Estimators of Influence Function

ABSTRACT

This article discusses estimators of influence function with some new counter-examples and tries to uphold their usefulness mathematically as well as through simulation. It is suggested that some estimators of influence function of uniformly Fréchet differentiable functional has more desirable properties.     

Sample Sizes in the Interval Estimation of the Correlation Coefficient

An expression is derived for the maximum length of the interval estimator of the correlation coefficient, p, under bivariate normal assumptions. The prespecification of this minimum attainable precision and the confidence level results in an expression for the sample size required. An approximate expression for the sample size is proposed and is numerically shown to be as good as or better than that based on the Fisher's Z transformation.     

Exact Confidence Intervals,Based on the Z Statistic,for the Difference Between Two Proportions

We show that the confidence interval version of the extended exact unconditional Z test of Suissa and Shuster (1985) for testing the equality of two binomial proportions is due to general results of Buehler (1957), Sudakov and references cited there (1974), and Harris and Soms (1984). We apply these results to obtain exact unconditional confidence intervals for the difference between two proportions, deriving an explicit solution for the “best” outcome, make some comments on Buehler's (1957) method and give a numerical example. The Appendix contains a listing of the necessary FORTRAN programs.     

The Large Sample Distribution of the Weighted Log Rank Statistic Under General Local Alternatives

We derive the large sample distribution of the weighted log rank statistic under a general class of local alternatives in which both the cure rates and the conditional distribution of time to failure among those who fail are assumed to vary in the two treatment arms. The analytic result presented here is important to data analysts who are designing clinical trials for diseases such as non-Hodgkins lymphoma, leukemia and melanoma, where a significant proportion of patients are cured. We present a numerical illustration comparing powers obtained from the analytic result to those obtained from simulations. This revised version was published online in July 2006 with corrections to the Cover Date.     

划类选点抽样调查方法在现阶段统计调查方法中的地位和作用

文章讨论了划类选点抽样调查方法的意义和特点,以及在现阶段交通统计中的重要作用与实际效果。     

国际国内市场原油及成品油价格关系分析

根据中国加入WTO的承诺,中国成品油市场将在2006年底之前全面开放。面对激烈的竞争,对国内原油成品油价格与国际原油成品油价格进行对比分析是非常有意义的。文章通过互相关性分析和对数回归分析揭示了国内和国际两个市场的差异和原因。当前成品油和原油价格体制改革的关键是尽快深化原油价格定价机制的改革。     

Influence Function Based Variance Estimation and Missing Data Issues in Case-Cohort Studies

Recognizing that the efficiency in relative risk estimation for the Cox proportional hazards model is largely constrained by the total number of cases, Prentice (1986) proposed the case-cohort design in which covariates are measured on all cases and on a random sample of the cohort. Subsequent to Prentice, other methods of estimation and sampling have been proposed for these designs. We formalize an approach to variance estimation suggested by Barlow (1994), and derive a robust variance estimator based on the influence function. We consider the applicability of the variance estimator to all the proposed case-cohort estimators, and derive the influence function when known sampling probabilities in the estimators are replaced by observed sampling fractions. We discuss the modifications required when cases are missing covariate information. The missingness may occur by chance, and be completely at random; or may occur as part of the sampling design, and depend upon other observed covariates. We provide an adaptation of S-plus code that allows estimating influence function variances in the presence of such missing covariates. Using examples from our current case-cohort studies on esophageal and gastric cancer, we illustrate how our results our useful in solving design and analytic issues that arise in practice.     

The Dynamic Relationship Between the Dollar and Components of U.S. Trade

The dynamic response of the nominal trade account to changes in the value of the U.S. dollar has been posited to follow a J-curve pattern. Recent experience calls this into question. The aggregate nominal trade balance is decomposed here into four components, both prices and volumes of imports and exports. Time series specification tests and Granger tests of causal priority are employed to identify the existence and nature of the response of each individual component to dollar movements. Surprisingly weak and delayed responses of both import prices and volumes are found, suggesting a new view of trade-balance evolution.     

The Fisher Transform of the Pearson Product Moment Correlation Coefficient and Its Square: Cumulants,Moments, and Applications

This paper extends results on the distribution of the Fisher transform of the correlation coefficient (Fisher, 1921 Fisher , R. A. ( 1921 ). On the “probable error” of a coefficient of correlation deduced from a small sample . Metron 1 : 1 – 32 . [Google Scholar]). Approaches to obtain exact moments of the Fisher transform for both null and non-null correlations are presented. We extend the classic series expansion formulae of Hotelling (1953 Hotelling , H. ( 1953 ). New light on the correlation coefficient and its transforms . Journal of the Royal Statistical Society, Ser. B 15 : 192 – 232 . [Google Scholar]) for the moments of the Fisher transform. These results are considered in the context of quadratic functions of the Fisher transform. Some applications of these results are discussed in the context of correlational hypothesis tests and confidence intervals, and a Monte Carlo experiment is used to demonstrate how application of these results impact the small sample performance of select tests on correlations.     

On a Class of Distributions Defined by the Relationship Between Their Density and Distribution Functions

Knowledge concerning the family of univariate continuous distributions with density function f and distribution function F defined through the relation f(x) = F ^α(x)(1 ? F(x))^β, α, β ? , is reviewed and modestly extended. Symmetry, modality, tail behavior, order statistics, shape properties based on the mode, L-moments, and—for the first time—transformations between members of the family are the general properties considered. Fully tractable special cases include all the complementary beta distributions (including uniform, power law and cosine distributions), the logistic, exponential and Pareto distributions, the Student t distribution on 2 degrees of freedom and, newly, the distribution corresponding to α = β = 5/2. The logistic distribution is central to some of the developments of the article.     

消费者信心指数与消费需求关系研究——以南京市居民消费需求为例

传统消费理论主要是注重对消费者收入与消费支出的关系研究,而新的西方消费者信心指数理论指出,消费者信心指数对消费支出有重要影响,特别是在当前国内消费需求增长缓慢的情况下,如何提高居民消费信心对刺激居民消费支出的增长有着重要的现实意义。运用计量经济模型,并结合南京市居民消费支出增长与消费者信心指数的数量关系,通过实证研究印证了两者间存在紧密相关性,从而建立南京市居民消费函数的主观因素拓展模型,并进而提出提高居民消费需求的政策建议,即在注重对影响消费的客观因素——收入的研究外,还要注重对影响消费的主观因素——消费者信心指数的研究。     

On robustness and efficiency of certain statistics involving the empirical characteristic function

Summary The polar coordinates of the empirical characteristic function have been used effectively in estimation and testing. Here the efficiency and robustness properties of two simple but useful statistics involving the polar coordinates of the empirical characteristic function are examined.     

中国工业基础金属期货价格与现货价格动态关系研究

文章借助VAR模型、协整检验、误差修正模型、格兰杰因果检验、脉冲响应函数、方差分解等方法,以上海期货交易所的工业基础金属铜和铝期货品种为例,研究了工业基础金属期货价格与现货价格之间的动态关系,定量地刻画出了期货市场在价格发现中作用的大小。其研究结果显示:铜和铝期货价格与它们的现货价格都存在相互引导关系;而且期货与现货价格之间也存在长期均衡关系;上海铜和铝金属期货市场在价格发现功能中发挥了主导作用。     

Choosing Between the Sample-Selection Model and the Multi-Part Model

Hay and Olsen (1984) incorrectly argue that a multi-part model, the two-part model used in Duan et al. (1982,1983), is nested within the sample-selection model. Their proof relies on an unmentioned restrictive assumption that cannot be satisfied. We provide a counterexample to show that the propensity to use medical care and the level of expense can be positively associated in the two-part model, contrary to their assertion. The conditional specification in the multi-part model is preferable to the unconditional specification in the selection model for modeling actual (v. potential) outcomes. The selection model also has poor statistical and numerical properties and relies on untestable assumptions. Empirically the multi-part estimators perform as well as or better than the sample selection estimator for the data set analyzed in Duan et al. (1982, 1983).     

Between stability and higher-order asymptotics

We discuss the effects of model misspecifications on higher-order asymptotic approximations of the distribution of estimators and test statistics. In particular we show that small deviations from the model can wipe out the nominal improvements of the accuracy obtained at the model by second-order approximations of the distribution of classical statistics. Although there is no guarantee that the first-order robustness properties of robust estimators and tests will carry over to second-order in a neighbourhood of the model, the behaviour of robust procedures in terms of second-order accuracy is generally more stable and reliable than that of their classical counterparts. Finally, we discuss some related work on robust adjustments of the profile likelihood and outline the role of computer algebra in this type of research.     

外商直接投资与我国资本形成的协整研究

文章首先概述了经济学界对外商直接投资与资本形成一般关系的分析,然后运用协整理论对数据进行了平稳性检验、格兰杰原因检验,建立了外商直接投资和全社会固定资产投资的协整模型和误差修正模型,认为在长期和短期内外商直接投资对我国的资本形成都有促进作用。     

Robustness and monotonicity properties of generalized correlation coefficients

A new class of generalized correlation coefficients that contains the Pearson and Kendall statistics as special cases was defined by Chinchilli et al. (2005) and applied to the estimation of correlations coefficients within the context of 2×2 cross-over designs for clinical trials. In this paper, we determine the infinitesimal robustness and local stability properties of these generalized correlation coefficients by deriving their corresponding influence functions. For cases in which the population distribution is a bivariate normal or a mixture of bivariate normal distributions we obtain explicit formulas, and establish monotonicity and sign-reverse rule properties of the generalized correlation coefficients.     

Efficiency of t-Test and Hotelling's T 2-Test After Box-Cox Transformation

Early investigations of the effects of non-normality indicated that skewness has a greater effect on the distribution of t-statistic than does kurtosis. When the distribution is skewed, the actual p-values can be larger than the values calculated from the t-tables. Transformation of data to normality has shown good results in the case of univariate t-test. In order to reduce the effect of skewness of the distribution on normal-based t-test, one can transform the data and perform the t-test on the transformed scale. This method is not only a remedy for satisfying the distributional assumption, but it also turns out that one can achieve greater efficiency of the test. We investigate the efficiency of tests after a Box-Cox transformation. In particular, we consider the one sample test of location and study the gains in efficiency for one-sample t-test following a Box-Cox transformation. Under some conditions, we prove that the asymptotic relative efficiency of transformed t-test and Hotelling's T ²-test of multivariate location with respect to the same statistic based on untransformed data is at least one.     

Estimation and Simulation for the M-Wright Function

In this article, a structural form of an M-Wright distributed random variable is derived. The mixture representation then led to a random number generation algorithm. A formal parameter estimation procedure is also proposed. This procedure is needed to make the M-Wright function usable in practice. The asymptotic normality of the estimator is established as well. The estimator and the random number generation algorithm are then tested using synthetic data.     

On Sensitivity of Inverse Response Plot Estimation and the Benefits of a Robust Estimation Approach

Abstract. Inverse response plots are a useful tool in determining a response transformation function for response linearization in regression. Under some mild conditions it is possible to seek such transformations by plotting ordinary least squares fits versus the responses. A common approach is then to use nonlinear least squares to estimate a transformation by modelling the fits on the transformed response where the transformation function depends on an unknown parameter to be estimated. We provide insight into this approach by considering sensitivity of the estimation via the influence function. For example, estimation is insensitive to the method chosen to estimate the fits in the initial step. Additionally, the inverse response plot does not provide direct information on how well the transformation parameter is being estimated and poor inverse response plots may still result in good estimates. We also introduce a simple robustified process that can vastly improve estimation.