Classification performance of mathematical programming techniques in discriminant analysis: Results for small and medium sample sizes

The performance on small and medium-size samples of several techniques to solve the classification problem in discriminant analysis is investigated. The techniques considered are two widely used parametric statistical techniques (Fisher's linear discriminant function and Smith's quadratic function), and a class of recently proposed nonparametric estimation techniques based on mathematical programming (linear and mixed-integer programming). A simulation study is performed, analyzing the relative performance of the above techniques in the two-group case, for various small sample sizes, moderate group overlap and across six different data conditions. Training samples as well as validation samples are used to assess the classificatory performance of the techniques. The degree of group overlap and sample sizes selected for analysis in this paper are of interest in practice because they closely reflect conditions of many real data sets. The results of the experiment show that Smith's nonlinear quadratic function tends to be superior on the training samples and validation samples when the variances–covariances across groups are heterogeneous, while the mixed-integer technique performs best on the training samples when the variances–covariances are equal, and on validation samples with equal variances and discrete uniform independent variables. The mixed-integer technique and the quadratic discriminant function are also found to be more sensitive than the other techniques to the sample size, giving disproportionally inaccurate results on small samples.     

Estimating demand with distance functions: Parameterization in the primal and dual

Our purpose is to investigate the ability of different parametric forms to ‘correctly’ estimate consumer demands based on distance functions using Monte Carlo methods. Our approach combines economic theory, econometrics and quadratic approximation. We begin by deriving parameterizations for transformed quadratic functions which are linear in parameters and characterized by either homogeneity or which satisfy the translation property. Homogeneity is typical of Shephard distance functions and expenditure functions, whereas translation is characteristic of benefit/shortage or directional distance functions. The functional forms which satisfy these conditions and include both first- and second-order terms are the translog and quadratic forms, respectively. We then derive a primal characterization which is homogeneous and parameterized as translog and a dual model which satisfies the translation property and is specified as quadratic. We assess functional form performance by focusing on empirical violations of the regularity conditions. Our analysis corroborates results from earlier Monte Carlo studies on the production side suggesting that the quadratic form more closely approximates the ‘true’ technology or in our context consumer preferences than the translog.     

On estimating nonnegative definite quadratic forms

It is often required to estimate a quadratic form in survey sampling, especially when one has to estimate the mean squared error of a linear estimator of the population total. In this note we consider the problem of obtaining uniformly nonnegative quadratic unbiased estimators for nonnegative definite quadratic forms. The estimators considered here are necessarily quadratic. Received January 1997     

Statistical Decision Problems and Bayesian Nonparametric Methods

This paper considers parametric statistical decision problems conducted within a Bayesian nonparametric context. Our work was motivated by the realisation that typical parametric model selection procedures are essentially incoherent. We argue that one solution to this problem is to use a flexible enough model in the first place, a model that will not be checked no matter what data arrive. Ideally, one would use a nonparametric model to describe all the uncertainty about the density function generating the data. However, parametric models are the preferred choice for many statisticians, despite the incoherence involved in model checking, incoherence that is quite often ignored for pragmatic reasons. In this paper we show how coherent parametric inference can be carried out via decision theory and Bayesian nonparametrics. None of the ingredients discussed here are new, but our main point only becomes evident when one sees all priors—even parametric ones—as measures on sets of densities as opposed to measures on finite-dimensional parameter spaces.     

A non-parametric statistical model for the control of Italian insurance companies

The problem of evaluating the solvency of insurance companies is tackled through the use of a non-parametric statistical model, constructed using decision-tree techniques. The model is tested on a sample of Italian non-life insurance companies and its performance over the test period compared with those of linear and quadratic parametric models.

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Riassunto Il problema della valutazione della solvibilità delle imprese di assicurazione è affrontato con l'impiego di un modello statistico non parametrico, costruito con le tecniche degli alberi delle decisioni. Viene proposta una sperimentazione del modello su un campione di imprese assicuratrici italiane operanti nei rami nonvita ed effettuata una analisi comparata intertemporale con gli standards di efficienza registrati su modelli parametrici lineare e quadratico.

     

Designs for estimating an extremal point of quadratic regression models in a hyperball

This paper is devoted to studying optimal designs for estimating an extremal point of a multivariate quadratic regression model in the unit hyperball. The problem of estimating an extremal point is reduced to that of estimating certain parameters of a corresponding nonlinear (in parameters) regression model. For this reduced problem truncated locally D-optimal designs are found in an explicit form. The result is a generalization of the results of Fedorov and Müller (1997) for onedimensional quadratic regression function in the unit segment. Received February 2002     

Testing a linear regression model against nonparametric alternatives

Summary As a test statistic for testing goodness-of-fit of a linear regression model, we propose a ratio of quadratic forms measuring the distance between parametric and nonparametric fits, relative to the estimated error variance. The test statistic is a modification of the statistic suggested by H?rdle and Mammen (1988). The asymptotic distribution under the hypothesis is established. The finite sample behaviour of the test is investigated in a Monte Carlo study, and is illustrated for two applications.     

LINEX Loss Functions with Applications to Determining the Optimum Process Parameters

In the classical Taguchi quality model, the symmetric quadratic loss function has been used to measure the loss of quality. However, there are a number of situations in which the symmetric quadratic loss may be inappropriate. In this paper, we proposed an asymmetric loss function, called linear exponential (LINEX) loss function, to determine optimum process parameters for the product quality. When the coefficient of LINEX loss function is small, it will be close to the quadratic loss. Moreover, the trade-off problem between quality and cost will be discussed.     

Decomposition of the Koopmans-Beckmann Problem

Subsequent findings indicate that the Koopmans-Beckmann linear programming representation of the quadratic assignment problem may be more useful in solving for the best integer assignment and a set of sustaining prices than their initial results suggest. In response to the renewed interest in the quadratic assignment problem, this brief paper presents the Koopmans-Beckmann linear program in general matrix form, demostrates that the program is decomposable, and shows that the primal subprogram constitutes a simple linear assignment problem whose optimal solution set always contains at least one integer assignment of plants to locations.     

Quasi-maximum likelihood estimation of volatility with high frequency data

This paper investigates the properties of the well-known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise, by extending the classic asymptotic results of quasi-maximum likelihood estimation. When trying to estimate the integrated volatility and the variance of noise, this parametric approach remains consistent, efficient and robust as a quasi-estimator under misspecified assumptions. Moreover, it shares the model-free feature with nonparametric alternatives, for instance realized kernels, while being advantageous over them in terms of finite sample performance. In light of quadratic representation, this estimator behaves like an iterative exponential realized kernel asymptotically. Comparisons with a variety of implementations of the Tukey–Hanning₂ kernel are provided using Monte Carlo simulations, and an empirical study with the Euro/US Dollar future illustrates its application in practice.     

An optimization model for the allocation of resources in family planning programs

In this paper we present a quadratic programming model for the allocation of resources in a family planning program. A quadratic cost function is minimized subject to three sets of linear constraints: one for a diffusion model of the acceptance, switching and dropout behavior of the target population; one for manpower limitations; and one for incorporating government policies on desired levels of birth rate. There are two important features to be noted. First, a recruitment cost parameter is introduced to account for the effects of market saturation on new acceptance. Second, by treating the problem as one of cost minimization rather than one of birth rate minimization, the model can be utilized to evaluate the feasibility and the minimum cost requirement of government policies on desired reductions in the birth rate. Computational considerations are also discussed.     

On scale-invariant parametric families of functions

In this paper, we attempt to characterize parametric families of functions such that the statement “a function is an element of the parametric family” is meaningful with respect to a given scale of measurement (a statement is said to be meaningful if its truth or falsity is unchanged when admissible transformations are applied to all of the scales in the statement). A few special cases of the problem are solved for nominal, ordinal, and some quantitative scales. As economic applications, axiomatizations of homothetic production functions and the Cobb–Douglas production function are given.     

Coalitional manipulations in a bankruptcy problem

In a bankruptcy problem framework we consider rules immune to possible manipulations by the creditors involved in the problem via merging or splitting of their individual claims. The paper provides characterization theorems for the non manipulable rules, the no advantageous merging parametric rules and the no advantageous splitting parametric rules. Received: 24 February 1998 / accepted: 9 March 1999     

Exact computation of GMM estimators for instrumental variable quantile regression models

We show that the generalized method of moments (GMM) estimation problem in instrumental variable quantile regression (IVQR) models can be equivalently formulated as a mixed‐integer quadratic programming problem. This enables exact computation of the GMM estimators for the IVQR models. We illustrate the usefulness of our algorithm via Monte Carlo experiments and an application to demand for fish.     

Compensation systems for green strategy implementation: parametric and non‐parametric approaches

This paper presents a design for compensation systems for green strategy implementation based on parametric and non‐parametric approaches. The purpose of the analysis is to use formal modeling to explain the issues that arise with the multi‐task problem of implementing an environmental strategy in addition to an already existing profit‐oriented strategy. For the first class of compensation systems (parametric), a multi‐task model is used as a basis. For the second class of compensation systems (non‐parametric), data envelopment analysis is applied.Copyright © 2003 John Wiley & Sons, Ltd. and ERP Environment     

Large-sample estimation strategies for eigenvalues of a Wishart matrix

The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.     

Testing for Seasonal Unit Roots in Monthly Panels of Time Series*

We consider the problem of testing for seasonal unit roots in monthly panel data. To this aim, we generalize the quarterly cross‐sectionally augmented Hylleberg–Engle–Granger–Yoo (CHEGY) test to the monthly case. This parametric test is contrasted with a new non‐parametric test, which is the panel counterpart to the univariate record unit–root seasonal (RURS) test that relies on counting extrema in time series. All methods are applied to an empirical data set on tourism in Austrian provinces. The power properties of the tests are evaluated in simulation experiments that are tuned to the tourism data.     

Quadratic risk domination of restricted least squares estimators via Stein-ruled auxiliary constraints

This paper develops an estimator that under the standard assumption of the General Linear Model, including normality of disturbances, can be designed to dominate the Restricted Least Squares estimator in quadratic risk under very general conditions. The domination is achieved for any choice of symmetric positive definite weighting matrix used in defining the quadratic risk function, regardless of the correctness of the constraints used to define the restricted least squares estimator. The general problem conditions under which the estimator exists, and the risk behavior of the estimator over the parameter space are identified.     

基于运筹优化的物流与设施规划课程教学研究

在传统教学中,物流与设施规划课程的物流分析、设施布局等内容的一些经典问题采用启发式方法或试验法来解决。为了更精确地解决这些问题,采用数学规划方法提供解决方案。主要围绕建模技巧、应用软件教学和解决物流与设施规划专业案例等方面培养学生运用优化原理与方法构建运筹优化模型的能力以及运用ILOG OPL优化软件解决实际优化问题的能力。以二次分配问题和多产品工艺过程图优化问题为案例阐述了数学规划建模和编程求解的过程。     

区间规划在证券投资组合问题中的运用

本文基于区间规划的方法研究了摩擦市场的投资组合选择问题。文中把风险证券的收益率、投资风险及证券的流动性用区间数来描述,并结合绝对偏差风险函数的思想建立了一种关于区间数的证券投资组合选择模型。最后利用区间数的两种序关系将所提出的模糊线性规划问题转化为普通的参数线性规划问题进而求其解。