A generalization of Friedman's rank statistic

Abstract  In this paper a very natural generalization of the two-way analysis of variance rank statistic of F riedman is given. The general distribution-free test procedure based on this statistic for the effect of J treatments in a random block design can be applied in general two-way layouts without interactions and with different numbers of the continuous observations per cell provided the design scheme is connected. The asymptotic distribution under the null hypothesis of the test statistic is derived. A comparison with the method of m rankings of B enard and van E lteren is made. The disadvantage of B enard and van E lteren's test procedure is that the number of observations per block does influence the statistic twice, namely firstly by the number itself, as it should, and see ondly by the level of the ranks which will be different in different blocks if the numbers of observations per block are different. The proposed test statistic is not sensitive to differences in the levels of the ranks caused by the different numbers of observations per block. The test is derived from considerhg the K ruskal -W allis statistics per block.
Finally, the results of simulation experiments are given. The simulation is carried out for three designs and a number of normal location alternatives and gives some information about the power of the suggested test procedure. A comparison is made with B enard and van E lteren's test and with the classical analysis of variance technique. For some simple orthogonal designs the exact null distributions of B enard and van E lteren's test and the proposed test are compared.     

RANK TESTS IN 2X2 DESIGNS

Abstract. In literature numerous attempts can be found for the evaluation of two factor designs with fixed effects by means of rank tests. The aim of the present article is to show the limits of these methods and to give some new procedures for 2X2 designs. First, functionals of distribution functions shall be defined whose relations to the usual parameters of the linear model are analysed. These functionals are free of nuissance parameters under the respective hypothesis; they are estimated by special ranks of the data. The asymptotic distribution of these statistics is derived by a generalization of the Chemoff–Savage theorem for correlated random variables. The asymptotic variance depends on the parent distribution function but it can be estimated by using special rank methods. Thus, one obtains asymptotically distribution–free tests for two–factor designs with fixed effects. Some counter examples show why it is not possible to construct suitable rank tests for greater designs than the 2X2 design. The paper closes with a discussion of the drawbacks of the well known rank transform.     

Tests for balanced incomplete block ranked data with ties

We consider balanced incomplete block data when ties occur and propose new statistics for testing (a) differences in mean ranks, (b) differences in distributions of ranks, (c) differences in nonlinear effects of ranks and (d) linear contrasts. A sensory evaluation example where the data are ranks is given.     

THE ASYMPTOTIC BEHAVIOR OF SIMPLE LINEAR RANK STATISTICS

The concept of a simple linear rank statistic is essential in nonparametric theory. These statistics may be used for testing the null-hypothesis that all observations are independent and identically distributed against classes of alternatives indicated by the choice of regression constants and scores. Special cases are the two-sample Linear rank statistics and Spearman's rank correlation coefficient rho under the null-hypothesis of independence.
In this paper we shall give a survey of the asympotic distribution theory of simple linear rank statistics.     

城镇密集区等级研究

城镇密集区是城市区域化和区域城市化过程中出现的一种独特地域空间组织形式.无论在国内还是国外,无论是在经济发达地区还是经济欠发达地区,城镇密集区均已成为区域经济发展的主体,在经济信息化和全球化的当今,其又成为全球经济体系的重要节点区域.城镇密集区的等级高低直接影响着其在全球经济分工中的地位,而决定城镇密集区等级的要素很多,但最     

U-type and column-orthogonal designs for computer experiments

U-type designs and orthogonal Latin hypercube designs (OLHDs) have been used extensively for performing computer experiments. Both have good spaced filling properties in one-dimension. U-type designs may not have low correlations among the main effects, quadratic effects and two-factor interactions. On the other hand, OLHDs are hard to be found due to their large number of levels for each factor. Recently, alternative classes of U-type designs with zero or low correlations among the effects of interest appear in the literature. In this paper, we present new classes of U-type or quantitative \(3\) -orthogonal designs for computer experiments. The proposed designs are constructed by combining known combinatorial structures and they have their main effects pairwise orthogonal, orthogonal to the mean effect, and orthogonal to both quadratic effects and two-factor interactions.     

Tests of Bivariate Exchangeability

We discuss the hypothesis of bivariate exchangeability and show that testing bivariate exchangeability is related to the two-sample testing of equality of distribution functions. We consider three test statistics based on the ordering of the Euclidean interpoint distances. The runs test of exchangeability counts the runs among the observations and their mirror images on the minimal spanning tree. The nearest neighbour test of exchangeability is based on the number of nearest neighbour type coincidences among the observations and their folded images on the plane. The rank test of exchangeability compares the within and between ranks of the interpoint distances. We also consider the sign test of exchangeability, which uses the signs of the observations in specific regions, and a bootstrap test of exchangeability based on the maximum distance between the mirror images. We compare the power of these methods in a Monte Carlo study which shows different power orderings of the methods, depending on the alternative hypothesis.     

A GENERALIZED SIGNED–RANK TEST FOR COMPARISON OF P TREATMENTS

Abstract. Several generalizations of the signed–rank test for testing of equality of p treatments in p related samples against general and trend alternative are discussed. A new generalized test is described and exemplified.     

First and second order rotatability of experimental designs,moment matrices,and information surfaces

Summary We place the well-known notion of rotatable experimental designs into the more general context of invariant design problems. Rotatability is studied as it pertains to the experimental designs themselves, as well as to moment matrices, or to information surfaces. The distinct aspects become visible even in the case of first order rotatability. The case of second order rotatability then is conceptually similar, but technically more involved. Our main result is that second order rotatability may be characterized through a finite subset of the orthogonal group, generated by sign changes, permutations, and a single reflection. This is a great reduction compared to the usual definition of rotatability which refers to the full orthogonal group. Our analysis is based on representing the second order terms in the regression function by a Kronecker power. We show that it is essentially the same as using the Schl?flian powers, or the usual minimal set of second order monomials, but it allows a more transparent calculus.     

A directional test of exponentiality based on maximum correlations

In Fortiana and Grané (J Stat Plann Infer 108:85–97), we study a scale-free statistic, based on Hoeffding’s maximum correlation, for testing exponentiality. This statistic admits an expansion along a countable set of orthogonal axes, originating a sequence of statistics. Linear combinations of a given number p of terms in this sequence can be written as a quotient of L-statistics. In this paper, we propose a scale-free adaptive statistic for testing exponentiality with optimal power against a specific alternative and obtain its exact distribution. An empirical power study shows that the test based on this new statistic has the same level of performance than the best tests in the statistical literature.     

Linear rank tests for comparing treatments with a control when data are subject to unequal patterns of censorship

A class of asymptotically distribution-free tests is considered for comparing several treatments with a control when the, data are subject to unequal right-censorship. A particular member of this class is proposed for use in practice and an illustrative numerical example is given. A general result for the Pitman efficacy of a test based on an asymptotically normal test statistic is proved for the multiparameter case and using this result the efficacy of the proposed class of tests is obtained under sequences of translation and proportional hazards alternatives. Simulation studies are conducted to compare the performance of the proposed test, in terms of power, with some other tests.     

Construction of nearly orthogonal Latin hypercube designs

Latin hypercube designs have found wide application in computer experiments. A number of methods have recently been proposed to construct orthogonal Latin hypercube designs. In this paper, we propose an approach for expanding the orthogonal Latin hypercube design in Sun et al. (Biometrika 96:971–974, 2009) to a nearly orthogonal Latin hypercube design of a larger column size. The newly added part has half number of columns of the original part. It can be shown that the upper bound of the maximum correlation between any two distinct columns of the resulting design is very small. Our method also works for expanding any symmetric Latin hypercube designs.     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

班组作业手等级方案的设定

针对上海市电力公司岗位说明书出台后出现的岗位评级问题,结合营业室作业手等级管理这一现状,设计了量化的考核评估手段,力争在公开、公正、公平、民主的基础上,运用统一管理、分块考核,提供营业人员从低级作业手向高级作业手晋升的通道,形成高分高级、竞争上岗的公平合理的竞争激励机制。     

The analysis of ranked data in blocked factorial experiments

A non-parametric method for the analysis of blocked factorial experiments, based on ranking within blocks, is proposed and shown to be equivalent to partitioning Friedman's test statistic into a set of contrasts reflecting polynomial components of the main effects and interaction. A slightly modified version of the procedure is suggested to partially overcome the problem of loss of power to detect one component when the model includes other components. This alternative procedure is shown to be equivalent to applying a standard normal theory analysis of variance to the ranks. The null distributions and power comparisons are investigated using simulation methods, and it is shown that the non-parametric methods are almost as powerful as the analysis of variance. Received: February 1999     

非参数统计视角下的异方差检验设计及其应用

以数据生成过程为导向,探讨了异方差来源的基本类型;依据非参数统计的基本思想,设计了切实可行的Mood方差检验方法与平方秩检验方法,并针对异方差来源类型,分析了相应的检验思路。蒙特卡罗模拟表明,Mood方差检验方法在异方差检验方面具有很高的检验效力;平方秩检验方法在异常经济现象情形下检验效力较低,而在其他情形下检验效力很高。同时,进一步阐释了纠正异方差的基本逻辑。     

A non-local perspective on the power properties of the CUSUM and CUSUM of squares tests for structural change

We consider the power properties of the CUSUM and CUSUM of squares (CUSQ) tests in the presence of a one-time change in the parameters of a linear regression model. A result due to Ploberger and Krämer [1990. The local power of the cusum and cusum of squares tests. Econometric Theory 6, 335–347.] is that the CUSQ test has only trivial asymptotic local power in this case, while the CUSUM test has non-trivial local asymptotic power unless the change is orthogonal to the mean regressor. The main theme of the paper is that such conclusions obtained from a local asymptotic framework are not reliable guides to what happens in finite samples. The approach we take is to derive expansions of the test statistics that retain terms related to the magnitude of the change under the alternative hypothesis. This enables us to analyze what happens for non-local to zero breaks. Our theoretical results are able to explain how the power function of the tests can be drastically different depending on whether one deals with a static regression with uncorrelated errors, a static regression with correlated errors, a dynamic regression with lagged dependent variables, or whether a correction for non-normality is applied in the case of the CUSQ. We discuss in which cases the tests are subject to a non-monotonic power function that goes to zero as the magnitude of the change increases, and uncover some curious properties. All theoretical results are verified to yield good guides to the finite sample power through simulation experiments. We finally highlight the practical importance of our results.     

Relative Efficiency of the Fuzzy p-Value Approach to Hypothesis Testing

In missing data problems, it is often the case that there is a natural test statistic for testing a statistical hypothesis had all the data been observed. A fuzzy  p -value approach to hypothesis testing has recently been proposed which is implemented by imputing the missing values in the "complete data" test statistic by values simulated from the conditional null distribution given the observed data. We argue that imputing data in this way will inevitably lead to loss in power. For the case of scalar parameter, we show that the asymptotic efficiency of the score test based on the imputed "complete data" relative to the score test based on the observed data is given by the ratio of the observed data information to the complete data information. Three examples involving probit regression, normal random effects model, and unidentified paired data are used for illustration. For testing linkage disequilibrium based on pooled genotype data, simulation results show that the imputed Neyman Pearson and Fisher exact tests are less powerful than a Wald-type test based on the observed data maximum likelihood estimator. In conclusion, we caution against the routine use of the fuzzy  p -value approach in latent variable or missing data problems and suggest some viable alternatives.     

A REVIEW AND COMPARISON OF TESTS OF CROSS-SECTION INDEPENDENCE IN PANELS

Abstract.  In this paper we review and compare diagnostic tests of cross-section independence in the disturbances of panel regression models. We examine tests based on the sample pairwise correlation coefficient or on its transformations, and tests based on the theory of spacings. The ultimate goal is to shed some light on the appropriate use of existing diagnostic tests for cross-equation error correlation. Our discussion is supported by means of a set of Monte Carlo experiments and a small empirical study on health. Results show that tests based on the average of pairwise correlation coefficients work well when the alternative hypothesis is a factor model with non-zero mean loadings. Tests based on spacings are powerful in identifying various forms of strong cross-section dependence, but have low power when they are used to capture spatial correlation.     

THE DYNAMICS OF REAL EXCHANGE RATES: A RECONSIDERATION

In this paper we offer a bootstrap‐based version of the Cox specification test for non‐nested hypothesis to discriminate between ESTAR and MSAR models. Both models are commonly used for modeling real exchange rates dynamics. We show that the test has good size and power properties in finite samples. In an application, we analyze several major real exchange rates to shed light on the question of which model describes these processes best. This allows us to draw conclusions about the driving forces of real exchange rates. Copyright © 2013 John Wiley & Sons, Ltd.