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Testing for dependence between pairs of random variables is a fundamental problem in statistics. In some applications, data are subject to selection bias that can create spurious dependence. An important example is truncation models, in which observed pairs are restricted to a specific subset of the X-Y plane. Standard tests for independence are not suitable in such cases, and alternative tests that take the selection bias into account are required. Here, we generalize the notion of quasi-independence with respect to the sampling mechanism, and study the problem of detecting any deviations from it. We develop two tests statistics motivated by the classic Hoeffding's statistic, and use two approaches to compute their distribution under the null: (i) a bootstrap-based approach, and (ii) a permutation-test with nonuniform probability of permutations. We also handle an important application to the case of censoring with truncation, by estimating the biased sampling mechanism from the data. We prove the validity of the tests, and show, using simulations, that they improve power compared to competing methods for important special cases. The tests are applied to four datasets, two that are subject to truncation, with and without censoring, and two to bias mechanisms related to length bias. [ABSTRACT FROM AUTHOR]

We propose a novel class of independence measures for testing independence between two random vectors based on the discrepancy between the conditional and the marginal characteristic functions. The relation between our index and other similar measures is studied, which indicates that they all belong to a large framework of reproducing kernel Hilbert space. If one of the variables is categorical, our asymmetric index extends the typical ANOVA to a kernel ANOVA that can test a more general hypothesis of equal distributions among groups. In addition, our index is also applicable when both variables are continuous. We develop two empirical estimates and obtain their respective asymptotic distributions. We illustrate the advantages of our approach by numerical studies across a variety of settings including a real data example. for this article are available online. [ABSTRACT FROM AUTHOR]

Capturing dependence among a large number of high-dimensional random vectors is a very important and challenging problem. By arrangingnrandom vectors of lengthpin the form of a matrix, we develop a linear spectral statistic of the constructed matrix to test whether thenrandom vectors are independent or not. Specifically, the proposed statistic can also be applied tonrandom vectors, each of whose elements can be written as either a linear stationary process or a linear combination of independent random variables. The asymptotic distribution of the proposed test statistic is established for the case ofasn→ ∞. To avoid estimating the spectrum of each random vector, a modified test statistic, which is based on splitting the originalnvectors into two equal parts and eliminating the term that contains the inner structure of each random vector or time series, is constructed. The facts that the limiting distribution is normal and there is no need to know the inner structure of each investigated random vector result in simple implementation of the constructed test statistic. Simulation results demonstrate that the proposed test is powerful against several commonly used dependence structures. An empirical application to detecting dependence of the closed prices from several stocks in the S&P500 also illustrates the applicability and effectiveness of our provided test. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Although nonparametric analyses of bivariate failure times under independent censoring have been widely studied, nonparametric analyses of bivariate competing-risks data have not yet been investigated. Such analyses are important in familial association studies, where multiple interacting failure types may violate the independent censoring assumption. We develop nonparametric estimators for the bivariate cause-specific hazards function and the bivariate cumulative incidence function that are natural analogs of their univariate counterparts and make no assumptions about the dependence of the risks. The estimators are shown to be uniformly consistent and to converge weakly to Gaussian processes. They provide the basis for novel time-dependent association measures, with the associated inferences yielding tests of cause-specific independence in pairs. The methodology performs well in simulations with realistic sample sizes, Its practical utility is illustrated in an analysis of dementia in the Cache County Study, where the nonparametric methods indicate that mother-child disease associations are strongly time-varying. [ABSTRACT FROM AUTHOR]

New test statistics are proposed for testing whether two random vectors are independent. Gieser and Randles, as well as Taskinen, Kankainen, and Oja have introduced and discussed multivariate extensions of the quadrant test of Blomqvist. This article serves as a sequel to this work and presents new multivariate extensions of Kendall's tau and Spearman's rho statistics. Two different approaches are discussed. First, interdirection proportions are used to estimate the cosines of angles between centered observation vectors and between differences of observation vectors. Second, covariances between affine-equivariant multivariate signs and ranks are used. The test statistics arising from these two approaches appear to be asymptotically equivalent if each vector is elliptically symmetric. The spatial sign versions are easy to compute for data in common dimensions, and they provide practical, robust alternatives to normal-theory methods. Asymptotic theory is developed to approximate the finite-sample null distributions as welt, as to calculate limiting Pitman efficiencies. Small-sample null permutation distributions are also described. A simple simulation study is used to compare the proposed tests with the classical Wilks test. Finally, the theory is illustrated by an example. [ABSTRACT FROM AUTHOR]

Presents information on a study about the rank test for testing independence. Test statistics; Difference between the power of the test and the maximal power of the tests under consideration at the given alternative; Asymptomatic null distribution and consistency.

The Haugh (1976) test for independence employs the univariate residual cross-correlation function. However, it ignores information about a possible pattern in successive cross-correlation coefficients. An asymptotic test is developed that incorporates this information and includes the Haugh test as a special case. A Monte Carlo study indicates that the proposed test is more powerful than the Haugh s and regression F tests for certain models. Two empirical examples are presented showing the simplicity of applying this test and its ability to recognize relationships that the Haugh test may fail to detect. [ABSTRACT FROM AUTHOR]

Independent observations Y[sub 1], Y[sub 2],..., are generated via the general linear model Y[sub i] = X[sub i]beta + epsilon[sub i], X[sub i] being either a random or a sequentially designed matrix. The parameter beta is estimated sequentially using the least squares estimator subject to a loss structure that is the sum of a cost due to sampling and a loss due to estimation error. If epsilon[sub 1], epsilon[sub 2] .... are iid N(0, sigma[sup2] I) with 0. unknown, no fixed-sample procedure will minimize the risk for all 0 < sigma < Infinity. The sequential procedure presented here performs nearly as well as the constant risk best fixed-sample procedure when sigma[sup 2] is known. Its risk and the regret in not knowing sigma[sup 2] are calculated up to terms that are o(c), as the cost of one observation c arrow right 0 (or as sigma arrow right Infinity). The asymptotic moments of the stopping time are given and the normalized stopping time's limiting distribution is shown to be normal. Its first two moments are calculated up to second-order terms. [ABSTRACT FROM AUTHOR]

A model specifying independence within each of two disjoint groups of rows is presented as an alternative to complete independence in a two-dimensional contingency table with r rows and two columns. If the measure used to assess the improvement in fit is a member of a class of order-producing (OP) functions, then the partition that maximizes the measure can be located after checking only (r - 1) of the (2[sup r-1] - 1) possible partitions. The likelihood ratio statistic and corresponding Pearson goodness-of-fit statistics for independence in the collapsed table are two such OP functions. In the case of equal row totals the asymptotic distribution of the Pearson statistic is approximately that of 2(pi - 2)/pi times a chi-squared variable with r/(pi - 2) degrees of freedom. [ABSTRACT FROM AUTHOR]

Consider r is greater than or equal to 2 random variables X[sub 1, sup (t)], ..... X[sub r, sup(t)], each taking values in [0, t], but subject to the constraint SIGMA[sup r, sub j = 1] X[sub j, sup(t)] less than or equal to t. The concept of F-independence for such variables is a modification of the concept of independence, which takes the constraint into account, and is applicable when (X[sub 1, sup (t)],.... X[sub r, sup (t)]) is an element of {(X[sub 1, sup (t)],..... X[sub r, sup (t)]); t' is an element of l} for some index set l [2, 3]. We consider F-independence properties as t arrow right Infinity. We show that F independence properties become independence properties or neutrality properties [1, 5], respectively, when the variables converge properly, or are asymptotically proportional to the bound. Examples of important families of distributions which satisfy the convergence criteria a re given. [ABSTRACT FROM AUTHOR]

A double sampling scheme is considered for estimation and testing under misclassification in both margins of a 2 x 2 contingency table; the matrix of misclassification probabilities is assumed to be unknown and nonsingular. Maximum likelihood estimators are given for the error-free and misclassification probabilities. Numerical methods are utilized to obtain maximum likelihood estimates under the independence hypothesis and the regularity conditions are affirmed for the asymptotic properties of the estimates to hold, The log-likelihood ratio statistic is shown to be asymptotically distributed as a chi-square. Computer simulations indicated important considerations for testing under double margin misclassification. [ABSTRACT FROM AUTHOR]

Extensions of time-series modeling procedures of Box and Jenkins [5] reveal that numerous economic variables which are generally regarded as being strongly interrelated may with equal validity, based on recent empirical evidence, be regarded as independent or only weakly related. Differences between these results and the bulk of econometric literature are attributed to the failure of the latter to satisfactorily account for autocorrelation. Due to limitations in the data, it is concluded that in many instances where economic relationships clearly do exist, econometric or other empirical means cannot reliably enable their existence to be ascertained. [ABSTRACT FROM AUTHOR]

This empirical study of the chi-square approximations as commonly encountered in behavioral research involved: (1) both tests of goodness-of-fit and of independence, (2) uniform distributions and two levels of departure from uniform, (3) sample sizes ranging from 10 to 100. Excellent approximations were obtained with average expected frequencies of one or two in tests of goodness-off-it to uniform; slightly higher expected frequencies were required with the nonuniform cases. Tests of independence were strikingly robust with respect to Type I errors; in almost all cases the errors were in the conservative direction. [ABSTRACT FROM AUTHOR]

Some interesting examples of uncorrelated simple linear rank statistics are listed. Consideration is then given to conditions under which general rank statistics are uncorrelated or have joint distributions which enjoy certain symmetric properties. Two simple linear rank statistics that are uncorrelated are found, under suitable restrictions, to be asymptotically independent. This is followed by some illustrations in which two rank statistics are exactly independent. The importance of these results about independence in actual testing is then noted. [ABSTRACT FROM AUTHOR]

Let X[sub n] and X[sub 1] be the largest and smallest order statistics, respectively, of a random sample of size n. Quite generally, X[sub n] and X[sub 1] are approximately independent for n sufficiently large. Minimum n for attaining at least specified levels of independence are developed. Level of independence is measured by the maximum difference between the true values of P(X[sub 1] is less than or equal to x[sub 1], X[sub n] is less than or equal to x[sub n]) and the corresponding values assuming independence of X[sub n] and X[sub 1]. The results apply to all possible distributions for the population sampled. The value of minimum n is the smallest allowable n for the continuous case but can be too large otherwise. Minimum n is finite for all nonzero differences. [ABSTRACT FROM AUTHOR]

Concepts of independence for nonnegative continuous random variables, X[sub 1],..., X[sub k], subject to the constraint SIGMA X[sub i] = 1 are developed. These concepts provide a means of modeling random vectors of proportions which is useful in analyzing certain kinds of data; and which may be of interest in quantifying prior opinions about multinomial parameters. A generalization of the Dirichlet distribution is given, and its relation to the Dirichlet is simply indicated by means of the concepts. The concepts are used to obtain conclusions of biological interest for data on bone composition in rats and scute growth in turtles. [ABSTRACT FROM AUTHOR]

The authors comment on the works of researchers Michael Hudgens and Thomas Lemieux who have expressed their views on articles on statistical research published in this issue. Topics discussed include the use of conditional independence assumption (CIA) and regression discontinuity (RD) in statistical analysis; directed acyclic graphs (DAGs); and the stable unit value assumption (SUTVA) method.

When discussing stochastic independence in a course on probability theory, it is customary to give an example of three identically distributed random variables X, Y and Z which are pairwise independent but not mutually independent. The purpose of this article is to show that, among the absolutely continuous distributions which have the closed unit interval for their support, the rectangular distribution is the only one for which these conclusions hold. The characterizations given by the theorem have analogs for discrete rectangular distributions. The applications of these characterizations to random number generation and the study of rounding error are under investigation.

The article presents information on independence of sample mean and translation invariant statistics of samples from multivariate normal populations. In this case it could be said that the sample has circularly symmetric type dependence. Statistical analysis of normal sample dependent in the preceding fashion has been considered in the multivariate case by I. Olkin and in the univariate case by L. Guttman. The main result is a theorem giving a necessary and sufficient condition for the independence of the sample mean and a translation invariant statistic of a normal sample. The sample mean is independent of any translation invariant statistic, if the sample is an equicorrelated normal sample or a normal sample with circularly symmetric type dependence.