Two-Group Classification Using Neural Networks*

Artificial neural networks are new methods for classification. We investigate two important issues in building neural network models; network architecture and size of training samples. Experiments were designed and carried out on two-group classification problems to find answers to these model building questions. The first experiment deals with selection of architecture and sample size for different classification problems. Results show that choice of architecture and choice of sample size depend on the objective: to maximize the classification rate of training samples, or to maximize the generalizability of neural networks. The second experiment compares neural network models with classical models such as linear discriminant analysis and quadratic discriminant analysis, and nonparametric methods such as k-nearest-neighbor and linear programming. Results show that neural networks are comparable to, if not better than, these other methods in terms of classification rates in the training samples but not in the test samples.     

ON USING LINEAR PROGRAMMING IN DISCRIMINANT PROBLEMS

This commentary evaluates the usefulness of the Freed and Glover [6] linear programming approach to the discriminant problem, relates linear programming to other parametric and nonparametric approaches, and evaluates the linear programming approach.     

FOUR APPROACHES TO THE CLASSIFICATION PROBLEM IN DISCRIMINANT ANALYSIS: AN EXPERIMENTAL STUDY*

Four discriminant models were compared in a simulation study: Fisher's linear discriminant function [14], Smith's quadratic discriminant function [34], the logistic discriminant model, and a model based on linear programming [17]. The study was conducted to estimate expected rates of misclassification for these four procedures when observations were sampled from a variety of normal and nonnormal distributions. In contrast to previous research, data were taken from four types of Kurtotic population distributions. The results indicate the four discriminant procedures are robust toward data from many types of distributions. The misclassification rates for both the logistic discriminant model and the formulation based on linear programming consistently decreased as the kurtosis in the data increased. The decreases, however, were of small magnitude. None of these procedures yielded statistically significant lower rates of misclassification under nonnormality. The quadratic discriminant function produced significantly lower error rates when the variances across groups were heterogeneous.     

Efficient Semiparametric Estimation of the Fama–French Model and Extensions

This paper develops a new estimation procedure for characteristic‐based factor models of stock returns. We treat the factor model as a weighted additive nonparametric regression model, with the factor returns serving as time‐varying weights and a set of univariate nonparametric functions relating security characteristic to the associated factor betas. We use a time‐series and cross‐sectional pooled weighted additive nonparametric regression methodology to simultaneously estimate the factor returns and characteristic‐beta functions. By avoiding the curse of dimensionality, our methodology allows for a larger number of factors than existing semiparametric methods. We apply the technique to the three‐factor Fama–French model, Carhart's four‐factor extension of it that adds a momentum factor, and a five‐factor extension that adds an own‐volatility factor. We find that momentum and own‐volatility factors are at least as important, if not more important, than size and value in explaining equity return comovements. We test the multifactor beta pricing theory against a general alternative using a new nonparametric test.     

QUALITATIVE VARIABLE DISCRIMINANT ANALYSIS AND ITS USE IN PRODUCT VERSION SELECTION

The purpose of this article is to illustrate the use of multiple discriminant analysis in those cases in which all discriminating variables are qualitative. The authors will show that an appropriate qualitative variable discriminant analysis model is a reformulation of the Bayesian decision theoretic model. Rules for the use of multiple discriminant analysis also are suggested for the cases in which some variables are qualitative and some are measurable on at least an interval scale. The qualitative variable discriminant analysis model is illustrated as an appropriate device for selecting a product version that will minimize a manufacturer's risk of product introduction.     

TOWARD UNDERSTANDING THE CONTENTS OF THE “BLACK BOX” FOR PREDICTING COMPLEX DECISION-MAKING OUTCOMES

The purpose of this research is to show the usefulness of three relatively simple nonlinear classification techniques for policy-capturing research where linear models have typically been used. This study uses 480 cases to assess the decision-making process used by 24 experienced national bank examiners in classifying commercial loans as acceptable or questionable. The results from multiple discriminant analysis (a linear technique) are compared to those of chi-squared automatic interaction detector analysis (a search technique), log-linear analysis, and logit analysis. Results show that while the four techniques are equally accurate in predicting loan classification, chi-squared automatic interaction detector analysis (CHAID) and log-linear analysis enable the researcher to analyze the decision-making structure and examine the “human” variable within the decision-making process. Consequently, if the sole purpose of research is to predict the decision maker's decisions, then any one of the four techniques turns out to be equally useful. If, however, the purpose is to analyze the decision-making process as well as to predict decisions, then CHAID or log-linear techniques are more useful than linear model techniques.     

A Comparison of Linear Programming and Parametric Approaches to the Two-Group Discriminant Problem*

Recent simulation-based studies of linear programming models for discriminant analysis have used the Fisher linear discriminant function as the benchmark for parametric methods. This article reports experimental evidence which suggests that, while some linear programming models may match or even exceed the Fisher approach in classification accuracy, none of the fifteen models tested is as accurate on normally distributed data as the Smith quadratic discriminant function. At the minimum, further testing is warranted with an emphasis on data sets that arise from significantly non-Gaussian populations.     

单指标非参数期权定价——改进的非参数定价方法

本文建立一种改进的非参数期权定价模型,称为单指标非参数期权定价模型。相比现有非参数回归期权定价模型是期权价格关于各个因素的多元回归函数,本模型通过变量变换把期权价格多个因素指标转换为一个综合变量——单指标,得到期权价格关于单指标的一元非参数回归方程。改进的模型实现了多元非参数期权定价模型的降维和简化了模型计算;还通过多个期限期权的单指标组合解决了非参数估计的样本数量问题;以及通过期限平滑解决了现有非参数定价模型中的日历效应问题。选取上证50ETF期权数据实证分析表明,无论是样本内的估计结果还是样本外的预测结果都比传统的Black-Scholes模型、半参数Black-Scholes模型和多元非参数回归期权定价模型估计效果有提高。     

Estimating Semiparametric ARCH(∞) Models by Kernel Smoothing Methods1

We investigate a class of semiparametric ARCH(∞) models that includes as a special case the partially nonparametric (PNP) model introduced by Engle and Ng (1993) and which allows for both flexible dynamics and flexible function form with regard to the “news impact” function. We show that the functional part of the model satisfies a type II linear integral equation and give simple conditions under which there is a unique solution. We propose an estimation method that is based on kernel smoothing and profiled likelihood. We establish the distribution theory of the parametric components and the pointwise distribution of the nonparametric component of the model. We also discuss efficiency of both the parametric part and the nonparametric part. We investigate the performance of our procedures on simulated data and on a sample of S&P500 index returns. We find evidence of asymmetric news impact functions, consistent with the parametric analysis.     

A use of a nonparametric statistic for DEA frontier shift: the Kruskal and Wallis rank test

A new use of the nonparametric statistic, referred to as the “Kruskal and Wallis rank test”, is proposed in this study. The nonparametric statistic examines whether or not any frontier shift occurs among observed periods. To document its practicality, the proposed statistic is incorporated into the framework of Window Malmquist Analysis (WMA) that is structured by combining Data Envelopment Analysis (DEA) window analysis with the Malmquist index approach. As an important case study, this research applies the new technique to examine the performance of Japanese postal services from 1983 to 1997. Two policy implications are derived from the empirical study.     

A Note on Discriminant Analysis Using LAD

Using a regression approach to discriminant analysis is often incorrect because it forces the use of a binary dependent variable which violates virtually any distributional assumption for a linear model. However, assuming a Laplace distribution in an LP framework leads to a theoretical foundation for MSD discriminant analysis.     

Solving the Classification Problem in Discriminant Analysis Via Linear and Nonlinear Programming Methods*

This paper demonstrates the feasibility of applying nonlinear programming methods to solve the classification problem in discriminant analysis. The application represents a useful extension of previously proposed linear programming-based solutions for discriminant analysis. The analysis of data obtained by conducting a Monte Carlo simulation experiment shows that these new procedures are promising. Future research that should promote application of the proposed methods for solving classification problems in a business decision-making environment is discussed.     

A Variable Selection Criterion in the Linear Programming Approaches to Discriminant Analysis*

There are numerous variable selection rules in classical discriminant analysis. These rules enable a researcher to distinguish significant variables from nonsignificant ones and thus provide a parsimonious classification model based solely on significant variables. Prominent among such rules are the forward and backward stepwise variable selection criteria employed in statistical software packages such as Statistical Package for the Social Sciences and BMDP Statistical Software. No such criterion currently exists for linear programming (LP) approaches to discriminant analysis. In this paper, a criterion is developed to distinguish significant from nonsignificant variables for use in LP models. This criterion is based on the “jackknife” methodology. Examples are presented to illustrate implementation of the proposed criterion.     

An Empirical,Nonparametric Simulator for Multivariate Random Variables with Differing Marginal Densities and Nonlinear Dependence with Hydroclimatic Applications

Multivariate simulations of a set of random variables are often needed for risk analysis. Given a historical data set, the goal is to develop simulations that reproduce the dependence structure in that data set so that the risk of potentially correlated factors can be evaluated. A nonparametric, copula‐based simulation approach is developed and exemplified. It can be applied to multiple variables or to spatial fields with arbitrary dependence structures and marginal densities. The nonparametric simulator uses logspline density estimation in the univariate setting, together with a sampling strategy to reproduce dependence across variables or spatial instances, through a nonparametric numerical approximation of the underlying copula function. The multivariate data vectors are assumed to be independent and identically distributed. A synthetic example is provided to illustrate the method, followed by an application to the risk of livestock losses in Mongolia.     

Nonparametric Bayesian Methods for Benchmark Dose Estimation

The article proposes and investigates the performance of two Bayesian nonparametric estimation procedures in the context of benchmark dose estimation in toxicological animal experiments. The methodology is illustrated using several existing animal dose‐response data sets and is compared with traditional parametric methods available in standard benchmark dose estimation software (BMDS), as well as with a published model‐averaging approach and a frequentist nonparametric approach. These comparisons together with simulation studies suggest that the nonparametric methods provide a lot of flexibility in terms of model fit and can be a very useful tool in benchmark dose estimation studies, especially when standard parametric models fail to fit to the data adequately.     

A NOTE ON THE ASSESSMENT OF THE RESULTS IN A DISCRIMINANT ANALYSIS *

This paper describes two statistical measures that can be applied to the analysis of the results in a discriminant analysis. These measures, similar to R² and R² in multiple regression, assess the “goodness of fit” of the model or the degree of separation established by the discriminant functions among the groups in the sample and in the population.     

Structural Nonparametric Cointegrating Regression

Nonparametric estimation of a structural cointegrating regression model is studied. As in the standard linear cointegrating regression model, the regressor and the dependent variable are jointly dependent and contemporaneously correlated. In nonparametric estimation problems, joint dependence is known to be a major complication that affects identification, induces bias in conventional kernel estimates, and frequently leads to ill‐posed inverse problems. In functional cointegrating regressions where the regressor is an integrated or near‐integrated time series, it is shown here that inverse and ill‐posed inverse problems do not arise. Instead, simple nonparametric kernel estimation of a structural nonparametric cointegrating regression is consistent and the limit distribution theory is mixed normal, giving straightforward asymptotics that are useable in practical work. It is further shown that use of augmented regression, as is common in linear cointegration modeling to address endogeneity, does not lead to bias reduction in nonparametric regression, but there is an asymptotic gain in variance reduction. The results provide a convenient basis for inference in structural nonparametric regression with nonstationary time series when there is a single integrated or near‐integrated regressor. The methods may be applied to a range of empirical models where functional estimation of cointegrating relations is required.     

Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals

This paper studies nonparametric estimation of conditional moment restrictions in which the generalized residual functions can be nonsmooth in the unknown functions of endogenous variables. This is a nonparametric nonlinear instrumental variables (IV) problem. We propose a class of penalized sieve minimum distance (PSMD) estimators, which are minimizers of a penalized empirical minimum distance criterion over a collection of sieve spaces that are dense in the infinite‐dimensional function parameter space. Some of the PSMD procedures use slowly growing finite‐dimensional sieves with flexible penalties or without any penalty; others use large dimensional sieves with lower semicompact and/or convex penalties. We establish their consistency and the convergence rates in Banach space norms (such as a sup‐norm or a root mean squared norm), allowing for possibly noncompact infinite‐dimensional parameter spaces. For both mildly and severely ill‐posed nonlinear inverse problems, our convergence rates in Hilbert space norms (such as a root mean squared norm) achieve the known minimax optimal rate for the nonparametric mean IV regression. We illustrate the theory with a nonparametric additive quantile IV regression. We present a simulation study and an empirical application of estimating nonparametric quantile IV Engel curves.     

基于非参数估计框架的期望效用最大化最优投资组合

本文基于期望效用最大化和非参数估计框架研究了最优投资组合选择问题。和以往大多文献假定资产收益率服从某些特定分布不同资产收益率的分布类型无需作任何假设。首先在一般效用函数下,利用组合收益率密度函数的非参数核估计给出了期望效用的基本非参数估计公式,并建立了期望效用最大化投资组合选择问题的基本框架。然后,在投资者具有幂效用函数的假定下,给出了期望效用具体的非参数计算公式,并给出了求解最大期望效用的数值算法。最后,利用中国证券交易所11支股票日收益率的真实数据给出了一个数值算例。本文提出的非参数估计框架具有一般性,还可以进一步用来研究各种现实条件下（如各种现实不等式约束和具有交易成本）的投资组合管理问题。