Uncertain Numbers and Uncertainty in the Selection of Input Distributions—Consequences for a Probabilistic Risk Assessment of Contaminated Land

Risks from exposure to contaminated land are often assessed with the aid of mathematical models. The current probabilistic approach is a considerable improvement on previous deterministic risk assessment practices, in that it attempts to characterize uncertainty and variability. However, some inputs continue to be assigned as precise numbers, while others are characterized as precise probability distributions. Such precision is hard to justify, and we show in this article how rounding errors and distribution assumptions can affect an exposure assessment. The outcome of traditional deterministic point estimates and Monte Carlo simulations were compared to probability bounds analyses. Assigning all scalars as imprecise numbers (intervals prescribed by significant digits) added uncertainty to the deterministic point estimate of about one order of magnitude. Similarly, representing probability distributions as probability boxes added several orders of magnitude to the uncertainty of the probabilistic estimate. This indicates that the size of the uncertainty in such assessments is actually much greater than currently reported. The article suggests that full disclosure of the uncertainty may facilitate decision making in opening up a negotiation window. In the risk analysis process, it is also an ethical obligation to clarify the boundary between the scientific and social domains.     

应用白纸测图技术测绘大比例尺草地群落分布图的探讨

在牧区生产方面 ,大比例尺草地植被分布图是非常重要的基础资料 ,对于划区轮牧、草场轮刈以及草地权属落实等都是最基础的档案 ,然而大比例尺植被图 ( 1:50 0 0 )的测绘却很少有人探讨 ,本文从生产发展需要出发 ,试用测绘学方法白纸测图技术 ,进行 1:50 0 0植物群落分布图的测绘研究 ,此方法可以概括为草群分类、样地选择、界线勾绘、建立控制网、样地扩展、分板实测、联绘成图等步骤     

M-quantile models with application to poverty mapping

Over the last decade there has been growing demand for estimates of population characteristics at small area level. Unfortunately, cost constraints in the design of sample surveys lead to small sample sizes within these areas and as a result direct estimation, using only the survey data, is inappropriate since it yields estimates with unacceptable levels of precision. Small area models are designed to tackle the small sample size problem. The most popular class of models for small area estimation is random effects models that include random area effects to account for between area variations. However, such models also depend on strong distributional assumptions, require a formal specification of the random part of the model and do not easily allow for outlier robust inference. An alternative approach to small area estimation that is based on the use of M-quantile models was recently proposed by Chambers and Tzavidis (Biometrika 93(2):255–268, 2006) and Tzavidis and Chambers (Robust prediction of small area means and distributions. Working paper, 2007). Unlike traditional random effects models, M-quantile models do not depend on strong distributional assumption and automatically provide outlier robust inference. In this paper we illustrate for the first time how M-quantile models can be practically employed for deriving small area estimates of poverty and inequality. The methodology we propose improves the traditional poverty mapping methods in the following ways: (a) it enables the estimation of the distribution function of the study variable within the small area of interest both under an M-quantile and a random effects model, (b) it provides analytical, instead of empirical, estimation of the mean squared error of the M-quantile small area mean estimates and (c) it employs a robust to outliers estimation method. The methodology is applied to data from the 2002 Living Standards Measurement Survey (LSMS) in Albania for estimating (a) district level estimates of the incidence of poverty in Albania, (b) district level inequality measures and (c) the distribution function of household per-capita consumption expenditure in each district. Small area estimates of poverty and inequality show that the poorest Albanian districts are in the mountainous regions (north and north east) with the wealthiest districts, which are also linked with high levels of inequality, in the coastal (south west) and southern part of country. We discuss the practical advantages of our methodology and note the consistency of our results with results from previous studies. We further demonstrate the usefulness of the M-quantile estimation framework through design-based simulations based on two realistic survey data sets containing small area information and show that the M-quantile approach may be preferable when the aim is to estimate the small area distribution function.     

A Trivariate Distribution for the Height,Weight, and Fat of Adult Men*

Using exploratory data analysis, probability plots, scatterplots, and computer animations to rotate and visualize the data, we fit a trivariate Normal distribution to data for the height, the natural logarithm of body weight, and the body fat for 646 men between the ages of 50 and 80 years as reported by the medical staff of the U.S. Veterans Administration's “Normative Aging Study” in Boston, MA. Although these data do not include any children, women, or young men, the measurements represent the best data that we could find through a 4-year search. We believe that these data are well measured and reliable for men in the specified age range and that these data reveal an interesting statistical pattern for use in probabilistic PBPK models.     

Reconfiguring the Supply Network Using Current Performance Data

Reconfiguration of the supply chain network from time to time is essential for businesses to retain their competitive edge. This paper presents a methodology for reconfiguration of an existing supply chain network. The methodology is characterized by two decision levels. In the first level, the current network performance is evaluated and efficient practices are identified. In the next level, a model that incorporates efficient practices is developed to reconfigure the network. This integrated methodology allows for decision maker (DM) input throughout the process. The methodology has been implemented and tested in the reconfiguration of an outbound petroleum supply chain network for CountryMark Cooperative, Inc. In this case study, Data Envelopment Analysis (DEA) is used to analyze current operations and an integer programming (IP) model that incorporates efficiency metrics is developed for selection of distribution facilities and allocation of resources to the facilities. Use of this methodology can lead to improved operations and reduced operating expenses.     

Sequential confidence intervals for the number of cells in a multinomial population

A two-sided sequential confidence interval is suggested for the number of equally probable cells in a given multinomial population with prescribed width and confidence coefficient. We establish large-sample properties of the fixed-width confidence interval procedure using a normal approximation, and some comparisons are made. In addition, a simulation study is carried out in order to investigate the finite sample behaviour of the suggested sequential interval estimation procedure.     

On computer generation of gamma random variables by rejection and composition procedures 2

The paper presents various algorithms for generating gamma random variables, by combining rejection and composition procedures. Two efficient algorithms are given for the case when the parameter of the gamma distribution is 0<v<l. For the case vl, several algorithms are given but they (except one), work reliably only for small values of v. Results of some computer tests together with FORTRAN subroutines are also presented.     

A Recursive Formula for the Probability of Occurrence of at Least m out of N Events

A recursive same-sign relation is derived that reduces the probability of occurrence of at least m out of N independent events to the probability of occurrence of at least m out of N ? 1 of these N events.     

Online Control Charts for Process Averages Based on Repeated Median Filters

Two types of estimates of process level, namely repeated median estimates (Siegel, 1982 Siegel , A. F. ( 1982 ). Robust regression using repeated medians . Biometrika 69 : 242 – 244 .[Crossref], [Web of Science ®] , [Google Scholar]) and full online estimates (Gather et al., 2006 Gather , U. , Schettlinger , K. , Fried , R. ( 2006 ). Online signal extraction by robust linear regression . Computational Statistics 21 : 33 – 51 .[Crossref], [Web of Science ®] , [Google Scholar]) based on repeated median filters, are used to develop control charts. The distributional properties of the estimates are studied using simulation and these are found to closely follow normal distribution. The repeated median being robust against outliers with asymptotically 50% breakdown value and having small standard deviation is found to be useful as a basis for monitoring process averages. The control charts using repeated median estimates have been recommended for general use.     

Approximations to the multivariate normal integral

We propose a simple and efficient way to approximate multivariate normal probabilities using univariate and bivariate probabilities. The approximation is computationally tested for the trivariate and quadrivariate normal probabilities. A few problems of higher dimensions were also tested.