A two-stage sampling by unequal size samples from two normal populations with unknown variances

A two-stage procedure based on the statistic introduced by Scheffé (1943) to solve the Behrens-Fisher problem is considered in estimating the difference between the mean values of two normal distributions having unequal and unknown variances. Samples of unequal size are considered at each stage. A numerical example is given where the two-stage procedure is compared with the Satterthwaite (1946) approximate method.     

Optimal allocation in J-stage sampling with stratification from finite population

Summary The known results of optimal allocation in two-stage simple random sampling with stratification from finite population are generalized for J-stages. The optimalization is treated by minimizing the cost function for given variance as well as by minimizing the variance for given cost. It turns out that for both cases the optimal values differ only for the first-stage units and are the same for all the subsequent. The general formula for minimum variance of the unbiased estimator of the population mean per element is given.     

Optimal allocation in J-stage sampling with stratification from finite population

Summary  The known results of optimal allocation in two-stage simple random sampling with stratification from finite population are generalized for J-stages. The optimalization is treated by minimizing the cost function for given variance as well as by minimizing the variance for given cost. It turns out that for both cases the optimal values differ only for the first-stage units and are the same for all the subsequent. The general formula for minimum variance of the unbiased estimator of the population mean per element is given.     

禁忌搜索算法在车辆调度决策中的应用

论文探讨了交通线路可能被敌方毁坏情况下的车辆路径优化问题，建立了不确定的两阶段整数规划数学模型，分析了路径方案的最大和最小效用值。在线路方案评价值的计算中只计算最小值和最大值，简化了方案的目标值计算。设计了求解模型的双层禁忌搜索算法。最后通过应用示例验证了模型和算法的有效性。     

On the impact of the tests for serial correlation upon the test of significance for the regression coefficient

Monte Carlo methods are used to investigate the relationship between the power of different pretests for autocorrelation, and the Type I error and power of the significance test for a resulting two-stage estimate of the slope parameter in a simple regression. Our results suggest it may be preferable to always transform without pretesting. Moreover we find little room for improvement in the Type I errors and power of two-stage estimators using existing pretests for autocorrelation, compared with the results obtained given perfect knowledge about when to transform (i.e., given a perfect pretest). Rather, researchers should seek better estimators of the transformation parameter itself.     

Estimation in Two-Stage Models with Heteroscedasticity

A surprising number of important problems can be cast in the framework of estimating a mean and variance using data arising from a two-stage structure. The first stage is a random sampling of "units" with some quantity of interest associated with the unit. The second stage produces an estimate of that quantity and usually, but not always, an estimated standard error, which may change considerably across units. Heteroscedasticity in the estimates over different units can arise for a number of reasons, including variation associated with the unit and changing sampling effort over units. This paper presents a broad discussion of the problem of making inferences for the population mean and variance associated with the unobserved true values at the first stage of sampling. A careful discussion of the causes of heteroscedasticity is given, followed by an examination of ways in which inferences can be carried out in a manner that is robust to the nature of the within unit heteroscedasticity. Among the conclusions are that under any type of heteroscedasticity, an unbiased estimate of the mean and the variance of the estimated mean can be obtained by using the estimates as if they were true unobserved values from the first stage. The issue of using the mean versus a weighted average which tries to account for the heteroscedasticity is also discussed. An unbiased estimate of the population variance is given and the variance of this estimate and its covariance with the estimated mean is provided under various types of heteroscedasticity. The two-stage setting arises in many contexts including the one-way random effects models with replication, meta-analysis, multi-stage sampling from finite populations and random coefficients models. We will motivate and illustrate the problem with data arising from these various contexts with the goal of providing a unified framework for addressing such problems.     

On finite sample properties of alternative estimators of coefficients in a structural equation with many instruments

We compare four different estimation methods for the coefficients of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum empirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most linear models of interest and we include some heteroscedastic cases and nonlinear cases. We have found that the LIML estimator has good performance in terms of the bounded loss functions and probabilities when the number of instruments is large, that is, the micro-econometric models with “many instruments” in the terminology of recent econometric literature.     

A Simulation Study of Joint Uses of Data Envelopment Analysis and Statistical Regressions for Production Function Estimation and Efficiency Evaluation

A previous paper by Arnold, Bardhan, Cooper and Kumbhakar (1996) introduced a very simple method to estimate a production frontier by proceeding in two stages as follows: Data Envelopment Analysis (DEA) is used in the first stage to identify efficient and inefficient decision-making units (DMUs). In the second stage the thus identified DMUs are incorporated as dummy variables in OLS (ordinary least squares) regressions. This gave very satisfactory results for both the efficient and inefficient DMUs. Here a simulation study provides additional evidence. Using this same two-stage approach with Cobb-Douglas and CES (constant elasticity-of-substitution) production functions, the estimated values for the coefficients associated with efficient DMUs are found to be not significantly different from the true parameter values for the (known) production functions whereas the parameter estimates for the inefficient DMUs are significantly different. A separate section of the present paper is devoted to explanations of these results. Other sections describe methods for estimating input-specific inefficiencies from the first stage use of DEA in the two-stage approaches. A concluding section provides further directions for research and use.     

On a use of stratified sampling in optimal estimation

Summary Postulating an intra-class correlation structure for a finite population vector of variate — values optimal stratified and unstratified strategies for estimating the total are identified and relative efficiencies noted. Higher efficiency of stratified over unstratified sampling is demonstrated for usual estimators in ppswr case, in particular. In the uncorrelated case appropriately chosen two-stage strategies are observed to fare at par with optimal stratified ones.     

Two-stage stochastic integer programming: a survey

Stochastic integer programming is more complicated than stochastic linear programming, as will be explained for the case of the two-stage stochastic programming model. A survey of the results accomplished in this recent field of research is given.     

Economic design of a two-stage screening procedure with a prescribed outgoing quality

Summary An economic two-stage screening procedure based on screening and performance variables is proposed. A screening variable is used first to decide whether an item should be accepted, rejected, or undecided. The performance variable is then used to classify the undecided items. The two variables are assumed to have a bivariate normal distribution. A cost model is constructed which involves costs due to screening and performance inspections and misclassification errors. Optimal cutoff values on the screening variable minimizing the expected cost are obtained subject to the constraint that the outgoint quality exceeds a prespecified level. Methods of finding the optimal cutoff values are presented for both parameters-known and parameters-unknown cases.     

供应链中双方供应的竞争和合作博弈研究

针对当前供应链中供应方面存在的相关局限性。构造了双方供应和制造商之间的博弈模型。该模型主要特点是双方 供应之间的二阶段博弈所形成的质量和价格策略。在此条件下制造商和供应商之间进行讨价还价。以分配整体利润。指出了供 需双方在讨价还价时存在惟一的子博彝精炼纳什均衡。     

Two-stage optimal control problems and optimality conditions

When an optimal investment decision is studied for a firm whose capital goods are subject to a delivery lag, a control problem with a time-delay argument in the objective function emerges. Such a problem under a set of simple assumptions is shown to be reducible to a two-stage optimal control problem. The significance of this is that familiar tools from standard optimal control theory are applicable to the resulting two-stage problem. Necessary conditions are presented for a general two-stage problem with an adjustable switching time. Some specific results are also obtained for several special cases.     

Benchmarking by State Space Models

We have a monthly series of observations which are obtained from sample surveys and are therefore subject to survey errors. We also have a series of annual values, called benchmarks, which are either exact or are substantially more accurate than the survey observations; these can be either annual totals or accurate values of the underlying variable at a particular month. The benchmarking problem is the problem of adjusting the monthly series to be consistent with the annual values. We provide two solutions to this problem. The first of these is a two-stage method in which we first fit a state space model to the monthly data alone and then combine the results obtained at this stage with the benchmark data. In the second solution we construct a single series from the monthly and annual values together and fit a state space model to this series in a single stage. The treatment is extended to series which behave multiplicatively. The methods are illustrated by applying them to Canadian retail sales sereis.     

Two-stage contests with effort-dependent values of winning

We study two-stage all-pay contests in which synergy exists between the stages. The value of winning for each contestant is fixed in the first stage while it is effort-dependent in the second one. We assume that a player’s effort in the first stage either increases (positive synergy) or decreases (negative synergy) his value of winning in the second stage. The subgame perfect equilibrium of this contest is analyzed with either positive or negative synergy. We show, in particular, that whether the contestants are symmetric or asymmetric their expected payoffs may be higher under negative synergy than under positive synergy. Consequently, they prefer smaller values of winning (negative synergy) over higher ones (positive synergy).     

Benchmarking regression algorithms for loss given default modeling

The introduction of the Basel II Accord has had a huge impact on financial institutions, allowing them to build credit risk models for three key risk parameters: PD (probability of default), LGD (loss given default) and EAD (exposure at default). Until recently, credit risk research has focused largely on the estimation and validation of the PD parameter, and much less on LGD modeling. In this first large-scale LGD benchmarking study, various regression techniques for modeling and predicting LGD are investigated. These include one-stage models, such as those built by ordinary least squares regression, beta regression, robust regression, ridge regression, regression splines, neural networks, support vector machines and regression trees, as well as two-stage models which combine multiple techniques. A total of 24 techniques are compared using six real-life loss datasets from major international banks. It is found that much of the variance in LGD remains unexplained, as the average prediction performance of the models in terms of R² ranges from 4% to 43%. Nonetheless, there is a clear trend that non-linear techniques, and in particular support vector machines and neural networks, perform significantly better than more traditional linear techniques. Also, two-stage models built by a combination of linear and non-linear techniques are shown to have a similarly good predictive power, with the added advantage of having a comprehensible linear model component.     

Censored regression quantiles with endogenous regressors

This paper develops a semiparametric method for estimation of the censored regression model when some of the regressors are endogenous (and continuously distributed) and instrumental variables are available for them. A “distributional exclusion” restriction is imposed on the unobservable errors, whose conditional distribution is assumed to depend on the regressors and instruments only through a lower-dimensional “control variable,” here assumed to be the difference between the endogenous regressors and their conditional expectations given the instruments. This assumption, which implies a similar exclusion restriction for the conditional quantiles of the censored dependent variable, is used to motivate a two-stage estimator of the censored regression coefficients. In the first stage, the conditional quantile of the dependent variable given the instruments and the regressors is nonparametrically estimated, as are the first-stage reduced-form residuals to be used as control variables. The second-stage estimator is a weighted least squares regression of pairwise differences in the estimated quantiles on the corresponding differences in regressors, using only pairs of observations for which both estimated quantiles are positive (i.e., in the uncensored region) and the corresponding difference in estimated control variables is small. The paper gives the form of the asymptotic distribution for the proposed estimator, and discusses how it compares to similar estimators for alternative models.     

Approximately optimum two-stage sampling plans

Summary In order to compare two-stage sampling plans we use the minimax regret principle i.e. the minimax principle applied to the corresponding regret functions. In this paper approximation formulas for optimum two-stage sampling plans are derived in the case of sampling by attributes as well as in the case of sampling by variables; furthermore a method is presented how to improve the approximate solution in the first case.     

基于两阶段选址法的单点区域型配送中心选址研究

以单点区域型配送中心选址为研究对象,以系统理论、组合理论等为基础,运用定量、定性方法对区域型配送中心选址进行研究,提出了新的选址方法——两阶段单点选址法。借助Matlab等软件的强大功能,进行计算求解,并通过实例验证了该方法的有效性。     

Best response dynamics with level-n expectations in two-stage games

This paper analyzes behavior in repeatedly played two-stage games, where players choose actions in both stages according to best replies using level-n expectations about the opponent׳s actions in both stages. Level-n expectations are recursively defined in a way that a player holding level-n expectations correctly predicts the action of an opponent holding level-(n−1) expectations. A general conceptual framework to study such dynamics for two-stage games is developed and it is shown that, contrary to results for single-stage games, the fixed points of the dynamics depend on the level of the expectations. In particular, for level-0 expectation, fixed points correspond to a Nash equilibrium of a simultaneous move version of the game, whereas (under certain conditions) fixed points converge towards the subgame perfect equilibrium of the two-stage game if the level of expectations goes to infinity. The approach is illustrated using a two-stage duopoly game, where firms in the first stage invest in activities reducing their marginal costs and in the second stage engage in Cournot competition. Conditions for local stability of the fixed points are derived for different levels of expectations and it is shown that level-2 expectations are sufficient to move the fixed-point of the dynamics to a close neighborhood of the subgame perfect equilibrium. Furthermore, it is demonstrated that although firms benefit from unilateral increases in the level of expectations, an increase of n by all firms reduces all profits.