On the independence problem and kendall's tau

A modification of Kendall's test for independence is described which allows one to test for association in a bivariate distribution as measured by Kendall's tau, a property not shared by Kendall's procedure. The proposed procedure, however, still provides an exactly distribution-free test of independence. The test procedure is inverted to obtain a confidence interval for tau which has distinct advantages over the currently employed confidence interval.     

Amemiya‘s generalized least squares and tests of overidentification in simultaneous equation models with qualitative or limited dependent variables

Amemiya's generalized least squares method for the estimation of simultaneous equation modeis with qualitative or limited dependent variables is known to be efficient relative to many popular two stage estimators. This note points out that test statistics for overidentification restrictions can be obtained as by-products of Amerniya's generalized least squares procedure. Amemiya's procedure is shown to be a minimum chisquare method. The Amemiya procedure is valuable both for efficient estimation and for model evaluation of such models.     

An Adaptive Method of Variable Selection in Regression

An adaptive variable selection procedure is proposed which uses an adaptive test along with a stepwise procedure to select variables for a multiple regression model. We compared this adaptive stepwise procedure to methods that use Akaike's information criterion, Schwartz's information criterion, and Sawa's information criterion. The simulation studies demonstrated that the adaptive stepwise method is more effective than the traditional variable selection methods if the error distribution is not normally distributed. If the error distribution is known to be normally distributed, the variable selection method based on Sawa's information criteria appears to be superior to the other methods. Unless the error distribution is known to be normally distributed, the adaptive stepwise method is recommended.     

A modification of the partial sequential probability ratio test

For the problem of discriminating between two simple hypoth¬eses concerning a Koopman - Darmois parameter, a modification of the partial sequential probability ratio test is proposed where instead of drawing only one fixed sample, two fixed samples are drawn and then Wald's SPRT is started. The OC and the ASN func¬tions are derived. Numerical comparisons are made with Wald's and Read's procedures for testing the normal mean with known variance. For some parameter values, the test procedure has a lower ASN than that of Read's procedure.     

An alternative procedure for performing a power analysis of Mantel's test

This study proposes a simple way to perform a power analysis of Mantel's test applied to squared Euclidean distance matrices. The general statistical aspects of the simple Mantel's test are reviewed. The Monte Carlo method is used to generate bivariate Gaussian variables in order to create squared Euclidean distance matrices. The power of the parametric correlation t-test applied to raw data is also evaluated and compared with that of Mantel's test. The standard procedure for calculating punctual power levels is used for validation. The proposed procedure allows one to draw the power curve by running the test only once, dispensing with the time demanding standard procedure of Monte Carlo simulations. Unlike the standard procedure, it does not depend on a knowledge of the distribution of the raw data. The simulated power function has all the properties of the power analysis theory and is in agreement with the results of the standard procedure.     

An approximation for the hotelling-lawley trace in the noncentral case

A procedure for estimating power in conjunction with the Hotelling-Lawley trace is developed. By approximating a non-central Wishart distribution with a central Wishart, and using McKeon's (1974) F-type approximation, a relatively simple procedure for obtaining power estimates is obtained. The accuracy of the approximation is investigated by comparing the approximate results with those for a wide range of conditions given in Olson's (1973) extensive Monte Carlo study. Siotani's (1971) asymptotic expansion is used to provide further comparative assessments. It is demonstrated that the approximation is of sufficient accuracy to be used in practical applications.     

A nonparametric confidence interval for slope based on spearman's rho

An algorithm is presented for computing an exact nonparametric interval estimate of the slope parameter in a simple linear regression model. The confidence interval is obtained by inverting the hypothesis test for slope that uses Spearman's rho. This method is compared to an exact procedure based on Kendall's tau. The Spearman rho procedure will generally give exact levels of confidence closer to desired levels, especially in small samples. Monte carlo results comparing these two methods with the parametric procedure are given     

On some tests for independence in nonnormal situations: neyman'S C(A)test

A study is made of Neyman's C(a) test for testing independence in nonnormal situations. It is shown that it performs very well both in terms of the level of significance and the powereven for smallvalues of the samplesize. Also, in the case of the bivariate Polsson distribution, itis shown that Fisher's z and Student's t transforms of the sample correlation coefficient are good competitors for Neyman's procedure.

     

Simultaneous Confidence Intervals for the Mean of Multivariate Poisson Distribution: A Comparison

Haibing (2009) proposed a procedure for successive comparisons between ordered treatment effects in one-way layout and showed that the proposed procedure has greater power than the procedure proposed by Lee and Spurrier (1995). Critical constants required for the proposed procedure were estimated using Monte Carlo simulation and few values of the constants were tabulated which limit the applications of the proposed procedure. In this article, a numerical method, using recursive integration methodology, is discussed to compute the critical constants which work efficiently for a large number of treatments and extensive values of critical constants are tabulated for the use of practitioners. Power comparisons of Haibing's and Lee and Spurrier's procedure is also discussed.     

Comparison of Two Distribution-Free Procedures for Multiple Comparisons With a Control

Chakraborti and Desu (1988) presented a distribution-free procedure for testing that k (≥1) distributions are equal to a control distribution. They compared their procedure, a generalization of the test proposed by Mathisen (1943), to the procedure proposed by Slivka (1970). They asserted that their procedure has shorter expected duration than Slivka's procedure in life-testing experiments where observations become available in an ordered manner. Here it is proven that, in fact, Slivka's procedure has shorter duration in such circumstances. Normal approximations are presented which indicate that their procedure requires a smaller sample size to guarantee a specified power for Lehmann alternatives and proportional hazard alternatives when all observations are to be observed.     

Performance of nonparametric multiple comparison tests under heteroscedasticity,dependency, and skewed error distribution

In this article, an extensive Monte Carlo simulation study is conducted to evaluate and compare nonparametric multiple comparison tests under violations of classical analysis of variance assumptions. Simulation space of the Monte Carlo study is composed of 288 different combinations of balanced and unbalanced sample sizes, number of groups, treatment effects, various levels of heterogeneity of variances, dependence between subgroup levels, and skewed error distributions under the single factor experimental design. By this large simulation space, we present a detailed analysis of effects of the violations of assumptions on the performance of nonparametric multiple comparison tests in terms of three error and four power measures. Observations of this study are beneficial to decide the optimal nonparametric test according to requirements and conditions of undertaken experiments. When some of the assumptions of analysis of variance are violated and number of groups is small, use of stepwise Steel-Dwass procedure with Holm's approach is appropriate to control type I error at a desired level. Dunn's method should be employed for greater number of groups. When subgroups are unbalanced and number of groups is small, Nemenyi's procedure with Duncan's approach produces high power values. Conover's procedure successfully provides high power values with a small number of unbalanced groups or with a greater number of balanced or unbalanced groups. At the same time, Conover's procedure is unable to control type I error rates.     

Calibration for Randomized Response Estimators

In the present article, we consider the calibration procedure for the Warner's and Mangat–Singh's (:M–S) randomized response survey estimators using auxiliary information associated with the variable of interest. In the calibration procedure, we can use auxiliary information such as age, gender, and income for the respondents of RR questions from an external source, and then the classical RR estimators can be improved with respect to the problems of noncoverage or nonresponse. From the efficiency comparison study, we show that the calibration estimators are more efficient than those of Warner's and Mangat-Singh's when the known population cell and marginal counts of auxiliary information are used for the calibration procedure.     

Moran's I statistic-based nonparametric test with spatio-temporal observations

Moran's I statistic [Moran, (1950), ‘Notes on Continuous Stochastic Phenomena’, Biometrika, 37, 17–23] has been widely used to evaluate spatial autocorrelation. This paper is concerned with Moran's I-induced testing procedure in residual analysis. We begin with exploring the Moran's I statistic in both its original and extended forms analytically and numerically. We demonstrate that the magnitude of the statistic in general depends not only on the underlying correlation but also on certain heterogeneity in the individual observations. One should exercise caution when interpreting the outcome on correlation by the Moran's I-induced procedure. On the other hand, the effect on the Moran's I due to heterogeneity in the observations enables a regression model checking procedure with the residuals. This novel application of Moran's I is justified by simulation and illustrated by an analysis of wildfire records from Alberta, Canada.     

Nonparametric Applications of Shaffer's Extension of Dennett's Procedure

Shaffer's extensions and generalization of Dunnett's procedure are shown to be applicable in several nonparametric data analyses. Applications are considered within the context of the Kruskal-Wallis one-way analysis of variance (ANOVA) test for ranked data, Friedman's two-way ANOVA test for ranked data, and Cochran's test of change for dichotomous data.     

A subset selection procedure for multinomial distributions

A subset selection procedure is developed for selecting a subset containing the multinomial population that has the highest value of a certain linear combination of the multinomial cell probabilities; such population is called the ‘best’. The multivariate normal large sample approximation to the multinomial distribution is used to derive expressions for the probability of a correct selection, and for the threshold constant involved in the procedure. The procedure guarantees that the probability of a correct selection is at least at a pre-assigned level. The proposed procedure is an extension of Gupta and Sobel's [14] selection procedure for binomials and of Bakir's [2] restrictive selection procedure for multinomials. One illustration of the procedure concerns population income mobility in four countries: Peru, Russia, South Africa and the USA. Analysis indicates that Russia and Peru fall in the selected subset containing the best population with respect to income mobility from poverty to a higher-income status. The procedure is also applied to data concerning grade distribution for students in a certain freshman class.     

Empirical study of some non-parametric tests for dispersion of correlated data

Results of a computer simulation study of power and robustness of three competitor tests for comparing scales, for use with correlated data: Rothstein, Richardson and Bell (RRB), Arvesen, and Pitman, are presented. It is found that unless one could ímprove the approximate null distributions for Arvesen's and Pitman's test, RRB's procedure is best, having simulated probabilities of Type I error closest to the test's nominal α and being reasonably robust and powerful, for all distributions considered.     

A comparison of two procedures to select the best binomial population with sequential elimination of inferior populations

We compare the selection procedure of Levin and Robbins [1981. Selecting the highest probability in binomial or multinomial trials. Proc. Nat. Acad. Sci. USA 78, 4663–4666.] with the procedure of Paulson [1994. Sequential procedures for selecting the best one of k Koopman–Darmois populations. Sequential Analysis 13, 207–220.] to identify the best of several binomial populations with sequential elimination of unlikely candidates. We point out situations in which the Levin–Robbins procedure dominates the Paulson procedure in terms of the duration of the experiment, the expected total number of observations, and the expected number of failures. Because the Levin–Robbins procedure is also easier to implement than Paulson's procedure and gives a tighter guarantee for the probability of correct selection, we conclude that it holds a competitive edge over Paulson's procedure.     

On confidence regions for the mean of a multivariate time series

We consider the problem of setting up a confidence region for the mean of amultivariate timeseries ont he basis of a part-realisation of that series.A procedure for setting up a confidence interval for the mean of a univariate time series Is implicitin Jones(1976).We present an analogous procedure for setting up a confidence region for the mean of a multivariatet ime series.This procedure is base donastatistic which is an analogue of Hotelling'sT'.Our results are applied to a comparison of climate means obtained from experiments with a General Circulation Model of the earth's atmosphere.     

A nonparametric coherent confidence procedure

ABSTRACT

Holm's step-down testing procedure starts with the smallest p-value and sequentially screens larger p-values without any information on confidence intervals. This article changes the conventional step-down testing framework by presenting a nonparametric procedure that starts with the largest p-value and sequentially screens smaller p-values in a step-by-step manner to construct a set of simultaneous confidence sets. We use a partitioning approach to prove that the new procedure controls the simultaneous confidence level (thus strongly controlling the familywise error rate). Discernible features of the new stepwise procedure include consistency with individual inference, coherence, and confidence estimations for follow-up investigations. In a simple simulation study, the proposed procedure (treated as a testing procedure), is more powerful than Holm's procedure when the correlation coefficient is large, and vice versa when it is small. In the data analysis of a medical study, the new procedure is able to detect the efficacy of Aspirin as a cardiovascular prophylaxis in a nonparametric setting.     

A clarifying comparison of methods for controlling the false discovery rate

Traditional multiple hypothesis testing procedures fix an error rate and determine the corresponding rejection region. In 2002 Storey proposed a fixed rejection region procedure and showed numerically that it can gain more power than the fixed error rate procedure of Benjamini and Hochberg while controlling the same false discovery rate (FDR). In this paper it is proved that when the number of alternatives is small compared to the total number of hypotheses, Storey's method can be less powerful than that of Benjamini and Hochberg. Moreover, the two procedures are compared by setting them to produce the same FDR. The difference in power between Storey's procedure and that of Benjamini and Hochberg is near zero when the distance between the null and alternative distributions is large, but Benjamini and Hochberg's procedure becomes more powerful as the distance decreases. It is shown that modifying the Benjamini and Hochberg procedure to incorporate an estimate of the proportion of true null hypotheses as proposed by Black gives a procedure with superior power.