Asymptotically optimal estimation of smooth functionals for interval censoring, part 2

Estimation in the interval censoring model is considered. A class of smooth functionals is introduced, of which the mean is an example. We consider case 2, with two observation times for each unobservable event time, in the situation that the observation times cannot become arbitrarily close to each other. It is proved that the nonparametric maximum likelihood estimator of the functional asymptotically reaches the information lower bound.     

Two heuristics for the deterministic,single operator,multiple machine,multiple run cyclic scheduling problem

The semiautomatic nature of machinery often makes it economical to assign more than one machine to a single operator. Multi-machine assignments are common in the textile, tooling and molding industries. They are also found where numerically controlled (NC) machinery are used and, more recently, where flexible manufacturing systems (FMS) are employed. Previous work on deterministic cyclic scheduling models has focused on determining the optimal number of usually identical machines with single runs to assign to an operator. In practice, the schedules are represented by man/machine charts.When, due to production requirements, we mix different types of runs with known times on non-identical machines, we have the deterministic, single operator, multiple machine, multiple run, cyclic scheduling problem. We present two heuristics for solving this more realistic generalization of earlier problems. For the nonidentical, multiple run case, the scheduling of the runs is crucial in minimizing the cycle time of the system. The integer programming formulation of small problems is large, and solving it directly could require excessive computation time on a large mainframe computer.Heuristic 1 selects the next machine run to schedule by minimizing the total immediate waiting cost of the operator and machines. The hourly machine costs reflect the relative merit of utilizing certain machines over others. Heuristic 2 first schedules the machine with the longest automatic processing time of run one. It then follows Heuristic 1 until the long processing time of the first run has ended. The next available run with the longest automatic processing time is then scheduled, and the process repeats. The underlying notion is that many short runs may be performed during the long automatic run of a machine.The heuristics are polynomially bounded, can be easily implemented on a mini- or micro-computer and in practice should be much faster than integer programming methods. In addition to the heuristics, we compute a lower bound on the cycle time. We use this bound as a measure of the effectiveness of a solution. If for a given schedule, the cycle time equals its lower bound, then the solution is optimal.Both heuristics were coded in FORTRAN on a CDC-6600 computer. An interactive version was also developed for a DEC PDP11/70. A detailed computational study is presented. In it, both heuristics solved 50 machine, 5 run problems in less than 10 CPU seconds on the CDC-6600. Computational experience indicates that the heuristics are efficient and often find schedules which have cycle times within 10% the lower bound.     

RESTORING THE PRINCIPLE OF MINIMUM DIFFERENTIATION IN PRODUCT POSITIONING

Most research on product positioning supports the idea of differentiation. Product standardization (i.e., minimum differentiation) occurs only under very limiting assumptions. Yet, similar products are often observed in the marketplace. We attempt to restore the case for standardization by using more realistic assumptions than in previous work. We assume that consumers consider not only observable attributes in brand choice, but also attributes that are unobservable by the firms. We find that standardization is an equilibrium when consumers exhibit sufficient heterogeneity along the unobservable attributes under both positioning with exogenously given prices and price competition, We also show that, under insufficient heterogeneity along the unobservable attribute, our results coincide with past research that argues in favor of differentiation.     

Instrumental variables estimators of nonparametric models with discrete endogenous regressors

This paper discusses estimation of nonparametric models whose regressor vectors consist of a vector of exogenous variables and a univariate discrete endogenous regressor with finite support. Both identification and estimators are derived from a transform of the model that evaluates the nonparametric structural function via indicator functions in the support of the discrete regressor. A two-step estimator is proposed where the first step constitutes nonparametric estimation of the instrument and the second step is a nonparametric version of two-stage least squares. Linear functionals of the model are shown to be asymptotically normal, and a consistent estimator of the asymptotic covariance matrix is described. For the binary endogenous regressor case, it is shown that one functional of the model is a conditional (on covariates) local average treatment effect, that permits both unobservable and observable heterogeneity in treatments. Finite sample properties of the estimators from a Monte Carlo simulation study illustrate the practicability of the proposed estimators.     

Asymptotic normality of the NPMLE of linear functionals for interval censored data, case 1

We give a new proof of the asymptotic normality of a class of linear functionals of the nonparametric maximum likelihood estimator (NPMLE) of a distribution function with "case 1" interval censored data. In particular our proof simplifies the proof of asymptotic normality of the mean given in Groeneboom and Wellner (1992). The proof relies strongly on a rate of convergence result due to van de Geer (1993), and methods from empirical process theory.     

基于FTP的地基微波辐射计数据实时传输系统

本系统实现了地基微波辐射计单站与中心站的通信。将各个辐射计站点的观测资料、状态参数、定标数据和报警信息等数据实时上传至中心站计算机,方便中心站对管辖范围内的地基微波辐射计单站工作状态的控制,并对各站数据综合处理;同时各单站计算机也可以从中心站计算机中下载其他站点的数据信息,实现各单站计算机与中心站计算机之间的数据资源共享。在程序实现上,以VB6.0为开发平台,利用Microsoft Internet Transfer Control6.0等多种控件。文章给出了本数据传输系统的网络拓扑图、各模块的功能设置及实现功能使用的关键函数。主要功能包括:用户登录、文件存放路径设置、文件上传、文件下载。     

On sequential comparisons of means of first-order autoregressive models

The problem of estimating a linear combination,μ, of means ofp-independent, first-order autoregressive models is considered. Sequential procedures are derived (i) to estimateμ pointwise using the linear combination of sample means, subject to a loss function (squared error plus cost per observation), and (ii) to arrive at a fixed-width confidence interval forμ. It is observed that in the case of point estimation we do not require a sampling scheme, where as in the case of interval estimation we do require a sampling scheme and a scheme similar to the one given in Mukhopadhyay and Liberman (1989) is proposed. All the first order efficiency properties of the sequential procedures involved here are derived. This paper is an extension of results of Sriram (1987) involving one time series to multiple time series. Research supported by AFOSR Grant number 89-0225.     

Additivity with multiple priors

The functional defined as the `min' of integrals with respect to probabilities in a given non-empty closed and convex class appears prominently in recent work on uncertainty in economics. In general, such a functional violates the additivity of the expectations operator. We characterize the types of functions over which additivity of this functional is preserved. This happens exactly when `integrating' functions which are positive affine transformations of each other (or when one is constant). We show that this result is quite general by restricting the types of classes of probabilities considered. Finally, we prove that with a very peculiar exception, all the results hold more generally for functionals which are linear combinations of the `min' and the `max' functional.     

An information theoretic argument for the validity of the exponential model

Based on the Cramér-Rao inequality (in the multiparameter case) the lower bound of Fisher information matrix is achieved if and only if the underlying distribution is ther-parameter exponential family. This family and the lower bound of Fisher information matrix are characterized when some constraints in the form of expected values of some statistics are available. If we combine the previous results we can find the class of parametric functions and the corresponding UMVU estimators via Cramér-Rao inequality.     

Density estimation in the uniform deconvolution model

We consider the problem of estimating a probability density function based on data that are corrupted by noise from a uniform distribution. The (nonparametric) maximum likelihood estimator for the corresponding distribution function is well defined. For the density function this is not the case. We study two nonparametric estimators for this density. The first is a type of kernel density estimate based on the empirical distribution function of the observable data. The second is a kernel density estimate based on the MLE of the distribution function of the unobservable (uncorrupted) data.     

Endogenous Group Formation in Contests: Unobservable Sharing Rules

We study contests in which players compete by expending irreversible effort to win a prize, the prize is awarded to one of the players, the winner shares the prize with other players in his group, if any, and each group's sharing rule is unobservable to the other groups and the singletons, if any, when the players expend their effort. The number of groups, their sizes, and the number of singletons are exogenous in the first model, whereas they are endogenous in the second model. We show that group formation occurs if the number of players is four or smaller, but does not occur otherwise. We examine the effect of endogenous group formation on total effort level and the profitability of endogenous group formation. In each of the two models, comparing the outcomes of the case of unobservable sharing rules with those of the case of observable sharing rules, we show that the two cases yield quite different outcomes.     

Information provision and consumer search behavior for products with asymmetric uncertainty

When selling multiple products with asymmetric uncertainty, should the seller disclose product information so that customers do not have to incur any cost to resolve their uncertainties; if so, which product should the seller choose? To address these questions, we consider a monopolist selling two substitutable products to a group of consumers. Each consumer has asymmetric uncertainty regarding the two products. A total of four different information provision structures are considered based on whether the seller discloses information about each product with the aim of determining which strategy provides the seller with the greatest revenue. We derive several interesting results. First, the optimal information provision strategy depends on the magnitude of uncertainty in relation to the product with lower uncertainty. Specifically, if the uncertainty regarding the product with lower uncertainty is sufficiently small, it is optimal for the seller to provide information about the product with higher uncertainty, otherwise, the seller should provide information about both products. Second, when only one product's information should be revealed, it is optimal for the seller to choose the product with higher uncertainty and charge a higher price. Third, withholding information on both products is never optimal for the seller. Finally, our main model is extended by examining the Mean-Preserving Spread setting, and the robustness of our main results is confirmed. Furthermore, we examine the situation in which a monopolist sells a single product with two main attributes. We find that each of the four information provision strategies can be optimal under various scenarios.     

Uncertainty in measurement and control

Summary  Controlling is intervening in a situation on the basis of measurements. The three elements occurring in this definition may each contain an uncertainty that sets a limit to the control efficiency:
The measurements may be in error owing to both inaccuracies and sluggishness of the measurements. Proper and rapid data-processing is essential therefore.
The interventions may lose part of their effect through over-determinancy or dynamically unfavourable (sluggish) responses. In either case statistical calculations disclose the average control errors that are liable to be made, and also the way in which these can be minimized.
The situation may be unclear, i.e. the static and dynamic process characteristics are insufficiently known. In such cases regression and correlation techniques may aid in finding a solution.
Following a general review, we shall discuss in more detail those aspects that are bound up with the dynamics of the phenomena. Arguments from information theory reveal that the dynamic efficiency of control actions depends on correlation functions of time series (disturbances) and response curves of the systems (processes). The effects of disturbance correlation time, disturbance variance, sampling time, sample treatment, measuring errors, measuring time, rapidity of the intervention, etc., on the efficiency are elucidated by means of some formulae, graphs and examples.     

基于中心化存储和横向转运的选址——库存联合决策模型

针对单一产品、单周期、单一供应商、多配送中心、供应商和需求点位置已知的情况,假定（Q,R）库存检查策略、给定备选地址、无容量约束、各需求点的需求完全独立且均服从正态分布、配送中心级和需求级同时采取中心化安全库存策略、允许配送中心间及同一配送线路内的需求点间通过横向转运对缺货进行瞬间补充、不计缺货成本和转运时间,构建了二层规划模型。上层规划解决选址及需求点的分配问题;下层规划确定配送中心级和需求级的中心化存储方案。发现如果考虑库存持有成本、转运成本等因素的影响,安全库存的集中存放策略并不是聚集在某一个地方,而是在多个地方进行有限、适度的集聚,其成本更低。     

Efficiency of iterated WLS in the linear model with completely unknown heteroskedasticity

Iterated weighted least squares (IWLS) is investigated for estimating the regression coefficients in a linear model with symmetrically distributed errors. The variances of the errors are not specified; it is not assumed that they are unknown functions of the explanatory variables nor that they are given in some parametric way.
IWLS is carried out in a random number of steps, of which the first one is OLS. In each step the error variance at time t is estimated with a weighted sum of m squared residuals in the neighbourhood of t and the coefficients are estimated using WLS. Furthermore an estimate of the co-variance matrix is obtained. If this estimate is minimal in some way the iteration process is stopped.
Asymptotic properties of IWLS are derived for increasing sample size n . Some particular cases show that the asymptotic efficiency can be increased by allowing more than two steps. Even asymptotic efficiency with respect to WLS with the true error variances can be obtained if m is not fixed but tends to infinity with n and if the heteroskedasticity is smooth.     

Minimum return guarantees with fund switching rights—An optimal stopping problem

Recently, there is a growing trend to offer guaranteed products where the investor is allowed to shift her account/investment value between multiple funds. The switching right is granted a finite number of times per year, i.e. it is American style with multiple exercise possibilities. In consequence, the pricing and the risk management is based on the switching strategy which maximizes the value of the guarantee put-option. We analyze the optimal stopping problem in the case of one switching right within different model classes and compare the exact price with the lower price bound implied by the optimal deterministic switching time. We show that, within the class of log-price processes with independent increments, the stopping problem is solved by a deterministic stopping time if (and only if) the price process is in addition continuous. Thus, in a sense, the Black and Scholes model is the only (meaningful) pricing model where the lower price bound gives the exact price. It turns out that even moderate deviations from the Black and Scholes model assumptions give a lower price bound which is really below the exact price. This is illustrated by means of a stylized stochastic volatility model setup.     

A heuristic solution procedure for the multi-item,single-level,limited capacity,lot-sizing problem

The problem to be considered is that of determining lot-sizes for a group of products which are produced at a single workcentre. It is assumed that the requirements for each product are known, period by period, out to the end of some common time horizon. (A reasonable assumption in a Material Requirements Planning context when we are dealing with components of one or more other items already scheduled.) For each product there is a fixed setup cost incurred each time production takes place. Unit production and holding costs are linear. The time required to set up the machine is assumed to be negligible. All costs and production rates can vary from product to product but not with respect to time. In each period there is a finite amount of machine time available that can vary from period to period. The objective is to determine lot-sizes so that 1) costs are minimized, 2) no backlogging occurs and 3) capacity is not exceeded.An exact solution to this complex problem is out of the question. Therefore, a simple heuristic has been developed which guarantees a feasible solution, if one exists. Results of a large number of test problems, including three supplied by industrial sources, are presented. The results indicate that the heuristic will usually generate a very good solution with a relatively small amount of computational effort.     

Uncertainty in measurement and control

Summary Controlling is intervening in a situation on the basis of measurements. The three elements occurring in this definition may each contain an uncertainty that sets a limit to the control efficiency: The measurements may be in error owing to both inaccuracies and sluggishness of the measurements. Proper and rapid data-processing is essential therefore. The interventions may lose part of their effect through over-determinancy or dynamically unfavourable (sluggish) responses. In either case statistical calculations disclose the average control errors that are liable to be made, and also the way in which these can be minimized. The situation may be unclear, i.e. the static and dynamic process characteristics are insufficiently known. In such cases regression and correlation techniques may aid in finding a solution. Following a general review, we shall discuss in more detail those aspects that are bound up with the dynamics of the phenomena. Arguments from information theory reveal that the dynamic efficiency of control actions depends on correlation functions of time series (disturbances) and response curves of the systems (processes). The effects of disturbance correlation time, disturbance variance, sampling time, sample treatment, measuring errors, measuring time, rapidity of the intervention, etc., on the efficiency are elucidated by means of some formulae, graphs and examples.     

Optimal requirement allocation among quantity-discount quoting suppliers

We consider a problem motivated by a central purchasing organization for a major office products distributor. This purchasing organization must source a quantity of a particular resale item from a set of capacitated suppliers. In our case each supplier offers an incremental quantity discount purchase price structure. The purchaser’s objective is to obtain a quantity of a required item at minimum cost. The resulting problem is one of allocating order quantities among an approved supply base and involves minimizing the sum of separable piecewise linear concave cost functions. We develop a branch and bound algorithm that arrives at an optimal solution by generating linear knapsack subproblems with feasible solutions to the original problem. This research was partially supported by a 2007 Summer Research Grant awarded to Asoo J. Vakharia.