How Important Is the Effect of Rounding in Goodness-of-Fit

Abstract

In the area of goodness-of-fit there is a clear distinction between the problem of testing the fit of a continuous distribution and that of testing a discrete distribution. In all continuous problems the data is recorded with a limited number of decimals, so in theory one could say that the problem is always of a discrete nature, but it is a common practice to ignore discretization and proceed as if the data is continuous. It is therefore an interesting question whether in a given problem of test of fit, the “limited resolution” in the observed recorded values may be or may be not of concern, if the analysis done ignores this implied discretization. In this article, we address the problem of testing the fit of a continuous distribution with data recorded with a limited resolution. A measure for the degree of discretization is proposed which involves the size of the rounding interval, the dispersion in the underlying distribution and the sample size. This measure is shown to be a key characteristic which allows comparison, in different problems, of the amount of discretization involved. Some asymptotic results are given for the distribution of the EDF (empirical distribution function) statistics that explicitly depend on the above mentioned measure of degree of discretization. The results obtained are illustrated with some simulations for testing normality when the parameters are known and also when they are unknown. The asymptotic distributions are shown to be an accurate approximation for the true finite n distribution obtained by Monte Carlo. A real example from image analysis is also discussed. The conclusion drawn is that in the cases where the value of the measure for the degree of discretization is not “large”, the practice of ignoring discreteness is of no concern. However, when this value is “large”, the effect of ignoring discreteness leads to an exceded number of rejections of the distribution tested, as compared to what would be the number of rejections if no rounding is taking into account. The error made in the number of rejections might be huge.     

A NONPARAMETRIC TEST FOR THE EQUALITY OF DEPENDENT CORRELATION COEFFICIENTS UNDER NORMALITY

ABSTRACT

A nonparametric testing method for the equality of two correlation coefficients in trivariate normal distribution, namely, one of the variables are common, is discussed. Using a permutation test, we obtain asymptotically exact solutions. The performance of this test is compared with the likelihood ratio test and a method of using the limiting distribution of correlation coefficients.     

Testing the Stochastic Structure of Production: A Flexible Moment-Based Approach

Conventional production function specifications are shown to impose restrictions on the probability distribution of output that cannot be tested with the conventional models. These restrictions have important implications for firm behavior under uncertainty. A flexible representation of a firm's stochastic technology is developed based on the moments of the probability distribution of output. These moments are a unique representation of the technology and are functions of inputs. Large-sample estimators are developed for a linear moment model that is sufficiently flexible to test the restrictions implied by conventional production function specifications. The flexible moment-based approach is applied to milk production data. The first three moments of output are statistically significant functions of inputs. The cross-moment restrictions implied by conventional models are rejected.     

Comment

Some considerations relating to the post–data selection of models are discussed. These include some difficulties with orthodox theory, implementation of the likelihood principle, and Bayesian tests of hypotheses.     

A monte carlo study on two methods of calculating the mle's covariance matrix in a seemingly unrelated nonlinear regression. *

Econometric techniques to estimate output supply systems, factor demand systems and consumer demand systems have often required estimating a nonlinear system of equations that have an additive error structure when written in reduced form. To calculate the ML estimate's covariance matrix of this nonlinear system one can either invert the Hessian of the concentrated log likelihood function, or invert the matrix calculated by pre-multiplying and post multiplying the inverted MLE of the disturbance covariance matrix by the Jacobian of the reduced form model. Malinvaud has shown that the latter of these methods is the actual limiting distribution's covariance matrix, while Barnett has shown that the former is only an approximation.

In this paper, we use a Monte Carlo simulation study to determine how these two covariance matrices differ with respect to the nonlinearity of the model, the number of observations in the dataet, and the residual process. We find that the covariance matrix calculated from the Hessian of the concentrated likelihood function produces Wald statistics that are distributed above those calculated with the other covariance matrix. This difference becomes insignificant as the sample size increases to one-hundred or more observations, suggesting that the asymptotics of the two covariance matrices are quickly reached.     

A Transformation Characterizing the Normal Distribution

In this article we propose a new transformation of random variables (RVs) which characterizes the normal distribution. It allows us to transform n i.i.d. normal RVs whose mean and variance are unknown into new n ? 2 i.i.d. new normal variables with zero mean while maintaining the same unknown variance. This belongs to the class of transformations designed to reduce the number of unknown parameters or remove them altogether.

Some historical remarks concerning methods for removing parameters in the normal distribution are given and two possible applications of the new transformation are described.     

公共预算转移支付反贫困瞄准：以低保为例的ROC方法分析

本文使用中国家庭动态跟踪调查（CFPS）数据,基于代理工具测试模型（Proxy means testing,PMT）并结合ROC曲线方法（Receiver Operating Characteristics）研究低保的反贫困瞄准问题。结果显示：城市单个贫困指数以卫生间类型、电脑拥有情况来衡量较为有效,农村单个贫困指数则以户主年龄、冰箱拥有情况来衡量较为有效。更改贫困概率门槛值会影响公共预算转移支付贫困瞄准结果,对于一个较低的贫困概率门槛值,其对应的覆盖率（公共预算转移支付覆盖贫困人口）和漏损率（非贫困人口被纳入公共预算转移支付）都比较高。当政策制定者把对覆盖贫困人口和排除非贫困人口的目标赋予同等权重时,城乡贫困概率门槛值约为0.5左右。当贫困率较低且使用与贫困率相同的瞄准率时,基于PMT模型的贫困瞄准较差。在贫困率给定的条件下,随着受益比率（包含率）的增加,贫困瞄准的精确性在提高。使用全覆盖所需预算的百分比下降时,覆盖率和漏损率也都呈现下降态势,贫困率逐渐上升,预算中做覆盖之用的比例上升,而预算中漏损部分的比例下降。贫困线的变动会影响覆盖率、漏损率。     

Reexamining a Model for Evaluating Information Center Success Using a Structural Equation Modeling Approach

Structural equation modeling is a technique that has been widely used for instrument validation and model testing in research in marketing and organizational behavior. The technique has also been introduced to MIS researchers and used in several studies recently reported in the literature. This article offers an example of how the technique can be used for instrument validation and model testing. The illustration is made through a reexamination of a model for evaluating information center (IC) success. With the growth of end-user computing, the success of an IC is more important than ever. Obtaining a valid model for measuring IC success is thus crucial to organizations with ICs. The results of this study highlight the importance of a strong theoretical base in developing such a valid model, and management should be cautious when using these models to assess the performance of their ICs.