Modification of Tukey's additivity test

In this paper we discuss testing for an interaction in the two-way ANOVA with just one observation per cell. The known results are reviewed and a simulation study is performed to evaluate type I and type II risks of the tests. It is shown that the Tukey and Mandel additivity tests have very low power in case of more general interaction scheme. A modification of Tukey's test is developed to resolve this issue. All tests mentioned in the paper have been implemented in R package Additivity Tests.     

Assessing additivity in nonparametric models —A kernel‐based method

In this article, we address the testing problem for additivity in nonparametric regression models. We develop a kernel‐based consistent test of a hypothesis of additivity in nonparametric regression, and establish its asymptotic distribution under a sequence of local alternatives. Compared to other existing kernel‐based tests, the proposed test is shown to effectively ameliorate the influence from estimation bias of the additive component of the nonparametric regression, and hence increase its efficiency. Most importantly, it avoids the tuning difficulties by using estimation‐based optimal criteria, while there is no direct tuning strategy for other existing kernel‐based testing methods. We discuss the usage of the new test and give numerical examples to demonstrate the practical performance of the test. The Canadian Journal of Statistics 39: 632–655; 2011. © 2011 Statistical Society of Canada     

A joint test for arch and bilinearity in the regression model

In this paper we argue that a simultaneous test for ARCH and bilinearity should be used to test for the possible nonlinearity of the error process in the regression model. We suggest such a joint test statistic. An empirical example shows that the individual tests of ARCH and bilinearity may not be conclusive while a joint test clearly rejects the linearity hypothesis. Our results are also applicable to pure time series models.     

Finding Jack Michael’s Footprints in the Sand

Jack Michael left a legacy of fine discriminations among technical terms, a careful parsing of theoretical concepts, and a variety of behavioral interpretations of complex behavioral phenomena. When pushing into relatively unfamiliar territory in areas such as “memory,” automatic reinforcement, multiple control, and additivity of response strength, I have been surprised and delighted to find evidence that Jack had once explored the same topics, in the same way, and had come to the same conclusions toward which I was groping.     

On Two Forms of Fisher's Measure of Information

ABSTRACT

Fisher's information number is the second moment of the “score function” where the derivative is with respect to x rather than Θ. It is Fisher's information for a location parameter, and also called shift-invariant Fisher information. In recent years, Fisher's information number has been frequently used in several places regardless of parameters of the distribution or of their nature. Is this number a nominal, standard, and typical measure of information? The Fisher information number is examined in light of the properties of classical statistical information theory. It has some properties analogous to those of Fisher's measure, but, in general, it does not have good properties if used as a measure of information when Θ is not a location parameter. Even in the case of location parameter, the regularity conditions must be satisfied. It does not possess the two fundamental properties of the mother information, namely the monotonicity and invariance under sufficient transformations. Thus the Fisher information number should not be used as a measure of information (except when Θ a location parameter). On the other hand, Fisher's information number, as a characteristic of a distribution f(x), has other interesting properties. As a byproduct of its superadditivity property a new coefficient of association is introduced.     

Axiomatization of a class of share functions for n-person games

The Shapley value is the unique value defined on the class of cooperative games in characteristic function form which satisfies certain intuitively reasonable axioms. Alternatively, the Banzhaf value is the unique value satisfying a different set of axioms. The main drawback of the latter value is that it does not satisfy the efficiency axiom, so that the sum of the values assigned to the players does not need to be equal to the worth of the grand coalition. By definition, the normalized Banzhaf value satisfies the efficiency axiom, but not the usual axiom of additivity.In this paper we generalize the axiom of additivity by introducing a positive real valued function on the class of cooperative games in characteristic function form. The so-called axiom of -additivity generalizes the classical axiom of additivity by putting the weight (v) on the value of the gamev . We show that any additive function determines a unique share function satisfying the axioms of efficient shares, null player property, symmetry and -additivity on the subclass of games on which is positive and which contains all positively scaled unanimity games. The axiom of efficient shares means that the sum of the values equals one. Hence the share function gives the shares of the players in the worth of the grand coalition. The corresponding value function is obtained by multiplying the shares with the worth of the grand coalition. By defining the function appropiately we get the share functions corresponding to the Shapley value and the Banzhaf value. So, for both values we have that the corresponding share functions belong to this class of share functions. Moreover, it shows that our approach provides an axiomatization of the normalized Banzhaf value. We also discuss some other choices of the function and the corresponding share functions. Furthermore we consider the axiomatization on the subclass of monotone simple games.     

Optimal domain estimation under summation restriction

The domain estimators that do not sum up to the population total (estimated or known) are considered. In order to achieve their additivity, the theory of the general restriction (GR)-estimator [Knottnerus P., 2003. Sample Survey Theory: Some Pythagorean Perspectives. Springer, New York] is used. The elaborated domain GR-estimators are optimal, they have the minimum variance in a class of estimators that satisfy summation restriction. Furthermore, their variances are smaller than the variances of the corresponding initial domain estimators. The variance/covariance formulae of the domain GR-estimators are explicitly given.The ratio estimators as representatives of the non-additive domain estimators are considered. Their design-based covariance matrix, being crucial for the GR-estimator, is presented. Its structure simplifies under certain assumptions on sampling design (and population model). The corresponding simpler forms of the domain GR-estimators are elaborated as well. The hypergeometric [Traat I., Ilves M., 2007. The hypergeometric sampling design, theory and practice. Acta Appl. Math. 97, 311–321] and the simple random sampling designs are considered in more detail. The results are illustrated in a simulation study where the optimal domain estimator displays its superiority among other meaningful domain estimators. It is noteworthy that due to the imposed restrictions also these other estimators, though not optimal, can be much more precise than the initial estimators.     

多重视角下的最大化双边特征法及其加权形式

在国际经济比较中,可加性和特征性是购买力平价汇总方法应该具备的重要性质。针对新近发展的、基于双边指数所构建的兼具可加性与特征性的多边比较方法———MBC法,本文对其进行了拓展研究,发展出更为完备的MBC法:一方面,鉴于原始极值最优视角下的MBC法主要考虑的是存在较多“奇异”型价格结构的国家双边比较的特征性,为弥补该视角下的MBC法难以实现整体最优的不足,本文从最小二乘和对数最小二乘的视角分别构建MBC法;另一方面,进一步在MBC法中考虑不同双边比较的可靠性差异问题,分别构建多重视角下的加权MBC法,并引入多种权重函数供应用者参考。采用 ICP2011数据的实证研究结果充分说明,多重视角下的MBC法较好地整合了官方GEKS法的特征性优点和 GK法的可加性优点,相比已有满足可加性的方法,MBC法替代偏差最小,是较理想的既适合于实际物量比较、又可用作支出结构分析的方法。实证研究还发现,最小二乘和对数最小二乘视角下的加权MBC法稳健性较好的优点。     

Randomization tests for small samples: an application for genetic expression data

Summary. An advantage of randomization tests for small samples is that an exact P -value can be computed under an additive model. A disadvantage with very small sample sizes is that the resulting discrete distribution for P -values can make it mathematically impossible for a P -value to attain a particular degree of significance. We investigate a distribution of P -values that arises when several thousand randomization tests are conducted simultaneously using small samples, a situation that arises with microarray gene expression data. We show that the distribution yields valuable information regarding groups of genes that are differentially expressed between two groups: a treatment group and a control group. This distribution helps to categorize genes with varying degrees of overlap of genetic expression values between the two groups, and it helps to quantify the degree of overlap by using the P -value from a randomization test. Moreover, a statistical test is available that compares the actual distribution of P -values with an expected distribution if there are no genes that are differentially expressed. We demonstrate the method and illustrate the results by using a microarray data set involving a cell line for rheumatoid arthritis. A small simulation study evaluates the effect that correlated gene expression levels could have on results from the analysis.