Nonparametric Tests Of Independence Research Articles

Nonparametric tests of independence of two autoregressive time series based on autoregression rank scores

We propose a nonparametric test of independence of two autoregressive time series. The test statistic is based on lagged cross-correlation coefficients computed from autoregression rank scores, and extends the traditional correlogram-based method of Haugh (1976). It is easily computable, asymptotically distribution-free, and, contrary to its traditional parametric competitor, it does not require any estimation of the unknown autoregression parameters. The test is applied in a study of the relations between outdoor temperature and the daily mortality related to cardio-vascular problems in Brussels, during the period 1980–1989.

A Nonparametric Test of Independence between Two Vectors

A new statistic, [Qcirc] n , based on interdirections is proposed for testing whether two vector-valued quantities are dependent. The statistic, which has an intuitive invariance property, reduces to the quadrant statistic when the two quantities are each univariate. Under the null hypothesis of independence, [Qcirc] n has a limiting chi-squared distribution when each vector is elliptically symmetric. The new statistic is compared to the classical normal theory competitor—Wilks' likelihood ratio criterion—and a componentwise quadrant statistic. Using a novel model of dependence between the vectors, Pitman asymptotic relative efficiencies (ARE's) are computed. The Pitman ARE's indicate that [Qcirc] n compares favorably to Wilks' likelihood ratio criterion when the vectors have heavy-tailed elliptically symmetric distributions and is uniformly better than the componentwise quadrant statistic when the vectors are spherically symmetric. A simulation study demonstrates that [Qcirc] n performs better than the others for heavy-tailed distributions and is competitive for distributions with moderate tail weights. Finally, an example illustrates that [Qcirc] n is resistant to outliers.

Determining delay times for phase space reconstruction with application to the [formula omitted] exchange rate

Economists have widely applied the correlation integral of Grassberger and Procaccia (1983) to determine the dimension of a nonlinear dynamical system. A key judgmental input into this procedure is the choice of delay time for reconstructing a possible attractor. The literature, however, lacks a rigorous procedure for choosing the delay time. In this paper, I apply a simple nonparametric test for independence developed in Mizrach (1995a) to determine the appropriate lag for reconstruction and dimension estimation. In an application to the Lorenz equations, I obtain the best estimates of the correlation dimension at the lag chosen by the test. I then use the method to uncover nonlinear structure in the FF DM exchange rate.

Bahadur efficiency and local asymptotic optimality of certain nonparametric tests for independence

One computes the Bahadur efficiency of nonparametric tests for the verification of the independence hypothesis, based on a Kolmogorov type statistic.

A note on the weak convergence of an estimator of monotonic dependence function of two random variables

Asymptotic distribution of an estimator of monotonic dependence function is derived under the hypothesis of independence. Also a nonparametric test of independence for two quadrantly dependent random variables is suggested.

Tests for independence of exponential variables

Some small sample results are given for two parametric tests of independence of a pair of exponential random variables and two non-parametric tests of independence. Critical values are given for three of the test statistics for sample sizes n ≦ 20 and various approximations to them are developed and evaluated. The small sample power performance of the tests against a specific bivariate exponential alternative is evaluated using monte carlo results. Finally, asymptotic distribution results are given and used to develop power approximations based on the use of variance stabilising transformations.

Asymptotic Normality of Nonparametric Tests for Independence

Asymptotic normality of linear rank statistics for testing the hypothesis of independence is established both under fixed alternatives (or the null hypothesis) and under converging alternatives. The results of Ruymgaart, Shorack and van Zwet [14] are used to obtain a further weakening of the smoothness conditions on the score functions. In the present case the score functions are allowed to have a finite number of discontinuities of the first kind. The results of the present paper and of the paper [14] will be summarized in the author's thesis [13].

A Note on a Nonparametric Test of Independence between Two Vectors

A Note on a Nonparametric Test of Independence between Two Vectors

On a Class of Nonparametric Tests for Independence—Bivariate Case(1)

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) be n mutually independent pairs of random variables with absolutely continuous (hereafter, a.c.) pdf given by(1)where f(ρ) denotes the conditional pdf of X given Y, g(y) the marginal pdf of Y, e(ρ)→ 1 and b(ρ)→0 as ρ→0 and,(2)We wish to test the hypothesis(3)against the alternative(4)For the two-sided alternative we take — ∞< b < ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.

Asymptotic Normality of Nonparametric Tests for Independence

Asymptotic normality of linear rank statistics for testing the hypothesis of independence is established under fixed alternatives. A generalization of a result of Bhuchongkul [1] is obtained both with respect to the conditions concerning the orders of magnitude of the score functions and with respect to the smoothness conditions on these functions.

Percentage Points of Some Non-Parametric Tests for Independence and Empirical Power Comparisons

Percentage Points of Some Non-Parametric Tests for Independence and Empirical Power Comparisons

Percentage Points of Some Non-Parametric Tests for Independence and Empirical Power Comparisons

Abstract A class of nonparametric tests based on the third quadrant layer ranks has recently been studied by Woodworth [12] in connection with the problem of testing for independence in a bivariate distribution. In this article exact one-sided rejection regions are tabulated for the normal score layer rank test which is asymptotically locally most powerful for positive dependence in the bivariate normal distribution. The cut-off points are tabulated for sample sizes n = 4(1)9 and significance levels α = .10, .05, .025 and .01. Normal and Edgeworth approximations for the significance probabilities are also given. A simplified version of the normal score test is proposed, and its rejection regions are tabulated. These tests are compared with the correlation coefficient test, Kendall's τ test, Fisher-Yates (normal-score) correlation test and Spearman's rank correlation test for independence by means of Monte Carlo evaluation of power employing 10,000 trials from two different types of bivariate distributions...

A Class of Nonparametric Tests for Independence in Bivariate Populations

This paper is concerned with the problem of testing the independence of two random variables $X, Y$ on the basis of a random sample, $(X_1, Y_1), (X_2, Y_2), \cdots, (X_N, Y_N)$. The joint distribution function $H$ and, consequently, the marginal distribution functions $F$ and $G$ are assumed to be absolutely continuous. The hypothesis to be tested may be stated as $H(x, y) = F(x)G(y)$. The usual parametric test of this hypothesis is based on the sample correlation coefficient. Several nonparametric tests have been proposed and studied by Kendall [11], Hoeffding [8], [9], Blomqvist [1], and others. Konijn [13] has investigated the asymptotic power properties of some of these tests. The class of rank tests for independence to be considered in this paper is based on the test statistics of the form \begin{equation*}\tag{1.1}T_N = N^{-1}\sum^N_{i = 1} E_{N,r_i}E'_{N,s_i}Z'_{N,r_i}Z'_{N,s_i}\end{equation*} where $\{E_{N,i}\}, \{E'_{N,i}\}, i = 1, 2, \cdots, N$, are two sets of constants satisfying certain restrictions to be stated below, and $Z_{N,r_i} = 1 (Z'_{N,s_i} = 1)$ when $X_i(Y_i)$ is the $r_i$th ($s_i$th) smallest of the $X$'s ($Y$'s) and $Z_{N,r_i} = 0 (Z'_{N,s_i} = 0)$ otherwise. By taking $E_{N,r_i}(E'_{N,s_i})$ to be the expected value of the $r_i$th ($s_i$th) standard normal order statistic from a sample of size $N$, we get the normal scores test statistic which belongs to the class of $c_1$-statistics considered by Fisher and Yates [4], Hoeffding [10] and Terry [17]. If we put $E_{N,r_i} = r_i$ and $E_{N,s_i} = s_i$, the resulting test statistic is equivalent to the Spearman rank correlation statistic. The normal scores test is shown to be (a) the locally most powerful rank test and (b) asymptotically as efficient as the parametric correlation coefficient $\mathscr{R}$-test for the alternatives (4.1) and (4.2) when the underlying distributions are normal. It is at least as efficient as the $\mathscr{R}$-test when the alternative belongs to the class (4.1) and when $H$ satisfies the restrictions stated in Theorem 3. Tables 1 and 2 give the exact null distribution and some critical values of the normal scores statistic for $N \leqq 6$. Table 2 provides a comparison of the $t$-approximation with the exact distribution.

A Class of Statistics with Asymptotically Normal Distribution

Let $X_1, \cdot, X_n$ be $n$ independent random vectors, $X_\nu = (X^{(1)}_\nu, \cdots, X^{(r)}_\nu),$ and $\Phi(x_1, \cdots, x_m)$ a function of $m(\leq n)$ vectors $x_\nu = (x^{(1)}_\nu , \cdots, x^{(r)}_\nu)$. A statistic of the form $U = \sum"\Phi(X_{\alpha 1}, \cdots, X_{\alpha_m})/n(n - 1) \cdots (n - m + 1),$ where the sum $\sum"$ is extended over all permutations $(\alpha_1, \cdots, \alpha_m)$ of $m$ different integers, $1 \leq \alpha_i \leq n$, is called a $U$-statistic. If $X_1, \cdots, X_n$ have the same (cumulative) distribution function (d.f.) $F(x), U$ is an unbiased estimate of the population characteristic $\theta(F) = \int \cdots \int\Phi(x_1, \cdots, x_m) dF(x_1) \cdots dF(x_m). \theta(F)$ is called a regular functional of the d.f. $F(x)$. Certain optimal properties of $U$-statistics as unbiased estimates of regular functionals have been established by Halmos [9] (cf. Section 4). The variance of a $U$-statistic as a function of the sample size $n$ and of certain population characteristics is studied in Section 5. It is shown that if $X_1, \cdots, X_n$ have the same distribution and $\Phi(x_1, \cdots, x_m)$ is independent of $n$, the d.f. of $\sqrt n(U - \theta)$ tends to a normal d.f. as $n \rightarrow \infty$ under the sole condition of the existence of $E\Phi^2(X_1, \cdots, X_m)$. Similar results hold for the joint distribution of several $U$-statistics (Theorems 7.1 and 7.2), for statistics $U'$ which, in a certain sense, are asymptotically equivalent to $U$ (Theorems 7.3 and 7.4), for certain functions of statistics $U$ or $U'$ (Theorem 7.5) and, under certain additional assumptions, for the case of the $X_\nu$'s having different distributions (Theorems 8.1 and 8.2). Results of a similar character, though under different assumptions, are contained in a recent paper by von Mises [18] (cf. Section 7). Examples of statistics of the form $U$ or $U'$ are the moments, Fisher's $k$-statistics, Gini's mean difference, and several rank correlation statistics such as Spearman's rank correlation and the difference sign correlation (cf. Section 9). Asymptotic power functions for the non-parametric tests of independence based on these rank statistics are obtained. They show that these tests are not unbiased in the limit (Section 9f). The asymptotic distribution of the coefficient of partial difference sign correlation which has been suggested by Kendall also is obtained (Section 9h).