sizes are at least 125 in each group. The NPJ test had unacceptable empirical Type I errors because it rejected far too often while the CLT and MCLT tests had low testing powers in some cases. Therefore, these three tests are not recommended for general use but the LR, Wald, and PJ tests all performed well in large sample applications.     

Recursive Estimation of GARCH Models

This article develops three recursive on-line algorithms, based on a two-stage least squares scheme for estimating generalized autoregressive conditionally heteroskedastic (GARCH) models. The first one, denoted by 2S-RLS, is an adaptation of the recursive least squares method for estimating autoregressive conditionally heteroskedastic (ARCH) models. The second and the third ones (denoted, respectively, by 2S-PLR and 2S-RML) are adapted versions of the pseudolinear regression (PLR) and the recursive maximum likelihood (RML) methods to the GARCH case. We show that the proposed algorithms give consistent estimators and that the 2S-RLS and the 2S-RML estimators are asymptotically Gaussian. These methods seem very adequate for modeling the sequential feature of financial time series, which are observed on a high-frequency basis. The performance of these algorithms is shown via a simulation study.     

A Bayes Theorem Urn Model Optimization Problem

An urn model is a finite collection of indistinguishable urns together with an arbitrary distribution of a finite number of balls (bills) of k colors (denominations) into the urns. A Bayes theorem expectation optimization problem associated with certain urn models is discussed.     

Lp (1 < p ⩽ 2) solutions of one-dimensional BSDEs whose generator is weakly monotonic in y and non-Lipschitz in z

ABSTRACT

In this paper, we start with establishing the existence of a minimal (maximal) L^p (1 < p ? 2) solution to a one-dimensional backward stochastic differential equation (BSDE), where the generator g satisfies a p-order weak monotonicity condition together with a general growth condition in y and a linear growth condition in z. Then, we propose and prove a comparison theorem of L^p (1 < p ? 2) solutions to one-dimensional BSDEs with q-order (1 ? q < p) weak monotonicity and uniform continuity generators. As a consequence, an existence and uniqueness result of L^p (1 < p ? 2) solutions is also given for BSDEs whose generator g is q-order (1 ? q < p) weakly monotonic with a general growth in y and uniformly continuous in z.     

A Decomposition of Copulas and Its Use

ABSTRACT

In this article, we create a decomposition that represents and describes the depen-dence structure between two variables. Since copulas provide a deep understanding of the dependence structure by eliminating the effects of the marginals, they play a key role in this study. We define a discretized copula density matrix and decompose it into a set of permutation matrices by using the Birkhoff–von Neumann theorem. This decomposition provides a way to effectively apply the concepts of copulas to solve problems in multivariate statistical data analysis.