A conditional distribution function-based measure for independence and [formula omitted]-sample tests in multivariate data

Testing homogeneity in high dimensional data through random projections

Some Thoughts About Data Type, Distribution, and Statistical Significance

Response to Zang et al. and Han et al.

Testing Independence for a Large Number of High-Dimensional Random Vectors

Genetic mapping of quantitative trait loci (QTLs) is performed typically by using a parametric approach, based on the assumption that the phenotype follows a normal distribution. Many traits of interest, however, are not normally distributed. In this paper, we present a nonparametric approach to QTL mapping applicable to any phenotypic distribution. The method is based on a statistic ZW, which generalizes the nonparametric Wilcoxon rank-sum test to the situation of whole-genome search by interval mapping. We determine the appropriate significance level for the statistic ZW, by showing that its asymptotic null distribution follows an Ornstein-Uhlenbeck process. These results provide a robust, distribution-free method for mapping QTLs.

Bootstrapping divergence statistics for testing homogeneity in multinomial populations

A Cramer von-Mises type statistic is proposed for testing the symmetry of a continuous distribution function. Its asymptotic null distribution is found explicitly, and its asymptotic distribution under a sequence of local alternatives is described. A Monte Carlo study indicates that the asymptotic formulae are accurate for sample sizes as small as twenty.

Stationarity Testing Under Endogenous Smooth Deterministic Components: Some Asymptotic Results

This paper proposes a residual based cointegration test with improved power. Based on the idea of Hansen (1995) and Elliott & Jansson (2003) in the unit root testing case, stationary covariates are used to improve the power of the residual based Augmented Dickey Fuller (ADF) test. The asymptotic null distribution contains difficult to estimate nuisance parameters for which there is no obvious method of estimation, therefore we propose a bootstrap methodology to obtain test critical values. Local-to-unity asymptotics and Monte Carlo simulations are used to evaluate the power of the test in large and small samples, respectively. These exercises show that the addition of covariates increases power relative to the ADF and Johansen tests, and that the power depends on the long-run correlation between the covariates and the cointegration candidates. The new test is used to test for cointegration between Credit Default Swap (CDS) and corporate bond spreads for a panel of U.S. firms during the 2007-2009 financial crisis. The new test finds stronger evidence for cointegration between the two spreads for more firms, relative to ADF and Johansen tests.

Testing for mutual independence among several random vectors is a challenging problem, and in recent years, it has gained significant attention in statistics and machine learning literature. Most of the existing tests of independence deal with only two random vectors, and they do not have straightforward generalizations for testing mutual independence among more than two random vectors of arbitrary dimensions. On the other hand, there are various tests for mutual independence among several random variables, but these univariate tests do not have natural multivariate extensions. In this article, we propose two general recipes, one based on inter-point distances and the other based on linear projections, for multivariate extensions of these univariate tests. Under appropriate regularity conditions, these resulting tests turn out to be consistent whenever we have consistency for the corresponding univariate tests. We carry out extensive numerical studies to compare the empirical performance of these proposed methods with the state-of-the-art methods.

Let $v(1),v(2), \cdots v(t), \cdots $ be a succession of random S-dimensional vectors, connected according to the arrangement of a homogeneous Markov chain. Let also \[ v^h (v_1^h ,v_2^h , \cdots ,v_s^h )\qquad (h = 1,2, \cdots ,m) \] be the possible values of the vectors $v(t)\,(t = 1,2, \cdots )$\[ {\bf P}(v(t + 1) = v^k |v(t) = v^h ) = p_{hk} ,\] where \[ p_{hk} > 0{\textit{and}}\sum\limits_{k = 1}^m {p_{hk} = 1.} \]The author considers the limiting distribution for the sum \[ w(t) = v(1) + v(2) + \cdots + v(t)\] for $t \to \infty $. It is established that under certain general conditions, after suitable normalization and centering, the random vector $w(t)$ will have an asymptotic normal distribution. Various degenerate cases are investigated for different relative values of s and m.

Statistical tests of a simple energy balance equation in a synthetic model of cotrending and cointegration

1. The main notions.- 2. The main lemmas.- 2.1. General lemmas on the approximation of distribution of an arbitrary random variable by the normal distribution.- 2.2. Proof of lemmas 2.1-2.4.- 3. Theorems on large deviations for the distributions of sums of independent random variables.- 3.1. Theorems on large deviations under Bernstein's condition.- a) Sums of non-identically distributed random variables.- b) Sums of weighted random variables.- 3.2. A theorem of large deviations in terms of Lyapunov's fractions.- 4. Theorems of large deviations for sums of dependent random variables.- 4.1. Estimates of the kth order centered moments of random processes with mixing.- 4.2. Estimates of mixed cumulants of random processes with mixing.- 4.3. Estimates of cumulants of sums of dependent random variables.- 4.4. Theorems and inequalities of large deviations for sums of dependent random variables.- 5. Theorems of large deviations for polynomial forms, multiple stochastic integrals and statistical estimates.- 5.1. Estimates of cumulants and theorems of large deviations for polynomial forms, polynomial Pitman estimates and U-statistics.- 5.2. Cumulants of multiple stochastic integrals and theorems of large deviations.- 5.3. Large deviations for estimates of the spectrum of a stationary sequence.- 6. Asymptotic expansions in the zones of large deviations.- 6.1. Asymptotic expansion for distribution density of an arbitrary random variable.- 6.2. Estimates for characteristic functions.- 6.3. Asymptotic expansion in the Cramer zone for distribution density of sums of independent random variables.- 6.4. Asymptotic expansions in integral theorems with large deviations.- 7. Probabilities of large deviations for random vectors.- 7.1. General lemmas on large deviations for a random vector with regular behaviour of cumulants.- 7.2. Theorems on large deviations for sums of random vectors and quadratic forms.- a) Sums of non-identically distributed random vectors.- b) Sums of weighted random vectors.- c) Sums of random number of random vectors.- d) Quadratic forms.- Appendices.- Appendix 1. Proof of inequalities for moments and Lyapunov's fractions.- Appendix 2. Proof of the lemma on the representation of cumulants.- Appendix 3. Leonov - Shiryaev's formula.- References.

LetX andY be two random vectors with values in ℝ k and ℝ∝, respectively. IfZ=(X T,Y T) T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X 1 , …,X t T ) T is splitted intot≥3 partsX 1, …,X t.

This paper presents a new specification test of independence between two random vectors by the kernel method. The test statistic is shown to have asymptotic standard normal distribution under the null hypothesis of independence, while diverging to infinity if the null is false. Hence, the test is consistent against any form of dependence between the two random vectors. The power of the test against local alternatives is also investigated.