Average rank correlation statistics in the presence of ties

The problem of n judges ranking r objects is considered in the situation where ties are permitted. Asymtotic distributions under the null hypothesis of complete randomness in the rankings are derived for the test statistics of average rank correlations between all pairs of ranking where the rank correlations are measured either by Spearman rho or Kendall tau. The relative efficeincies of these average rank correlatins are derived using approximate Bahadur slope and limiting pitman efficiency, and in both cases the Kendall statistic is shown to be more efficient. Some interpretatins of these and related results are also given.     

Approximate asymptotic Bahadur efficiency of independence tests with random sample size

We compare various kind of independence tests based on samples with random size, in order to provide practitioners with some guidance for their choice based on approximate Bahadur efficiency. Such results are obtained for a wide class of distributions of the random index; the efficiency slopes of the statistics we consider are then expressed in terms of the parameters of these same distributions.     

Kolmogorov–Smirnov test for life test data with hybrid censoring

This work considers goodness-of-fit for the life test data with hybrid censoring. An alternative representation of the Kolmogorov–Smirnov (KS) statistics is provided under Type-I censoring. The alternative representation leads us to approximate the limiting distributions of the KS statistic as a functional of the Brownian bridge for Type-II, Type-I hybrid, and Type-II hybrid censored data. The approximated distributions are used to obtain the critical values of the tests in this context. We found that the proposed KS test procedure for Type-II censoring has more power than the available one(s) in literature.     

Testing parallelism of regression lines against ordered alternatives

For the problem of testing the equality of slopes of several regression lines against the alternative that the slopes are in increasing (decreasing) order of magnitude, two types of tests are proposed. These are the likelihood ratio test and a test that depends on suitable linear combination of one group statistics. Rank analogues of the two tests are also examined.     

Approximate transformation trimmed mean methods to the test of simple linear regression slope equality

To deal with the problem of non-normality and heteroscedasticity, the current study proposes applying approximate transformation trimmed mean methods to the test of simple linear regression slope equality. The distribution-free slope estimates are first trimmed on both sides and then the test statistic t is transformed by Johnson's method for each group to correct non-normality. Lastly, an approximate test such as the James second-order test, the Welch test, or the DeShon-Alexander test, which are robust for heterogeneous variances, is applied to test the equality of regression slopes. Bootstrap methods and Monte Carlo simulation results show that the proposed methods provide protection against both unusual y values, as well as unusual x values. The new methods are valid alternatives for testing the simple linear regression slopes when heteroscedastic variances and nonnormality are present.     

A Test for the Equality of Means of Two Groups with Different Variances When the Sample Size and the Dimension are Large

We consider a class of test statistics including the Dempster trace criterion in the case of two groups without assuming equal covariance matrices. The test statistics in the class are valid when the dimension is larger than the sample size. We obtain asymptotic distributions of the test statistics in the class and use these distributions to derive the limiting power in each case. We obtain the most powerful test in the class with respect to this limiting power.     

Multivariate hypothesis testing using generalized and {2}-inverses – with applications

The use of generalized inverses in Wald's-type quadratic forms of test statistics having singular normal limiting distributions does not guarantee to obtain chi-square limiting distributions. In this article, the use of {2} -inverses for that problem is investigated. Alternatively, Imhof-based test statistics can also be defined, which converge in distribution to weighted sum of chi-square variables. The asymptotic distributions of these test statistics under the null and alternative hypotheses are discussed. Under fixed and local alternatives, the asymptotic powers are compared theoretically. Simulation studies are also performed to compare the exact powers of the test statistics in finite samples. A data analysis on the temperature and precipitation variability in the European Alps illustrates the proposed methods.     

Parametric bootstrap and approximate tests for two Poisson variates

The parametric bootstrap tests and the asymptotic or approximate tests for detecting difference of two Poisson means are compared. The test statistics used are the Wald statistics with and without log-transformation, the Cox F statistic and the likelihood ratio statistic. It is found that the type I error rate of an asymptotic/approximate test may deviate too much from the nominal significance level α under some situations. It is recommended that we should use the parametric bootstrap tests, under which the four test statistics are similarly powerful and their type I error rates are all close to α. We apply the tests to breast cancer data and injurious motor vehicle crash data.     

Nonstationary regression models with a lagged dependent variable

This paper studies regression models with a lagged dependent variable when both the dependent and independent variables are nonstationary, and the regression model is misspecified in some dimension. In particular, we discuss the limiting properties of leastsquares estimates of the parameters in such regression models, and the limiting distributions of their test statistics. We show that the estimate of the lagged dependent variable tends to unity asymptotically independent of its true value, while the estimates of the independent variables tend to zero. The limiting distributions of their test statistics are shown to diverge with sample size.     

Asymptotic properties of likelihood ratio test statistics in affected‐sib‐pair analysis

In an affected‐sib‐pair genetic linkage analysis, identical by descent data for affected sib pairs are routinely collected at a large number of markers along chromosomes. Under very general genetic assumptions, the IBD distribution at each marker satisfies the possible triangle constraint. Statistical analysis of IBD data should thus utilize this information to improve efficiency. At the same time, this constraint renders the usual regularity conditions for likelihood‐based statistical methods unsatisfied. In this paper, the authors study the asymptotic properties of the likelihood ratio test (LRT) under the possible triangle constraint. They derive the limiting distribution of the LRT statistic based on data from a single locus. They investigate the precision of the asymptotic distribution and the power of the test by simulation. They also study the test based on the supremum of the LRT statistics over the markers distributed throughout a chromosome. Instead of deriving a limiting distribution for this test, they use a mixture of chi‐squared distributions to approximate its true distribution. Their simulation results show that this approach has desirable simplicity and satisfactory precision.     

On blest's measure of rank correlation

Blest (2000) proposed a new nonparametric measure of correlation between two random variables. His coefficient, which is dissymmetric in its arguments, emphasizes discrepancies observed among the first ranks in the orderings induced by the variables. The authors derive the limiting distribution of Blest's index and suggest symmetric variants whose merits as statistics for testing independence are explored using asymptotic relative efficiency calculations and Monte Carlo simulations.     

Extensions to Basu's theorem,factorizations, and infinite divisibility

We define a notion of approximate sufficiency and approximate ancillarity and show that such statistics are approximately independent pointwise under each value of the parameter. We do so without mentioning the somewhat nonintuitive concept of completeness, thus providing a more transparent version of Basu's theorem. Two total variation inequalities are given, which we call approximate Basu theorems.     

Tests of fit for exponentiality based on a characterization via the mean residual life function

We study two new omnibus goodness of fit tests for exponentiality, each based on a characterization of the exponential distribution via the mean residual life function. The limiting null distributions of the tests statistics are the same as the limiting null distributions of the Kolmogorov-Smirnov and Cramér-von Mises statistics proposed when testing the simple hypothesis that the distribution of the sample variables is uniform on the interval [0, 1]. Work supported by the Deutsche Forschungsgemeinschaft     

Robust permutation tests for two samples

We consider robust permutation tests for a location shift in the two sample case based on estimating equations, comparing the test statistics based on a score function and an M-estimate. First we obtain a form for both tests so that the exact tests may be carried out using the same algorithms as used for permutation tests based on the mean. Then we obtain the Bahadur slopes of the tests in these two statistics, giving numerical results for two cases equivalent to a test based on Huber scores and a particular case of this related to a median test. We show that they have different Bahadur slopes with neither exceeding the other over the whole range. Finally, we give some numerical results illustrating the robustness properties of the tests and confirming the theoretical results on Bahadur slopes.     

A Nonparametric Test Procedure Based on Quantiles for the Right Censored Data

In this study, we propose nonparametric tests using the several quantile statistics simultaneously for the right censored data. First of all, we consider statistics of the quadratic form with estimated covariance matrices. Then we derive the limiting distribution using the large sample approximation theory. Also we consider different forms of statistics such as the maximal and summing types with their limiting distributions. Then we illustrate our procedure with examples and compare performance among tests with empirical powers through a simulation study. Also we comment briefly on some interesting features including re-sampling methods as concluding remarks. Finally in Appendices, we provide proofs for the theoretic results needed for the derivation of the limiting distributions of the proposed test statistics.     

One-sample location tests for multilevel data

In this paper, we consider testing the location parameter with multilevel (or hierarchical) data. A general family of weighted test statistics is introduced. This family includes extensions to the case of multilevel data of familiar procedures like the t, the sign and the Wilcoxon signed-rank tests. Under mild assumptions, the test statistics have a null limiting normal distribution which facilitates their use. An investigation of the relative merits of selected members of the family of tests is achieved theoretically by deriving their asymptotic relative efficiency (ARE) and empirically via a simulation study. It is shown that the performance of a test depends on the clusters configurations and on the intracluster correlations. Explicit formulas for optimal weights and a discussion of the impact of omitting a level are provided for 2 and 3-level data. It is shown that using appropriate weights can greatly improve the performance of the tests. Finally, the use of the new tests is illustrated with a real data example.     

Multiple Comparisons Among Mean Vectors When the Dimension is Larger Than the Total Sample Size

We consider pairwise multiple comparisons and multiple comparisons with a control among mean vectors for high-dimensional data under the multivariate normality. For such cases, the statistics based on the Dempster trace criterion are given, and also their approximate upper percentiles are derived by using the Bonferroni’s inequality. Finally, the accuracy of their approximate values is evaluated by Monte Carlo simulation.     

A limit theorem on moment convergence of centered spectral statistics of random matrices

ABSTRACT

The eigenvalues of a random matrix are a sequence of specific dependent random variables, the limiting properties of which are one of interesting topics in probability theory. The aim of the article is to extend some probability limiting properties of i.i.d. random variables in the context of the complete moment convergence to the centered spectral statistics of random matrices. Some precise asymptotic results related to the complete convergence of p-order conditional moment of Wigner matrices and sample covariance matrices are obtained. The proofs mainly depend on the central limit theorem and large deviation inequalities of spectral statistics.     

Alternate test statistics for a functional errors-in-variables model

Some test statistics for the structural coefficients of simultaneous equations model often referred to as the multivariate linear functional relationship model are proposed in this article. The following cases are considered: the covariance matrix of errors is either unknown, known up to a proportionality factor, or completely known. The exact and approximate distributions of the proposed test statistics, as well as those of some that are known, are also given.     

Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

For two or more multivariate distributions with common covariance matrix, test statistics for certain special structures of the common covariance matrix are presented when the dimension of the multivariate vectors may exceed the number of such vectors. The test statistics are constructed as functions of location‐invariant estimators defined as U‐statistics, and the corresponding asymptotic theory is used to derive the limiting distributions of the proposed tests. The properties of the test statistics are established under mild and practical assumptions, and the same are numerically demonstrated using simulation results with small or moderate sample sizes and large dimensions.