The effects of correlation among observations on the consistency property of sample variance

It is well known that even when the sample observations are correlated and not normal the sample variance, S² converges in probability to E(S²). But the required sample size for S² to be a consistent estimator of E(S²) is an open question. Some light is shed on this question in this paper. In particular the relation between the rate of convergence and the correlation property of the observations is explored. It is shown that the retardation to the rate of convergence is not appreciable if the correlation is moderate but it can be severe for extreme correlations.     

The Effectiveness of Risk Management: An Analysis of Project Risk Planning Across Industries and Countries

This article examines the effectiveness of current risk management practices to reduce project risk using a multinational, multi‐industry study across different scenarios and cultures. A survey was administered to 701 project managers, and their supervisors, in seven industries and three diverse countries (New Zealand, Israel, and Japan), in multiple languages during the 2002–2007 period. Results of this study show that project context—industry and country where a project is executed—significantly impacts perceived levels of project risk, and the intensity of risk management processes. Our findings also suggest that risk management moderates the relationship between risk level and project success. Specifically, we found that even moderate levels of risk management planning are sufficient to reduce the negative effect risk levels have on project success.     

Poliovirus Surveillance by Examining Sewage Water Specimens: Studies on Detection Probability Using Simulation Models

Efficiency of environmental surveillance of poliovirus circulation was studied using simulation models. First, three transmission models were defined for describing different scenarios of poliovirus infections in a large unstructured population. Second, environmental factors, such as the total volume of the sewage network and losses of viruses, were modeled for computing the virus output at the sewage sampling site. Third, the effect of sampling and laboratory procedures was accounted for in the probability of detection, given the amount of polioviruses in a specimen. The simulation model can be used for theoretical assessments of the likely efficiency of environmental surveillance, compared with acute flaccid paralysis (AFP) surveillance. Under reasonable assumptions in a vaccinated population, the AFP surveillance can be outperformed if the poliovirus outbreak is not large. However, this depends on the assumed case-to-infection ratio and on the sampling frequency of the sewage water specimens. Increasing the latter will lead to a higher detection probability, which will further enhance the method based on environmental surveillance.     

Using adaptive weighted least squares to reduce the lengths of confidence intervals

The author proposes an adaptive method which produces confidence intervals that are often narrower than those obtained by the traditional procedures. The proposed methods use both a weighted least squares approach to reduce the length of the confidence interval and a permutation technique to insure that its coverage probability is near the nominal level. The author reports simulations comparing the adaptive intervals to the traditional ones for the difference between two population means, for the slope in a simple linear regression, and for the slope in a multiple linear regression having two correlated exogenous variables. He is led to recommend adaptive intervals for sample sizes superior to 40 when the error distribution is not known to be Gaussian.     

Backward Induction Is Not Robust: The Parity Problem and the Uncertainty Problem

A cornerstone of game theory is backward induction, whereby players reason backward from the end of a game in extensive form to the beginning in order to determine what choices are rational at each stage of play. Truels, or three-person duels, are used to illustrate how the outcome can depend on (1) the evenness/oddness of the number of rounds (the parity problem) and (2) uncertainty about the endpoint of the game (the uncertainty problem). Since there is no known endpoint in the latter case, an extension of the idea of backward induction is used to determine the possible outcomes. The parity problem highlights the lack of robustness of backward induction, but it poses no conflict between foundational principles. On the other hand, two conflicting views of the future underlie the uncertainty problem, depending on whether the number of rounds is bounded (the players invariably shoot from the start) or unbounded (they may all cooperate and never shoot, despite the fact that the truel will end with certainty and therefore be effectively bounded). Some real-life examples, in which destructive behavior sometimes occurred and sometimes did not, are used to illustrate these differences, and some ethical implications of the analysis are discussed.     

A Generalization of the Childs-Moran Result for Orthoscheme Probabilities

van der Vaart (1953, 1955) introduced the orthoscheme probability R_n (c ₁,..., c_n−1 ), meaning the orthant probability of an n -dimensional normal random vector with zero mean and tridiagonal correlation matrix with elements c ₁,..., c_n−1 on the upper diagonal. Childs (1967) conjectured and Moran (1983) proved that the generating function of { R_n (½,...,½)} equals tan z + sin z . This paper derives the generating function of { R_n (τ,½,...,½)}.     

负二项分布的推广

本文主要是在二项分布,多项分布,负二项分布的基础上,把负二项分布进一步推广,给出负N项分布的定义,推导出它的概率分布,并计算出其数学期望和方差.     

On some probability distributions associated with random walks

The probability distribution of the total number of games to ruin in a gambler's ruin random walk with initial position n, the probability distribution of the total size of an epidemic starting with n cases and the probability distribution of the number of customers served during a busy period M/M/1 when the service starts with n waiting customers are identical. All these can be easily obtained by using Lagrangian expansions instead of long combinatorial methods. The binomial, trinomial, quadrinomial and polynomial random walks of a particle have been considered with an absorbing barrier at 0 when the particle starts its walks from a point n, and the pgfs. and the probability distributions of the total number of jumps (trials) before absorption at 0 have been obtained. The values for the mean and variance of such walks have also been given.     

Some aspects of probability forecasting

The problems of assessing, comparing and combining probability forecasts for a binary events sequence are considered. A Gaussian threshold model (analytically of closed form) is introduced which allows generation of different probability forecast sequences valid for the same events. Chi - squared type test statistics, and also a marginal-conditional method are proposed for the assessment problem, and an asymptotic normality result is given. A graphical method is developed for the comparison problem, based upon decomposing arbitrary proper scoring rules into certain elementary scoring functions. The special role of the logarithmic scoring rule is examined in the context of Neyman - Pearson theory.     

The neyman accuracy and the wolfowitz accuracy of the stein type confidence interval for the disturbance variance

In this paper we consider the Neyman accuracy and the Wolfowitz accuracy of the Stein type improved confidence interval I?_S for the disturbance variance in a linear regression model. The Neyman accuracy is a measure related to the unbiasedness of a confidence interval, and the Wolfowitz accuracy is related to the closeness of the endpoints to the true parameter. We show that I?_S is not unbiased and give some numerical results for the Neyman accuracy. As for the Wolfowitz accuracy we derive the sufficient condition for I?_S to improve on the usual confidence interval under this criterion and show numerically that a large degree of improvement can be obtainted.