SOLUTION PROCEDURES WITH LIMITED SAMPLE DATA FOR THE OPTIMAL REPLACEMENT PROBLEM

Recently we have developed solution procedures for the optimal replacement problem when the distribution of the time-to-failure (ttf) is partially specified by the first two moments, partial and complete. However, we have later learned, using Monte-Carlo simulation, that when moments are unknown and have to be estimated from sample data, the most accurate procedure developed therein is in practice extremely sensitive to sampling fluctuations. In this paper we modify the procedure to render it less susceptible to sampling variation. In addition, we introduce a new solution procedure that requires specification of only the median and the partial means of the ttf distribution. For both procedures, it is demonstrated that when the moments required for the distribution fitting are known, highly accurate optimal solutions are obtained. Conversely, when the moments are unknown and sample estimates based on small samples are used, both procedures result in stable solutions (low mean-squared-errors).     

MODIFIED STEPPING-STONE METHOD AS A TEACHING AID FOR CAPACITATED TRANSPORTATION PROBLEMS

This paper extends the standard stepping-stone method to the case of capacitated transportation problems. The purpose is to offer an alternative to network-oriented or simplex-based methods (such as the out-of-kilter and upper-bounding techniques) often used in handling the capacitated transportation problem but generally beyond the scope of introductory decision sciences courses. The proposed procedure is based on the simple stepping-stone method and requires only a few modifications in the cost/assignment matrix and some alterations in the pivoting rules. As such, it is much easier to understand and apply than the above-mentioned solution algorithms. It therefore can serve as an effective tool for teaching capacitated transportation problems. Two numerical examples illustrate applications of the proposed method, procedures for sensitivity analysis, and comparisons with linear programming.     

Estimating Moments of Subjectively Assessed Distributions*

Moment‐matching discrete distributions were developed by Miller and Rice (1983) as a method to translate continuous probability distributions into discrete distributions for use in decision and risk analysis. Using gaussian quadrature, they showed that an n‐point discrete distribution can be constructed that exactly matches the first 2n ‐ 1 moments of the underlying distribution. These moment‐matching discrete distributions offer several theoretical advantages over the typical discrete approximations as shown in Smith (1993), but they also pose practical problems. In particular, how does the analyst estimate the moments given only the subjective assessments of the continuous probability distribution? Smith suggests that the moments can be estimated by fitting a distribution to the assessments. This research note shows that the quality of the moment estimates cannot be judged solely by how close the fitted distribution is to the true distribution. Examples are used to show that the relative errors in higher order moment estimates can be greater than 100%, even though the cumulative distribution function is estimated within a Kolmogorov‐Smirnov distance less than 1%.     

Importance of Distributional Form in Characterizing Inputs to Monte Carlo Risk Assessments

The selection among distributional forms for inputs into uncertainty and variability (e.g., Monte Carlo) analyses is an important task. This paper considers the importance of distributional selection by examining the overall and tail behavior of the lognormal, Weibull, gamma, and inverse gaussian distributions. It is concluded that at low relative standard deviation (below 1), there is less of a difference between upper tail behavior among the distributions than at higher RSD values. Sample sizes in excess of 200 are required to reliably distinguish between distributional forms at the higher RSD values. The likelihood statistic appears to offer a reasonable approach to distributional discrimination, and it, or a similar approach, should be incorporated into distributional fitting procedures used in risk analysis.     

APPROXIMATIONS FOR THE GI/G/m QUEUE

Queueing models can usefully represent production systems experiencing congestion due to irregular flows, but exact analyses of these queueing models can be difficult. Thus it is natural to seek relatively simple approximations that are suitably accurate for engineering purposes. Here approximations for a basic queueing model are developed and evaluated. The model is the GI/G/m queue, which has m identical servers in parallel, unlimited waiting room, and the first-come first-served queue discipline, with service and interarrival times coming from independent sequences of independent and identically distributed random variables with general distributions. The approximations depend on the general interarrival-time and service-time distributions only through their first two moments. The main focus is on the expected waiting time and the probability of having to wait before beginning service, but approximations are also developed for other congestion measures, including the entire distributions of waiting time, queue-length and number in system. These relatively simple approximations are useful supplements to algorithms for computing the exact values that have been developed in recent years. The simple approximations can serve as starting points for developing approximations for more complicated systems for which exact solutions are not yet available. These approximations are especially useful for incorporating GI/G/m models in larger models, such as queueing networks, wherein the approximations can be components of rapid modeling tools.     

动态金融高阶矩建模:基于Generalized-t分布和Gram-Charlier展开分布的比较研究

动态时变高阶矩是金融收益率的一个重要特征。本文对比研究了主流的Generalized-t分布(GT)和Gram Charlier Expansion分布(GCE)在GJRGARCH模型下对动态高阶矩的拟合能力和Value-at-Risk的预测能力。基于2005-2014美国标普500股指和中国沪深300股指日收益率的实证结果显示,收益率的条件高阶矩存在显著的时变性和持续性,其中偏度参数的持续性参数达到0.9以上。从各种统计指标综合来看,这两种方法都具有较好的实证表现。尽管GCE分布具有某些高阶矩建模的便利性,GT分布的实证拟合能力更强,对极端概率Value-at-Risk的样本外预测也更加准确。     

CONCEPTS,THEORY, AND TECHNIQUES

The beta distribution is becoming more widely used in business applications and in economic model-building. A simple way to estimate the parameters of the univariate distribution and the general multivariate distribution would be useful but there appears to be no simple way of accomplishing this task. For example, the method of maximum likelihood does not yield tractable results. The method of moments presented in this paper is a practical way to obtain parameter estimates for the beta distribution. Explicit estimators are derived for the univariate and bivariate case and estimators are then inferred for the n-variate case. Empirical tests suggest that the method of moments may be used to establish useful estimates for these parameters. Also, estimates obtained by the method of moments may be used to develop bounds on the parameters so that computational techniques can be utilized with other estimation methods possessing more desirable properties.     

非合作-合作两型博弈的Shapley值纯策略纳什均衡解求解方法

在合作中又有竞争的"经济全球化"时代背景下,经济实体之间越来越多地体现出竞争与合作交织的特点,既有策略的选择,同时也有利益的分配或者成本的分摊,即竞争与合作相互联系。为此,Brandenburger和Stuart提出了非合作-合作两型博弈模型为这类博弈提供了有效的工具。目前非合作-合作两型博弈研究较少,且Brandenburger和Stuart提出的非合作-合作两型博弈存在一些不足：合作博弈用核心求解可能为空或者不唯一。Shapley值是一种重要的合作博弈单值解,满足匿名性、有效性、可加性和虚拟性,表达形式简单且唯一,对一些成本分摊问题和利益分配问题,给决策者提供了一个公平满意的分配方案。因此本文研究将Shapley值作为合作博弈的解时非合作-合作两型博弈解存在的条件。为了分析本文提出的基于Shapley值的非合作-合作两型博弈的新理论框架,首先给出了其特征函数满足的联盟无外部性条件。在满足此条件下,我们进一步证明了非合作-合作两型博弈解存在的条件及性质。结合数值实例比较分析合作博弈用核心和Shapley值求解非合作-合作两型博弈解的优缺点。研究表明：当用Shapley值求解合作博弈解,降低了非合作-合作两型博弈解存在条件。因此,本文的研究不仅弥补了Brandenburger和Stuart提出的非合作-合作两型博弈中合作博弈的核心为空或者不唯一的情况,而且为非合作-合作两型博弈的解提供新的理论框架,从而为既有竞争又有合作的博弈问题提供新的求解方法,因此,本文的研究具有一定的理论价值和应用价值。     

COMPETING FOR OWNERSHIP

We develop a tractable model of the allocation of ownership and control within firms operating in competitive markets. The model permits analysis of how the scarcity of assets in the market translates into ownership structures inside the organization. It identifies a price‐like mechanism whereby local liquidity or productivity shocks propagate and lead to widespread organizational restructuring. Firms will be more integrated when the terms of trade are more favorable to the short side of the market, when liquidity is unequally distributed among existing firms, and following a uniform increase in productivity. Shocks to the first two moments of the liquidity distribution have multiplier effects on the corresponding moments of the distribution of ownership structures. (JEL: D21, D31, D51, D86)     

Competing statistical methods for the fitting of normal species sensitivity distributions: recommendations for practitioners

A species sensitivity distribution (SSD) models data on toxicity of a specific toxicant to species in a defined assemblage. SSDs are typically assumed to be parametric, despite noteworthy criticism, with a standard proposal being the log-normal distribution. Recently, and confusingly, there have emerged different statistical methods in the ecotoxicological risk assessment literature, independent of the distributional assumption, for fitting SSDs to toxicity data with the overall aim of estimating the concentration of the toxicant that is hazardous to % of the biological assemblage (usually with small). We analyze two such estimators derived from simple linear regression applied to the ordered log-transformed toxicity data values and probit transformed rank-based plotting positions. These are compared to the more intuitive and statistically defensible confidence limit-based estimator. We conclude based on a large-scale simulation study that the latter estimator should be used in typical assessments where a pointwise value of the hazardous concentration is required.     

Concepts,Theory, and Techniques ON THE USE OF THE GEOMETRIC MEAN IN LONG-TERM INVESTMENT

Application of the geometric mean to holding-period returns is discussed from a statistical theory standpoint. The population geometric mean is considered a parameter of the probability distribution of returns its relationship to moments of the distribution is discussed. The sample geometric mean and its relation to sample moments is assessed through its sampling distribution it is viewed as an estimator of the population geometric mean. For application to long-term investment where a geometric mean is maximized, the distributional properties of the geometric mean should be used. The terms statistic, approximation, and parameter are differentiated.     

TSP in spreadsheets—A fast and flexible tool

The traveling salesman problem (TSP) is well-known and many specially developed solution procedures have been constructed to solve particular variants of it. This paper considers several different variants of TSP. However, developing tailored solution procedures for each is impractical. These problems are non-deterministic polynomial-time hard (NP hard). Solving them using standard linear programming/mixed integer programming (LP/MIP) solvers has therefore only been regarded to be feasible for very small problems. A careful consideration of the problem formulation may facilitate efficient software utilization, and for real-world problems this can have a considerable impact. Problems that were previously regarded as large and unwieldy are now easily solvable using spreadsheets, thanks to the recent advancement in general optimization software.     

A Simple Formula for Calculating the "Mass Density" of a Lognormally Distributed Characteristic: Applications to Risk Analysis

Statements such as "80% of the employees do 20% of the work" or "the richest 1% of society controls 10% of its assets" are commonly used to describe the distribution or concentration of a variable characteristic within a population. Analogous statements can be constructed to reflect the relationship between probability and concentration for unvarying quantities surrounded by uncertainty. Both kinds of statements represent specific usages of a general relationship, the "mass density function," that is not widely exploited in risk analysis and management. This paper derives a simple formula for the mass density function when the uncertainty and/or the variability in a quantity is lognormally distributed; the formula gives the risk analyst an exact, "back-of-the-envelope" method for determining the fraction of the total amount of a quantity contained within any portion of its distribution. For example, if exposures to a toxicant are lognormally distributed with σ_{in x}= 2, 50% of all the exposure is borne by the 2.3% of persons most heavily exposed. Implications of this formula for various issues in risk assessment are explored, including: (1) the marginal benefits of risk reduction; (2) distributional equity and risk perception; (3) accurate confidence intervals for the population mean when a limited set of data is available; (4) the possible biases introduced by the uncritical assumption that extreme "outliers" exist; and (5) the calculation of the value of new information.     

At Home on the Range? Lay Interpretations of Numerical Uncertainty Ranges

Numerical uncertainty ranges are often used to convey the precision of a forecast. In three studies, we examined how users perceive the distribution underlying numerical ranges and test specific hypotheses about the display characteristics that affect these perceptions. We discuss five primary conclusions from these studies: (1) substantial variation exists in how people perceive the distribution underlying numerical ranges; (2) distributional perceptions appear similar whether the uncertain variable is a probability or an outcome; (3) the variation in distributional perceptions is due in part to individual differences in numeracy, with more numerate individuals more likely to perceive the distribution as roughly normal; (4) the variation is also due in part to the presence versus absence of common cues used to convey the correct interpretation (e.g., including a best estimate increases perceptions that the distribution is roughly normal); and (5) simple graphical representations can decrease the variance in distributional perceptions. These results point toward significant opportunities to improve uncertainty communication in climate change and other domains.     

ON ESTIMATING PARAMETERS FOR BETA DISTRIBUTIONS

The purpose of this paper is to comment on and give historical perspective to two methdologies for estimating parameters of beta distributions. Fielitz and Myers [3] [4] developed and advocated a methodology using the method of moments, while Romesburg [20] advocated a methodology usingthe method of maximum likelihood. However, what Fielitz and Myers presented as new research and suggested as an area needing further study is ground already trampled. The authors have prepared a graph to underline the superiority of the maximum likelihood method in fitting beta distributions.     

OPTIMAL PORTFOLIO SELECTION WITH UPPER BOUNDS FOR INDIVIDUAL SECURITIES*

In this paper, we consider optimal portfolio selection with no short sales and with upper bounds for individual securities. The solution is reached by directy revising the optimal portfolio without upper bounds. Specifically, our analysis is based on the single-index model, as well as the general multi-index model that provides the return generating process for securities in the arbitrage pricing theory. As demonstrated in a simulation study, the proposed algorithm for optimal portfolio selection usually requires very few iterations. Also, since our approach is developed using intuitive reasoning and simple linear algebra, we are able to provide direct and intuitive justifications for the resulting portfolio choice. Therefore this paper should be of interest to both finance academics and practitioners in portfolio management.     

HEURISTICS FOR INTEGER PROGRAMMING USING SURROGATE CONSTRAINTS

This paper proposes a class of surrogate constraint heuristics for obtaining approximate, near optimal solutions to integer programming problems. These heuristics are based on a simple framework that illuminates the character of several earlier heuristic proposals and provides a variety of new alternatives. The paper also proposes additional heuristics that can be used either to supplement the surrogate constraint procedures or to provide independent solution strategies. Preliminary computational results are reported for applying one of these alternatives to a class of nonlinear generalized set covering problems involving approximately 100 constraints and 300–500 integer variables. The solutions obtained by the tested procedure had objective function values twice as good as values obtained by standard approaches (e.g., reducing the best objective function values of other methods from 85 to 40 on the average. Total solution time for the tested procedure ranged from ten to twenty seconds on the CDC 6600.     

A NOTE ON PROCEDURES FOR TESTING THE QUALITY OF A CLUSTERING OF A SET OF OBJECTS

Despite the increased application of cluster analysis in decision sciences, few attempts have been made to derive hypothesis-testing procedures for the evaluation of clustering solutions. In fact, the present paper shows that at least one such attempt failed to specify a meaningful sampling distribution for the test procedure. An alternative index based on the concept of point-biserial correlation is proposed as a possible recovery measure. The index is subsequently used to form the basis of a valid statistical test for the existence of cluster structure.     

A Practical Two‐Step Method for Testing Moment Inequalities

This paper considers the problem of testing a finite number of moment inequalities. We propose a two‐step approach. In the first step, a confidence region for the moments is constructed. In the second step, this set is used to provide information about which moments are “negative.” A Bonferonni‐type correction is used to account for the fact that, with some probability, the moments may not lie in the confidence region. It is shown that the test controls size uniformly over a large class of distributions for the observed data. An important feature of the proposal is that it remains computationally feasible, even when the number of moments is large. The finite‐sample properties of the procedure are examined via a simulation study, which demonstrates, among other things, that the proposal remains competitive with existing procedures while being computationally more attractive.     

Correcting the Errors: Volatility Forecast Evaluation Using High‐Frequency Data and Realized Volatilities

We develop general model‐free adjustment procedures for the calculation of unbiased volatility loss functions based on practically feasible realized volatility benchmarks. The procedures, which exploit recent nonparametric asymptotic distributional results, are both easy‐to‐implement and highly accurate in empirically realistic situations. We also illustrate that properly accounting for the measurement errors in the volatility forecast evaluations reported in the existing literature can result in markedly higher estimates for the true degree of return volatility predictability.