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Abstract Hypothesis testing for the proportionality of covariance matrices is a classical statistical problem and has been widely studied in the literature. However, there have been few treatments of this test in high-dimensional settings, especially for the case where the number of variables is larger than the sample size, despite high-dimensional statistical inference having recently received considerable attention. This paper studies hypothesis testing for the proportionality of two covariance matrices in the high-dimensional setting: m , n ≍ p δ for some δ ∈ (1 ∕ 2 , 1) , where m and n denote the sample sizes and p denotes the number of variables. A test statistic is proposed and its asymptotic distribution is derived under multivariate normality. The non-asymptotic performance of the proposed test procedure is numerically examined. [ABSTRACT FROM AUTHOR]

Abstract This paper considers estimation and testing problems for partial functional linear models when the covariates in the non-functional linear component are measured with additive error. A corrected profile, least-squares based, estimation procedure is developed for the parametric component. Asymptotic properties of the proposed estimators are established under some regularity conditions. To test a hypothesis on the parametric component, a statistic based on the difference between the corrected residual sums of squares under the null and alternative hypotheses is proposed; its limiting null distribution is shown to be a weighted sum of independent standard χ 1 2 variables. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for illustration. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the simultaneous estimation of k population means when one suspects that the k means are nearly equal. As an alternative to the preliminary test estimator based on the test statistics for testing hypothesis of equal means, we derive Bayesian and minimax estimators which shrink individual sample means toward a pooled mean estimator given under the hypothesis. It is shown that both the preliminary test estimator and the Bayesian minimax shrinkage estimators are further improved by shrinking the pooled mean estimator. The performance of the proposed shrinkage estimators is investigated by simulation. [ABSTRACT FROM AUTHOR]

Abstract To test whether two populations have the same mean vector in a high-dimensional setting, Chen and Qin (2010, Ann. Statist.) derived an unbiased estimator of the squared Euclidean distance between the mean vectors and proved the asymptotic normality of this estimator under local assumptions about the mean vectors. In this study, their results are extended without assumptions about the mean vectors. In addition, asymptotic normality is established in the class of general statistics including Chen and Qin's statistics and other important statistics under general moment conditions that cover both Chen and Qin's moment condition and elliptical distributional assumption. These asymptotic results are applied to the construction of simultaneous intervals for all pair-wise differences between mean vectors of k ≥ 2 groups. The finite-sample and dimension performance of the proposed methods is also studied via Monte Carlo simulations. The methodology is illustrated using microarray data. [ABSTRACT FROM AUTHOR]

This paper is concerned with tests for the parallelism and flatness hypotheses in multi-group profile analysis for high-dimensional data. We extend to elliptical distributions the procedures developed for normal populations by Harrar and Kong (2016). Specifically, we prove that their statistics continue to be asymptotically normal when the underlying population is elliptical, and we obtain new tests by improving their estimator of the asymptotic variance. Using asymptotic normality, we show that the asymptotic size of the proposed tests is equal to the nominal significance level, and we also derive the asymptotic power. Finally, we present simulation results and find that the power of the new tests is superior to that of the original Harrar–Kong procedure. [ABSTRACT FROM AUTHOR]

In recent years, with the rapid development of data collecting technologies, high-dimensional data have become increasingly prevalent. Much work has been done for testing hypotheses on mean vectors, especially for high-dimensional two-sample problems. Rather than considering a specific problem, we are interested in a general linear hypothesis testing (GLHT) problem on mean vectors of several populations, which includes many existing hypotheses about mean vectors as special cases. A few existing methodologies on this important GLHT problem impose strong assumptions on the underlying covariance matrix so that the null distributions of the associated test statistics are asymptotically normal. In this paper, we propose a simple and adaptive test based on the L 2 -norm for the GLHT problem. For normal data, we show that the null distribution of our test statistic is the same as that of a chi-squared type mixture which is generally skewed. Therefore, it may yield misleading results if we blindly approximate the underlying null distribution of our test statistic using a normal distribution. In fact, we show that the null distribution of our test statistic is asymptotically normal only when a necessary and sufficient condition on the underlying covariance matrix is satisfied. This condition, however, is not always satisfied and it is not an easy task to check if it is satisfied in practice. To overcome this difficulty, we propose to approximate the null distribution of our test statistic using the well-known Welch–Satterthwaite chi-squared approximation so that our new test is applicable without any assumption on the underlying covariance matrix. Simple ratio-consistent estimators of the unknown parameters are obtained. The asymptotic and approximate powers of our new test are also investigated. The methodologies are then extended for non-normal data. Four simulation studies and a real data application are presented to demonstrate the good performance of our new test compared with some existing testing procedures available in the literature. [ABSTRACT FROM AUTHOR]

In this article, we propose a test for making an inference about the block-diagonal covariance structure of a covariance matrix in non-normal high-dimensional data. We prove that the limiting null distribution of the proposed test is normal under mild conditions when its dimension is substantially larger than its sample size. We further study the local power of the proposed test. Finally, we study the finite-sample performance of the proposed test via Monte Carlo simulations. We demonstrate the relevance and benefits of the proposed approach for a number of alternative covariance structures. [ABSTRACT FROM AUTHOR]

Varying-coefficient models are very important tools to explore the hidden structure between the response variable and its predictors. This article focuses on the estimation of varying-coefficient models with discontinuous coefficient functions. Based on local linear smoothing and jump-preserving regression techniques, an adaptive jump-preserving (AJP) estimation procedure is proposed to estimate the coefficient functions with jumps. This method can automatically accommodate possible jumps of the coefficient functions without knowing the number and locations of jump points or performing any hypothesis tests. Under some mild conditions, the asymptotical properties of the resulting estimators can be established. Furthermore, several numerical studies are conducted to evaluate the finite sample performance of the proposed methodologies. Finally, an application to Australia consumer price index (CPI) data illustrates the validity of the proposed techniques. [ABSTRACT FROM AUTHOR]

We observe a sample of n independent p -dimensional Gaussian vectors with Toeplitz covariance matrix Σ = [ σ ∣ i − j ∣ ] 1 ≤ i , j ≤ p and σ 0 = 1 . We consider the problem of testing the hypothesis that Σ is the identity matrix asymptotically when n → ∞ and p → ∞ . We suppose that the covariances σ k decrease either polynomially ( ∑ k ≥ 1 k 2 α σ k 2 ≤ L for α > 1 / 4 and L > 0 ) or exponentially ( ∑ k ≥ 1 e 2 A k σ k 2 ≤ L for A , L > 0 ). We consider a test procedure based on a weighted U-statistic of order 2, with optimal weights chosen as solution of an extremal problem. We give the asymptotic normality of the test statistic under the null hypothesis for fixed n and p → + ∞ and the asymptotic behavior of the type I error probability of our test procedure. We also show that the maximal type II error probability, either tend to 0 , or is bounded from above. In the latter case, the upper bound is given using the asymptotic normality of our test statistic under alternatives close to the separation boundary. Our assumptions imply mild conditions: n = o ( p 2 α − 1 / 2 ) (in the polynomial case), n = o ( e p ) (in the exponential case). We prove both rate optimality and sharp optimality of our results, for α > 1 in the polynomial case and for any A > 0 in the exponential case. A simulation study illustrates the good behavior of our procedure, in particular for small n , large p . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the model checking problem of a partially linear model when some covariates are measured with error and some auxiliary variables are supplied. The often-used assumptions on the measurement error, such as a known error variance or a known distribution of the error variable, are not required. Also repeated measurements are not needed. Instead, a nonparametric calibration method is applied to deal with the measurement error. An estimating method for the null hypothetical model is proposed and the asymptotic properties of the proposed estimators are established. A testing method based on a residual-marked empirical process is then developed to check the null hypothetical partially linear model. The tests are shown to be consistent and can detect the alternative hypothesis close to the null hypothesis at the rate n − r with 0 ≤ r ≤ 1 / 2 . Simulation studies and real data analysis are conducted to examine the finite sample behavior of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of testing the sphericity of a covariance matrix in high-dimensional frameworks. A new test procedure is put forward by taking the maximum of two existing statistics which are proved weakly independent in our settings. Asymptotic distribution of the new statistic is derived for generally distributed population with a finite fourth moment. Extensive simulations demonstrate that the proposed test has a great improvement in robustness of power against various models under the alternative hypothesis. [ABSTRACT FROM AUTHOR]

Shifts in responses typically are obscured from users, so that regression proceeds as if unshifted. At issue is the infusion of such shifts into classical analysis. On projecting outliers into the “Regressor” and “Error” spaces of a model, findings here are that shifts in responses may account for shifts in the OLS solutions, or for inflated residuals, or both. These in turn impact estimation, prediction, and hypothesis tests, all of vital interest to users, and all considered here. Tools for identifying shifts are given. Case studies illustrate effects of shifts on regression, to include a reexamination of studies from the literature. [ABSTRACT FROM AUTHOR]

In the context of a heteroscedastic nonparametric regression model, we develop a test for the null hypothesis that a subset of the predictors has no influence on the regression function. The test uses residuals obtained from local polynomial fitting of the null model and is based on a test statistic inspired from high-dimensional analysis of variance. Using p -values from this test, and multiple testing ideas, a group variable selection method is proposed, which can consistently select even groups of variables with diminishing predictive significance. A backward elimination version of this procedure, called GBEAMS for Group Backward Elimination Anova-type Model Selection, is recommended for practical applications. Simulation studies, suggest that the proposed test procedure outperforms the generalized likelihood ratio test when the alternative is non-additive or there is heteroscedasticity. Additional simulation studies reveal that the proposed group variable selection procedure performs competitively against other variable selection methods, and outperforms them in selecting groups having nonlinear or dependent effects. The proposed group variable selection procedure is illustrated on a real data set. [ABSTRACT FROM AUTHOR]

Let X 1 , … , X n 1 + 1 ∼ iid N p ( μ 1 , Σ 1 ) and Y 1 , … , Y n 2 + 1 ∼ iid N p ( μ 2 , Σ 2 ) be two independent random samples, where p < n 2 . In this article, we propose a new test for the proportionality of two large p × p covariance matrices Σ 1 and Σ 2 . By applying modern random matrix theory, we establish the asymptotic normality property for the proposed test statistic as ( p , n 1 , n 2 ) → ∞ together with the ratios p / n 1 → y 1 ∈ ( 0 , ∞ ) and p / n 2 → y 2 ∈ ( 0 , 1 ) under suitable conditions. We further showed that these conclusions are still valid if normal populations are replaced by general populations with finite fourth moments. [ABSTRACT FROM AUTHOR]

We consider hypothesis testing for the null hypothesis being represented as an arbitrary-shaped region in the parameter space. We compute an approximate p-value by counting how many times the null hypothesis holds in bootstrap replicates. This frequency, known as bootstrap probability, is widely used in evolutionary biology, but often reported as biased in the literature. Based on the asymptotic theory of bootstrap confidence intervals, there have been some new attempts for adjusting the bias via bootstrap probability without direct access to the parameter value. One such an attempt is the double bootstrap which adjusts the bias by bootstrapping the bootstrap probability. Another new attempt is the multiscale bootstrap which is similar to the m-out-of-n bootstrap but very unusually extrapolating the bootstrap probability to m=-n. In this paper, we employ these two attempts at the same time, and call the new procedure as multiscale-double bootstrap. By focusing on the multivariate normal model, we investigate higher-order asymptotics up to fourth-order accuracy. Geometry of the region plays important roles in the asymptotic theory. It was known in the literature that the curvature of the boundary surface of the region determines the bias of bootstrap probability. We found out that the "curvature of curvature" determines the remaining bias of double bootstrap. The multiscale bootstrap removes these biases. The multiscale-double bootstrap is fourth order accurate with coverage probability erring only O(n-2), and it is robust against computational error of parameter estimation used for generating bootstrap replicates from the null distribution. [ABSTRACT FROM AUTHOR]

In this article, we propose a computationally efficient approach to estimate (large) p-dimensional covariance matrices of ordered (or longitudinal) data based on an independent sample of size n. To do this, we construct the estimator based on a k-band partial autocorrelation matrix with the number of bands chosen using an exact multiple hypothesis testing procedure. This approach is considerably faster than many existing methods and only requires inversion of (k+1)-dimensional covariance matrices. The resulting estimator is positive definite as long as k

Abstract: This paper is concerned with hierarchical data having three levels. The level 1 units are nested in the level 2 units or subclusters which are themselves nested in the level 3 clusters. The model for this data is assumed to fulfill some symmetry assumptions. The level 1 units within each subcluster are exchangeable and a permutation of the subclusters belonging to the same cluster leaves the model unchanged. We are interested in measuring the dependence associated to clusters and subclusters respectively. Two exchangeable Kendall’s tau are proposed as non parametric measures of these two associations and estimators for these measures are proposed. Their asymptotic properties are then investigated under the proposed hierarchical model for the data. These statistics are then used to estimate the intra-class correlation coefficients for data drawn from elliptical hierarchical distributions. Hypothesis tests for the cluster and subcluster effects based on the proposed estimators are developed and their performances are assessed using Pitman efficiencies and a Monte Carlo study. [Copyright &y& Elsevier]

Abstract: We propose a new class of nonparametric tests for the supposition of independence between two continuous random variables and . Given a size sample, let be the permutation which maps the ranks of the observations on the ranks of the observations. We identify the independence assumption of the null hypothesis with the uniform distribution on the permutation space. A test based on the size of the longest increasing subsequence of ( ) is defined. The exact distribution of is computed from Schensted’s theorem (Schensted, 1961). The asymptotic distribution of was obtained by Baik et al. (1999). As the statistic is discrete, there is a small set of possible significance levels. To solve this problem we define the statistic which is a jackknife version of , as well as the corresponding hypothesis test. A third test is defined based on the statistic which is a jackknife version of the longest monotonic subsequence of . On a simulation study we apply our tests to diverse dependence situations with null or very small correlations where the independence hypothesis is difficult to reject. We show that , and tests have very good performance on that kind of situations. We illustrate the use of those tests on two real data examples with small sample size. [Copyright &y& Elsevier]

Abstract: Many multiple testing procedures make use of the -values from the individual pairs of hypothesis tests, and are valid if the -value statistics are independent and uniformly distributed under the null hypotheses. However, it has recently been shown that these types of multiple testing procedures are inefficient since such -values do not depend upon all of the available data. This paper provides tools for constructing compound -value statistics, which are those that depend upon all of the available data, but still satisfy the conditions of independence and uniformity under the null hypotheses. Several examples are provided, including a class of compound -value statistics for testing location shifts. It is demonstrated, both analytically and through simulations, that multiple testing procedures tend to reject more false null hypotheses when applied to these compound -values rather than the usual -values, and at the same time still guarantee the desired type I error rate control. The compound -values are used to analyze a real microarray data set and allow for more rejected null hypotheses. [Copyright &y& Elsevier]

Abstract: The main results imply that the probability is Schur-concave/Schur-convex in provided that the indicator function of a set in is so, respectively; here, and is a standard normal random vector in . Moreover, it is shown that the Schur-concavity/Schur-convexity is strict unless the set is equivalent to a spherically symmetric set. Applications to testing hypotheses on multivariate means are given. [Copyright &y& Elsevier]

Abstract: In this article we derive an optimal test for testing the significance of covariance matrices of random-effects of two multivariate mixed-effects linear models. We compute the power of this newly derived test via simulation for various alternative hypotheses in a bivariate set up for unbalanced designs and observe that power responds sharply when sample size and alternative hypotheses are changed. For some balanced designs we compare power of the optimal test to that of the likelihood ratio test via simulation, and find that the proposed test has greater power than the likelihood ratio test. The results are illustrated using real data on human growth. Other relevant applications of the model are highlighted. [Copyright &y& Elsevier]

Abstract: Consider the problem of testing a linear hypothesis of regression coefficients in a general linear regression model with a covariance matrix involving several nuisance parameters. Then, the Bartlett-type adjustments of the Wald, Score, and modified Likelihood Ratio tests are derived for general consistent estimators of the unknown nuisance parameters. The adjusted test statistics have second-order corrections in type I errors. Simple parametric bootstrap methods are also suggested for estimating the Bartlett-type adjustments and it is shown that they have the second order accuracy. [Copyright &y& Elsevier]

Abstract: When it is not possible to obtain the analytical null distribution of a test statistic , Monte Carlo hypothesis tests can be used to perform the test. Monte Carlo tests are commonly used in a wide variety of applications, including spatial statistics, and biostatistics. Conventional Monte Carlo tests require the simulation of independent copies from under the null hypothesis, what is computationally intensive for large data sets. Truncated sequential Monte Carlo designs can be performed to reduce computational effort in such situations. Different truncated sequential procedures have been proposed. They work under restrictive assumptions on the distribution of aiming to bound the power loss and to reduce execution time. Since the use of Monte Carlo tests are based on the situations where the null distribution of is unknown, their results are not valid for the general case of any test statistic. In this paper, we derive an optimal scheme for truncated sequential Monte Carlo hypothesis tests. This scheme minimizes the expected number of simulations under any alternative hypothesis, and bounds the power loss in arbitrarily small values. The first advantage from this scheme is that the results concerning the power and the expected time are valid for any test statistic. Also, we present practical examples of optimal procedures for which the expected number of simulations are reduced by 60% in comparison with some of the best procedures in the literature. [Copyright &y& Elsevier]

Abstract: In this paper we propose a test for testing the equality of the mean vectors of two groups with unequal covariance matrices based on and independently distributed -dimensional observation vectors. It will be assumed that observation vectors from the first group are normally distributed with mean vector and covariance matrix . Similarly, the observation vectors from the second group are normally distributed with mean vector and covariance matrix . We propose a test for testing the hypothesis that . This test is invariant under the group of nonsingular diagonal matrices. The asymptotic distribution is obtained as and but and may go to zero or infinity. It is compared with a recently proposed non-invariant test. It is shown that the proposed test performs the best. [Copyright &y& Elsevier]

Abstract: This paper addresses, for a composite hypothesis about a subvector of the parameters in the parametric model, the issues posed by Rao and Mukerjee (1995) [22] and Li (2001) [14] on the power under a sequence of local alternatives. It is shown that a partially adjusted test statistic in a class of test statistics is equally sensitive (up to the third-order) to the change of the nuisance parameters. However, there exist infinitely many ways for improving the chi-squared approximation to the null distribution, which reveal, in general, the non-equivalence of the resulting third-order point-by-point local powers. To make a definitive conclusion, the average local power is then considered, from which the third-order asymptotic optimality of the Bartlett-type adjusted Rao test can be also established. [Copyright &y& Elsevier]

Abstract: What is the distribution of the product of given powers of independent uniform (0, 1) random variables? Is this distribution useful? Is this distribution commonly used in some contexts? Is this distribution somehow related to the distribution of the product of other random variables? Are there some test statistics with this distribution? This paper will give the answers to the above questions. It will be seen that the answer to the last four questions above is: yes! We will show how particular choices of the numbers of variables involved and their powers will result in interesting and useful distributions and how these distributions may help us to shed some new light on some well-known distributions and also how it may help us to address, in a much simpler way, some distributions usually considered to be rather complicated as is the case with the exact distribution of a number of statistics used in Multivariate Analysis, including some whose exact distribution up until now is not available in a concise and manageable form. [Copyright &y& Elsevier]

Abstract: In this article we propose two consistent hypothesis tests of -type for weakly dependent observations based on the empirical characteristic function. We consider a symmetry test and a goodness-of-fit test for the marginal distribution of a time series. The asymptotic behaviour under the null as well as fixed and certain local alternatives is investigated. Since the limit distributions of the test statistics depend on unknown parameters in a complicated way, we suggest to apply certain parametric bootstrap methods in order to determine critical values of the tests. [Copyright &y& Elsevier]

Abstract: We consider testing for discontinuities in a trend function when the residual process exhibits long memory. Using a wavelet decomposition of the estimated trend function into a low-resolution and a high-resolution component, a test statistic is proposed based on blockwise resampling of estimated residual variances. Asymptotic validity of the test is derived. A simulation study illustrates finite sample properties. [Copyright &y& Elsevier]

Abstract: Inference about the difference between two normal mean vectors when the covariance matrices are unknown and arbitrary is considered. Assuming that the incomplete data are of monotone pattern, a pivotal quantity, similar to the Hotelling statistic, is proposed. A satisfactory moment approximation to the distribution of the pivotal quantity is derived. Hypothesis testing and confidence estimation based on the approximate distribution are outlined. The accuracy of the approximation is investigated using Monte Carlo simulation. Monte Carlo studies indicate that the approximate method is very satisfactory even for moderately small samples. The proposed methods are illustrated using an example. [Copyright &y& Elsevier]

Abstract: We derive the asymptotical distributions of two-sample -statistics and two-sample empirical -quantiles in the case of weakly dependent data. Our results apply to observations that can be represented as functionals of absolutely regular processes, including e.g. many classical time series models as well as data from chaotic dynamical systems. Based on these theoretical results we propose a new robust nonparametric test for the two-sample location problem, which is constructed from the median of pairwise differences between the two samples. We inspect the properties of the test in the case of weakly dependent data and compare the performance with classical tests such as the -test and Wilcoxon’s two-sample rank test with corrections for dependencies. Simulations indicate that the new test offers better power than the Wilcoxon test in case of skewed and heavy tailed distributions, when at least one of the two samples is not very large. The test is then applied for detecting shifts of location in some weakly dependent time series, which are contaminated by outliers. [Copyright &y& Elsevier]

Abstract: Let be a closed convex cone which contains a linear subspace . We investigate the restricted likelihood ratio test for the null and alternative hypotheses based on an -dimensional, normally distributed random vector with unknown mean and known covariance matrix . We prove that if the true mean vector satisfies the alternative hypothesis , then the restricted likelihood ratio test is more powerful than the unrestricted test with larger alternative hypothesis . The proof uses isoperimetric inequalities for the uniform distribution on the -dimensional sphere and for -dimensional standard Gaussian measure. [Copyright &y& Elsevier]

Abstract: We consider the problem of testing whether the common mean of a single -vector of multivariate normal random variables with known variance and unknown common correlation is zero. We derive the standardized likelihood ratio test for known and explore different ways of proceeding with unknown. We evaluate the performance of the standardized statistic where is replaced with an estimate of and determine the critical value that controls the type I error rate for the least favorable in [0,1]. The constant increases with and this procedure has pathological behavior if depends on and converges to zero at a certain rate. As an alternate approach, we replace with the upper limit of a confidence interval chosen so that for all . We determine so that the type I error rate is exactly controlled for all in [0,1]. We also investigate a simpler approach where we bound the type I error rate. The former method performs well for all while the less powerful bound method may be a useful in some settings as a simple approach. The proposed tests can be used in different applications, including within-cluster resampling and combining exchangeable -values. [Copyright &y& Elsevier]

Abstract: The main objective of this work is to calculate and compare different measures of multivariate skewness for the skew-normal family of distributions. For this purpose, we consider the Mardia (1970) , Malkovich and Afifi (1973) , Isogai (1982) , Srivastava (1984) , Song (2001) , Móri et al. (1993) , Balakrishnan et al. (2007) and Kollo (2008) measures of skewness. The exact expressions of all measures of skewness, except for Song’s, are derived for the family of skew-normal distributions, while Song’s measure of shape is approximated by the use of delta method. The behavior of these measures, their similarities and differences, possible interpretations, and their practical use in testing for multivariate normal are studied by evaluating their power in the case of some specific members of the multivariate skew-normal family of distributions. [Copyright &y& Elsevier]

Abstract: Empirical likelihood for general estimating equations is a method for testing hypothesis or constructing confidence regions on parameters of interest. If the number of parameters of interest is smaller than that of estimating equations, a profile empirical likelihood has to be employed. In case of dependent data, a profile blockwise empirical likelihood method can be used. However, if too many nuisance parameters are involved, a computational difficulty in optimizing the profile empirical likelihood arises. Recently, Li et al. (2011) proposed a jackknife empirical likelihood method to reduce the computation in the profile empirical likelihood methods for independent data. In this paper, we propose a jackknife–blockwise empirical likelihood method to overcome the computational burden in the profile blockwise empirical likelihood method for weakly dependent data. [Copyright &y& Elsevier]

Abstract: Goodness-of-fit tests allow one to conclude that possible outcomes are not equally likely. In this paper, we develop an exact equivalence test that allows one to conclude that possible outcomes are approximately equally likely. We show that the power properties of the test compare favorably to those of possible alternative tests, and we develop an associated simultaneous confidence interval procedure. We apply the test to data sets on the digits of , winning roulette numbers, and winning numbers from the Pennsylvania Lottery. [Copyright &y& Elsevier]