Variances of the Foster–Stuart round-trip test statistics

The Foster–Stuart test is often used in the analysis of time series to determine trends. The test is based on calculating the lower and upper time series records x₁, …, x_n. Unlike other tests of randomness, tests based on records are not invariant under a reversal of the direction of the time variable. To construct invariant round-trip tests, it is necessary to count the records in both forward and backward time variables. Thus far, the construction of this test has been impossible because the variances of test statistics were not known when the null hypothesis was true. Variances for Foster–Stuart round-trip test statistics D and S have been found in the present paper. The test was designed to identify positive and negative trends in means and variances of x₁, …, x_n time series. Dispersions of S and D test statistics have been found under the H₀ randomness null hypothesis. Asymptotic approximations were obtained for dispersions. Thus, it has turned out to be possible to construct a full-fledged invariant test for two-sided alternatives. The Foster–Stuart round-trip test application example has been reviewed.     

Post-<Emphasis Type="Italic">J</Emphasis> test inference in non-nested linear regression models

This paper considers the post-J test inference in non-nested linear regression models. Post-J test inference means that the inference problem is considered by taking the first stage J test into account. We first propose a post-J test estimator and derive its asymptotic distribution. We then consider the test problem of the unknown parameters, and a Wald statistic based on the post-J test estimator is proposed. A simulation study shows that the proposed Wald statistic works perfectly as well as the two-stage test from the view of the empirical size and power in large-sample cases, and when the sample size is small, it is even better. As a result, the new Wald statistic can be used directly to test the hypotheses on the unknown parameters in non-nested linear regression models.     

A class of semiparametric rank-based tests for right-truncated data

A class of semiparametric rank-based tests is proposed for the two-sample problem with right-truncated data, where the truncation distribution is parameterized, while the lifetime distribution is left unspecified. The class contains as special cases the extension of the semiparametric Mann–Whitney test proposed by Bilker and Wang (1996) for right-truncated data. The asymptotic distribution theory of the test is presented. The small-sample performance of the test is investigated under a variety of situations by means of Monte Carlo simulations.     

On consistent testing for serial correlation of unknown form in vector time series models

Multivariate autoregressive models with exogenous variables (VARX) are often used in econometric applications. Many properties of the basic statistics for this class of models rely on the assumption of independent errors. Using results of Hong (Econometrica 64 (1996) 837), we propose a new test statistic for checking the hypothesis of non-correlation or independence in the Gaussian case. The test statistic is obtained by comparing the spectral density of the errors under the null hypothesis of independence with a kernel-based spectral density estimator. The asymptotic distribution of the statistic is derived under the null hypothesis. This test generalizes the portmanteau test of Hosking (J. Amer. Statist. Assoc. 75 (1980) 602). The consistency of the test is established for a general class of static regression models with autocorrelated errors. Its asymptotic slope is derived and the asymptotic relative efficiency within the class of possible kernels is also investigated. Finally, the level and power of the resulting tests are also studied by simulation.     

Test elements in pro-<Emphasis Type="Italic">p</Emphasis> groups with applications in discrete groups

Let G be a group. An element g ∈ G is called a test element of G if for every endomorphism ? : G → G, ?(g) = g implies ? is an automorphism. We prove that for a finitely generated profinite group G, g ∈ G is a test element of G if and only if it is not contained in a proper retract of G. Using this result we prove that an endomorphism of a free pro-p group of finite rank which preserves an automorphic orbit of a nontrivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-p groups and Demushkin groups. By relating test elements in finitely generated residually finite-p Turner groups to test elements in their pro-p completions, we provide new examples of test elements in free discrete groups and surface groups. Moreover, we prove that the set of test elements of a free discrete group of finite rank is dense in the profinite topology.     

Distribution-free lack-of-fit tests in balanced mixed models

Here we discuss the problem of fitting a parametric model to the regression function of the fixed effects in a class of balanced mixed effects models. The proposed test is based on the supremum of the Khmaladze transformation of a certain partial sum process of calibrated residuals, and the asymptotic null distribution of this transformed process turns out to be the same as that of a time transformed standard Brownian motion. Moreover, we show that this test is consistent against a large class of fixed alternatives and has non-trivial asymptotic power against a class of nonparametric local alternatives. Simulation studies are conducted to assess the finite sample performance of the proposed test.     

Testing renewal new better than used life distributions based on U-test

New test-statistic for testing exponentiality against renewal new better than used (RNBU) class of life distribution based on U-statistic is studied. Selected critical values are tabulated for sample size n = 5(5)50. The Pitman asymptotic efficiency (PAE) of the test is calculated and compared with, the (PAE) of the test for new renewal better than used (NRBU) class of life distribution [see M.A.W. Mahmoud, S.M. EL-arishy, L.S. Diab, A non-parametric test of new renewal better than used class of life distributions, in: Proceedings of the International Conference on Mathematics Trends and Developments, Cairo, Egypt, vol. 4, 2002, pp. 191–203]. The power of the test is estimated by simulation at 0.05 significant level. A real example is given to elucidate the use of the proposed test. The problem in case of right-censored data is also handled.     

一类新的共轭梯度法的全局收敛性

本文提出了一类与HS方法相关的新的共轭梯度法.在强Wolfe线搜索的条件下,该方法能够保证搜索方向的充分下降性,并且在不需要假设目标函数为凸的情况下,证明了该方法的全局收敛性.同时,给出了这类新共轭梯度法的一种特殊形式,通过调整参数ρ,验证了它对给定测试函数的有效性.     

Jarque–Bera normality test for the driving Lévy process of a discretely observed univariate SDE

In this paper, we study the Jarque–Bera test for a class of univariate parametric stochastic differential equations (SDE) dX _t  = b(X _t , α)dt + dZ _t , constructed based on the sample observed at discrete time points \({t^{n}_{i}=ih_{n}}\) , i = 1, 2, . . . , n, where Z is a nondegenerate Lévy process with finite moments and h is a sequence of positive real numbers with nh _n → ∞ and \({nh_{n}^{2} \to 0}\) as n → ∞. It is shown that under proper conditions, the Jarque–Bera test statistic based on the Euler residuals can be used to test for the normality of the unobserved Z and the proposed test is consistent against the presence of any nontrivial jump components. Our result indicates that the Jarque–Bera test is easy to implement and asymptotically distribution-free with no fine-tuning parameters. Simulation results to validate the test are given for illustration.     

Robust <Emphasis Type="Italic">U</Emphasis>-type test for high dimensional regression coefficients using refitted cross-validation variance estimation

This paper aims to develop a new robust U-type test for high dimensional regression coefficients using the estimated U-statistic of order two and refitted cross-validation error variance estimation. It is proved that the limiting null distribution of the proposed new test is normal under two kinds of ordinary models. We further study the local power of the proposed test and compare with other competitive tests for high dimensional data. The idea of refitted cross-validation approach is utilized to reduce the bias of sample variance in the estimation of the test statistic. Our theoretical results indicate that the proposed test can have even more substantial power gain than the test by Zhong and Chen (2011) when testing a hypothesis with outlying observations and heavy tailed distributions. We assess the finite-sample performance of the proposed test by examining its size and power via Monte Carlo studies. We also illustrate the application of the proposed test by an empirical analysis of a real data example.     

On the generalization of the Salem test

The assertion that the Salem test [5] for the uniform convergence of a trigonometric Fourier series is improvable, is proved. In particular, an example of a continuous function, which does not fulfill the condition of the Salem test but satisfies the condition of the generalized Salem test [10], is constructed.Besides, the theorem which improves Golubov’s [3,4] result for continuous functions of two variables, is given.     

Two-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approach

In this paper, a Bayesian nonparametric approach to the two-sample problem is proposed. Given two samples \(\text{X} = {X_1}, \ldots ,{X_{m1}}\;\mathop {\text~}\limits^{i.i.d.} F\) and \(Y = {Y_1}, \ldots ,{Y_{{m_2}}}\mathop {\text~}\limits^{i.i.d.} G\), with F and G being unknown continuous cumulative distribution functions, we wish to test the null hypothesis H ₀: F = G. The method is based on computing the Kolmogorov distance between two posterior Dirichlet processes and comparing the results with a reference distance. The parameters of the Dirichlet processes are selected so that any discrepancy between the posterior distance and the reference distance is related to the difference between the two samples. Relevant theoretical properties of the procedure are also developed. Through simulated examples, the approach is compared to the frequentist Kolmogorov–Smirnov test and a Bayesian nonparametric test in which it demonstrates excellent performance.     

轴对称问题的Wilson四边形单元

给出了一族轴对称问题的改进Ｗｉｌｓｏｎ元，利用强分片检验证明了其收敛性，讨论了单元函数结构·从而给出了一种构造收敛的轴对称非协调元方法·     

On Upper Class of Increments of Fractional Brownian Motion

A series test for the upper upper class and lower upper class of increments of fractional brownian motion is provided.     

Statistical inference in constrained latent class models for multinomial data based on $$\phi $$-divergence measures

In this paper we explore the possibilities of applying \(\phi \)-divergence measures in inferential problems in the field of latent class models (LCMs) for multinomial data. We first treat the problem of estimating the model parameters. As explained below, minimum \(\phi \)-divergence estimators (M\(\phi \)Es) considered in this paper are a natural extension of the maximum likelihood estimator (MLE), the usual estimator for this problem; we study the asymptotic properties of M\(\phi \)Es, showing that they share the same asymptotic distribution as the MLE. To compare the efficiency of the M\(\phi \)Es when the sample size is not big enough to apply the asymptotic results, we have carried out an extensive simulation study; from this study, we conclude that there are estimators in this family that are competitive with the MLE. Next, we deal with the problem of testing whether a LCM for multinomial data fits a data set; again, \(\phi \)-divergence measures can be used to generate a family of test statistics generalizing both the classical likelihood ratio test and the chi-squared test statistics. Finally, we treat the problem of choosing the best model out of a sequence of nested LCMs; as before, \(\phi \)-divergence measures can handle the problem and we derive a family of \(\phi \)-divergence test statistics based on them; we study the asymptotic behavior of these test statistics, showing that it is the same as the classical test statistics. A simulation study for small and moderate sample sizes shows that there are some test statistics in the family that can compete with the classical likelihood ratio and the chi-squared test statistics.     

p-adic 域上的拟微分算子

本文研究p-adic 域Q_p 上的一类拟微分算子{T^α : α∈R}, 其中算子T^α 在Q_p 的检验函数空 间S(Q_p) 以及相应的分布空间S′(Q_p) 中作为一个运算是封闭的. 本文构造了算子T^α 的卷积核, 指 出算子与其卷积核在解决p-adic 域上的相关问题中所起的重要作用. 最后研究算子T^α 的谱性质, 构 造了算子T^α 的全体固有值与固有函数, 并证明全体固有函数所成的集合组成L²(Q_p) 空间中的一组 完整正交基.     

A VIEW OF THE CHI-SQUARE STATISTIC

Statistical analysis of contingency tables is essentially a discrete multivariate problem. To test independence in a two-way contingency table, one can use different principles such as general class of distance or Mahalanobis distance to derive test statistic. Most of them result in the chi-square statistic, which is a simple test statistic. The disadvantage of the chi-square test is discussed from a multivariate point of view. A new test statistic is proposed for testing independence. This statistic is more sensitive to dependence than the chi-square statistic.     

Bahadur asymptotic efficiency of ω2 -type tests in the many-sample case

Consider independent samples of sizen ₁,n ₂,... ,n ₂ and the empirical distribtion constructed using these samples. We test the hypothesis that all the samples are drawn from a population with the same continuous distribution function F (x). The test statistic is the weight function) is defined as generalizes Kiefer's well-known statistic originally proposed for the case K=2 and q,= 1. A rough asymptotics is obtained for the probability of large deviations of the statistic , which allows to give explicit expressions for the Bahadur local exact slopes and to compare the statistics for various K and q in the sense of Bahadur efficiency. In conclusion, the question posed by Renyi is considered: to what extent is it advisable to use statistics of the type instead of pooling all the samples and using a one-sample test?     

Test Functions, Schur–Agler Classes and Transfer-Function Realizations: The Matrix-Valued Setting

Given a collection of test functions, one defines the associated Schur–Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur–Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy algebra (bounded analytic functions on the unit disk with derivative at the origin constrained to have zero value). Emphasis is on examples where the matrix-valued version is not obtained as a simple tensoring with ${{\mathbb C}^{N}}$ of the scalar-valued version.     

Global convergence and the Powell singular function

The Powell singular function was introduced 1962 by M.J.D. Powell as an unconstrained optimization problem. The function is also used as nonlinear least squares problem and system of nonlinear equations. The function is a classic test function included in collections of test problems in optimization as well as an example problem in text books. In the global optimization literature the function is stated as a difficult test case. The function is convex and the Hessian has a double singularity at the solution. In this paper we consider Newton’s method and methods in Halley class and we discuss the relationship between these methods on the Powell Singular Function. We show that these methods have global but linear rate of convergence. The function is in a subclass of unary functions and results for Newton’s method and methods in the Halley class can be extended to this class. Newton’s method is often made globally convergent by introducing a line search. We show that a full Newton step will satisfy many of standard step length rules and that exact line searches will yield slightly faster linear rate of convergence than Newton’s method. We illustrate some of these properties with numerical experiments.