Abstract
The probability integral transform of a continuous random variable
1 Introduction
Testing for independence is one of the most important tasks in statistics, for example, when constructing the joint distribution of a set of random variables or considering the conditional dependence of one variable in terms of other variables as in regression models. According to Herwatz and Maxand [23], one can consider the following tests of independence: bivariate (pairwise), mutual, and groupwise independence tests. Hereinafter, we only consider absolutely continuous random variables. A random variable with a density function with support on an interval of the real line is a linear random variable, and one with support on the unit circle is a circular random variable. The distribution function of a circular random is a periodic function, and then, it has an arbitrary starting direction. In our case, we considered that all the circular random variables are defined on the interval
and
for
where
Among the nonparametric tests of independence, there is a family of tests based on some functional of the empirical independence process [5], which is defined as the distance between the empirical joint distribution function and product of the empirical univariate distribution functions. Historically, the most used functionals have been the Cramérvon Mises and KolmogorovSmirnov functionals [3,6,7]. For example, Hoeffding [24] considered the Cramérvon Mises functional to generate a rank test of independence between two random variables. Modern rank tests of independence have been developed by Kallenberg and Ledwina [29]. Kernelbased methods have also been used to estimate the empirical independence process, as in the study by Pfister et al. [40]. Mardia and Kent [34] used the general Rao score test to generate independence tests. Csörgö [5] developed independence tests based on the multivariate empirical characteristic function, and Einmahl and McKeague [8] developed the tests based on the empirical likelihood. The measures of dependence derived from entropy were defined by Joe [27] and from mutual information by Berrett and Samworth [2].
When constructing tests of independence, one can take advantage of the characteristics of the multivariate joint distribution. For example, for the multivariate normal distribution, one can test for independence by testing for an identity correlation matrix. Some of these pairwise tests are the Pearson’s [37] product moment correlation coefficient test, Kendall’s [30] rank correlation coefficient test, and Spearman’s [46] rank correlation coefficient test. Of course, for a pair of Gaussian random variables, rejecting null correlation implies rejecting pairwise independence, but applying pairwise independence (correlation) tests is not adequate to test for mutual independence for a set with more than two Gaussian random variables. The Wilks test [49] is an optimal test of independence for multivariate Gaussian populations and for the case of a bivariate groupwise independence test for the vectors
where
For the circularcircular (angularangular) and circularlinear (angularlinear) cases, in which the objective is to test for bivariate independence between two circular random variables and one circular and one linear random variable, respectively, independence tests were developed by considering the specification of measures of dependence and studying their (asymptotic) distributions. By applying Kendall’s tau and Spearman’s rho general measures of dependence based on the concept of concordance, or the construction of distributionfree correlation coefficients based on ranks to a pair of circular random variables or a circular and a linear random variables, tests of independence were developed by Fisher and Lee [14–16] and reviewed by Fisher [17] and Mardia and Jupp [35].
The objective of this study is to develop a test of bivariate (pairwise) independence for two random variables by considering the angular probability transform of each variables, which correspond to circular uniform distributions on
This article is divided into six sections, including the introduction. In Section 2, Johnson and Wehrly’s [28] model is presented as a motivation for performing the test of bivariate independence for two random variables, and here, the theory of NNTS circular distributions is included. Section 3 presents the proposed bivariate independence test, a measure of dependence, and its application to simulated data to study the power of the test. The Section 4 includes a simulation study to evaluate the power of the proposed test in the linearlinear, circularlinear, and circularcircular cases. The Section 5 describes the application of the proposed independence test to real datasets. Finally, conclusions are presented in Section 6.
2 Bivariate Johnson and Wehrly model and NNTS family of circular densities
Sklar [45] theorem specifies that the joint cumulative distribution function of two continuous random variables,
where
where
and
FernándezDurán [9] identified the structure of the Johnson and Wehrly’s model in terms of the theory of copula functions through Sklar’s [45] theorem [36] satisfying
The function
Johnson and Wehrly derived bivariate circular–circular and circular–linear models by considering conditional arguments. When function
This property of the Johnson and Wehrly model motivated our independence test by approximating the circular density function
The circular density function based on NNTS for a circular (angular) random variable
where
where
3 Proposed test for bivariate independence
For absolutely continuous independent and identically distributed (i.i.d.) circular uniform random variables,
where
The critical values of the Pycke test are obtained via simulation.
The steps of the proposed independence test for two absolutely continuous random variables,
Derived from the fitting of an NNTS model with
where the correction term
4 Simulation study
In this section, we present a simulation study to compare the power of the proposed test with the Wilks test and a test of independence based on the empirical copula. We simulated the data from different multivariate distributions using known parameters that define the dependence structure and known marginal densities. We considered the sample sizes of 20, 50, 100, and 200. For a given significance level
4.1 Circularlinear models
Table 1 includes the powers of the proposed ART and APT, and the WT and ECT when simulating samples from the circularlinear model of Johnson and Wehrly with the circular marginal density being an NNTS density with
Marginals  SS 







RT  PT  WT  ECT  RT  PT  WT  ECT  RT  PT  WT  ECT  

20  0.7  0.61  91  89  29  43  89  88  22  32  68  64  8  3 

20  0.8  0.66  88  85  26  48  83  79  22  39  67  64  6  5 
20  0.9  0.62  86  85  39  47  83  80  21  35  58  61  8  11  
20  0.99  0.22  34  35  18  11  28  26  10  9  10  7  3  2  
20  0.9999  0.15  13  10  12  10  3  5  7  7  1  1  1  2  
50  0.7  0.46  100  100  33  79  100  99  20  57  97  97  11  28  
50  0.8  0.51  100  100  33  86  100  100  18  61  99  98  6  29  
50  0.9  0.5  100  100  43  89  100  100  34  72  99  98  9  48  
50  0.99  0.15  71  81  21  40  64  70  13  20  40  41  4  3  
50  0.9999  0.04  12  10  7  10  6  6  3  4  1  2  0  2  
100  0.7  0.45  100  100  51  100  100  100  39  100  100  100  19  89  
100  0.8  0.49  100  100  47  100  100  100  30  99  100  100  16  94  
100  0.9  0.47  100  100  72  100  100  100  56  100  100  100  26  90  
100  0.99  0.12  90  95  31  55  82  92  22  41  66  80  7  18  
100  0.9999  0.02  10  7  11  9  5  4  4  4  1  0  1  2  
200  0.7  0.44  100  100  71  100  100  100  64  100  100  100  42  100  
200  0.8  0.49  100  100  65  100  100  100  58  100  100  100  31  100  
200  0.9  0.45  100  100  86  100  100  100  76  100  100  100  56  100  
200  0.99  0.11  100  100  39  97  100  100  31  79  99  100  13  55  
200  0.9999  0.01  11  14  10  12  9  7  3  8  3  1  0  0  

20  0.7  0.62  89  86  13  36  79  80  7  25  55  52  2  4 

20  0.8  0.63  89  85  17  39  78  76  8  23  53  52  6  6 
20  0.9  0.64  94  88  20  48  86  78  8  37  62  57  1  9  
20  0.99  0.32  38  30  17  23  28  19  11  19  11  10  1  8  
20  0.9999  0.16  13  18  18  14  7  8  10  12  1  2  3  3  
50  0.7  0.56  100  100  19  92  100  100  8  77  99  99  1  34  
50  0.8  0.59  100  100  12  89  100  100  7  78  99  99  2  33  
50  0.9  0.49  100  100  29  85  100  100  18  64  98  99  7  30  
50  0.99  0.13  59  63  21  27  52  55  13  11  30  33  5  5  
50  0.9999  0.04  14  12  8  10  5  6  4  5  2  2  1  1  
100  0.7  0.43  100  100  12  100  100  100  7  100  100  100  2  95  
100  0.8  0.5  100  100  14  100  100  100  10  100  100  100  0  97  
100  0.9  0.48  100  100  44  100  100  100  29  98  100  100  13  93  
100  0.99  0.13  91  97  21  64  86  91  14  36  66  76  4  23  
100  0.9999  0.02  10  14  7  13  5  7  4  3  1  1  1  2  
200  0.7  0.44  100  100  18  100  100  100  9  100  100  100  4  100  
200  0.8  0.48  100  100  16  100  100  100  10  100  100  100  3  100  
200  0.9  0.46  100  100  64  100  100  100  56  100  100  100  31  100  
200  0.99  0.11  100  100  30  92  100  100  15  81  97  100  8  50  
200  0.9999  0.01  11  7  9  4  6  3  8  1  0  0  3  0  

20  0.7  0.52  81  75  17  28  73  66  13  14  46  46  6  6 

20  0.8  0.58  82  84  15  40  78  72  9  22  53  50  1  5 
20  0.9  0.6  81  79  22  34  70  71  14  22  56  52  3  9  
20  0.99  0.26  34  37  12  21  22  25  8  9  7  10  1  4  
20  0.9999  0.15  12  16  12  15  6  10  10  6  2  6  3  0  
50  0.7  0.49  100  100  26  84  100  100  17  68  99  99  1  35  
50  0.8  0.5  100  100  15  89  100  100  8  80  99  98  0  35  
50  0.9  0.48  100  100  13  88  100  100  6  67  99  99  2  32  
50  0.99  0.13  62  69  13  28  52  56  8  19  26  26  2  5  
50  0.9999  0.05  12  12  15  9  4  5  9  3  1  2  3  1  
100  0.7  0.45  100  100  13  100  100  100  10  100  100  100  4  89  
100  0.8  0.49  100  100  14  100  100  100  10  99  100  100  3  89  
100  0.9  0.47  100  100  12  100  100  100  7  97  100  100  2  83  
100  0.99  0.12  97  94  12  52  88  94  10  34  74  84  3  15  
100  0.9999  0.02  9  14  7  7  4  5  6  5  0  1  2  0  
200  0.7  0.43  100  100  17  100  100  100  11  100  100  100  0  100  
200  0.8  0.48  100  100  17  100  100  100  13  100  100  100  1  100  
200  0.9  0.46  100  100  15  100  100  100  7  100  100  100  1  100  
200  0.99  0.1  100  100  13  92  99  100  5  84  97  100  3  56  
200  0.9999  0.01  14  13  10  7  7  5  8  2  2  1  1  1 
The powers of the proposed test implemented using the Rayleigh (ART) and Pycke (APT) circular uniformity tests, the Wilks test (WT) and the empirical copula test (ECT) are compared when simulating 100 times samples of sizes 20, 50, 100, and 200 from a Johnson and Wehrly circularlinear density function constructed from an NNTS angular joining density with
4.2 Circularcircular models
For Johnson and Wehrly’s circularcircular model, we used the same angular joining density and one of the marginal circular densities as that used in the circularlinear model. Figure 1 depicts the plots of the marginal circular densities that correspond to NNTS densities with
Marginals  SS 







RT  PT  WT  ECT  RT  PT  WT  ECT  RT  PT  WT  ECT  

20  0.7  0.63  87  80  16  38  78  73  11  29  59  61  1  11 

20  0.8  0.6  84  81  13  41  72  74  9  28  57  57  3  10 
20  0.9  0.59  81  79  20  36  76  67  15  28  46  49  5  7  
20  0.99  0.25  34  33  19  15  26  23  13  12  10  8  7  4  
20  0.9999  0.13  11  13  12  10  7  7  7  10  1  0  1  1  
50  0.7  0.45  100  100  16  88  99  99  7  63  99  98  1  41  
50  0.8  0.5  100  100  15  91  100  100  5  70  98  98  0  38  
50  0.9  0.49  100  100  28  92  100  100  14  70  100  100  6  34  
50  0.99  0.14  70  73  25  36  55  63  16  17  36  36  7  6  
50  0.9999  0.05  17  14  9  12  6  6  4  3  0  1  3  1  
100  0.7  0.42  100  100  13  100  100  100  9  100  100  100  3  92  
100  0.8  0.47  100  100  17  100  100  100  9  100  100  100  2  93  
100  0.9  0.46  100  100  38  100  100  100  28  100  100  100  10  96  
100  0.99  0.12  90  95  32  60  83  92  23  41  64  85  10  15  
100  0.9999  0.02  9  14  17  8  4  7  7  1  3  1  1  0  
200  0.7  0.43  100  100  25  100  100  100  19  100  100  100  6  100  
200  0.8  0.48  100  100  18  100  100  100  12  100  100  100  6  100  
200  0.9  0.45  100  100  51  100  100  100  45  100  100  100  21  100  
200  0.99  0.1  100  100  42  86  100  100  31  80  99  99  15  40  
200  0.9999  0.01  8  10  12  9  4  6  6  4  0  0  2  1 
The powers of the proposed test implemented using the ART and APT circular uniformity tests, the WT, and the empirical copula test (ECT) are compared when simulating 100 times samples of sizes 20, 50, 100, and 200 from a Johnson and Wehrly circularcircular density function constructed from an NNTS angular joining density with
4.3 Linearlinear models
Tables 3 and 4 list the powers of different tests while simulating samples from a bivariate distribution in which both variables are linear. In the first case, a Gaussian copula was used, and in the second case, a Frank copula was used.
Marginals  SS 







ART  APT  WT  ECT  ART  APT  WT  ECT  ART  APT  WT  ECT  

20  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100 

20  0.75  0.57  75  69  98  97  68  60  92  92  50  45  83  75 
20  0.5  0.25  38  36  64  60  24  25  51  47  9  7  40  24  
20  0.25  0.15  17  20  33  29  10  8  24  15  3  4  9  3  
20  0  0.14  10  14  12  10  7  9  8  5  0  0  2  3  
50  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100  
50  0.75  0.48  99  98  100  100  99  96  100  100  91  89  100  100  
50  0.5  0.14  57  51  93  94  46  38  87  90  23  21  71  85  
50  0.25  0.06  18  17  52  42  10  7  42  28  2  1  14  15  
50  0  0.05  7  6  11  8  1  3  6  4  1  0  4  0  
100  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100  
100  0.75  0.43  100  100  100  100  100  100  100  100  99  99  100  100  
100  0.5  0.11  79  76  97  99  72  61  96  99  50  43  94  98  
100  0.25  0.03  32  30  71  69  20  15  58  55  4  5  37  36  
100  0  0.02  8  12  13  10  4  3  10  5  1  1  4  1  
200  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100  
200  0.75  0.4  100  100  100  100  100  100  100  100  100  100  100  100  
200  0.5  0.08  99  98  100  100  97  97  100  100  88  82  100  100  
200  0.25  0.02  33  27  90  94  21  24  86  90  10  7  63  74  
200  0  0.01  10  13  13  6  6  7  7  1  1  1  1  0  

20  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100 

20  0.75  0.6  81  75  99  97  70  64  99  93  50  37  94  72 
20  0.5  0.26  27  21  73  66  19  13  65  54  7  4  42  25  
20  0.25  0.14  13  10  38  21  4  7  25  12  1  2  11  2  
20  0  0.15  13  8  15  7  4  6  8  5  1  1  2  2  
50  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100  
50  0.75  0.49  99  99  100  100  98  97  100  100  96  92  100  100  
50  0.5  0.12  59  51  99  95  40  40  98  92  26  22  92  75  
50  0.25  0.05  19  15  58  45  8  11  47  32  1  1  28  13  
50  0  0.04  5  8  13  10  1  5  6  5  1  1  1  3  
100  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100  
100  0.75  0.44  100  100  100  100  100  100  100  100  100  100  100  100  
100  0.5  0.1  81  75  100  100  69  66  100  100  47  44  99  100  
100  0.25  0.03  23  18  80  71  12  11  75  56  3  4  57  33  
100  0  0.02  4  6  9  5  1  5  4  1  0  0  2  0  
200  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100  
200  0.75  0.4  100  100  100  100  100  100  100  100  100  100  100  100  
200  0.5  0.09  98  97  100  100  96  92  100  100  88  83  100  100  
200  0.25  0.02  41  28  98  93  22  19  93  90  9  8  83  79  
200  0  0.01  11  11  10  11  8  5  5  5  0  0  0  2  

20  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100 

20  0.75  0.57  75  69  85  97  68  60  77  92  50  45  60  80 
20  0.5  0.25  38  36  42  60  24  25  35  47  9  7  23  26  
20  0.25  0.15  17  20  18  30  10  8  15  14  3  4  12  6  
20  0  0.14  10  14  16  10  7  9  13  5  0  0  7  4  
50  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100  
50  0.75  0.48  99  98  93  100  99  96  90  100  91  89  81  100  
50  0.5  0.14  57  51  55  94  46  38  43  90  23  21  31  85  
50  0.25  0.06  18  17  18  42  10  7  15  28  2  1  9  15  
50  0  0.05  7  6  8  8  1  3  7  4  1  0  5  0  
100  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100  
100  0.75  0.43  100  100  96  100  100  100  95  100  99  99  88  100  
100  0.5  0.11  79  76  63  99  72  61  53  99  50  43  40  98  
100  0.25  0.03  32  30  21  69  20  15  16  55  4  5  11  36  
100  0  0.02  8  12  10  10  4  3  6  5  1  1  2  1  
200  0.99  1  100  100  100  100  100  100  100  100  100  100  100  100  
200  0.75  0.4  100  100  96  100  100  100  95  100  100  100  89  100  
200  0.5  0.08  99  98  67  100  97  97  62  100  88  82  44  100  
200  0.25  0.02  33  27  19  94  21  24  13  90  10  7  10  74  
200  0  0.01  10  13  5  6  6  7  5  1  1  1  2  0 
The powers of the proposed test implemented using the ART and APT circular uniformity tests, the WT and the ECT tests are compared when simulating 100 times samples of sizes 20, 50, 100, and 200 from a linearlinear density function constructed from a Gaussian copula and three different marginals (exponential, Gaussian, and Cauchy). The Gaussian copula is defined with an equicorrelated correlation matrix with five different common correlation values of 0, 0.25, 0.5, 0.75, and 0.99. The case with common correlation equal to zero corresponds to the null independence model.
Marginals  SS 







ART  APT  WT  ECT  ART  APT  WT  ECT  ART  APT  WT  ECT  

20  50  1  100  100  100  100  100  100  100  100  100  100  100  100 

20  15  0.95  100  100  100  100  100  99  100  100  99  97  98  100 
20  10  0.83  97  95  98  100  96  90  97  99  88  78  92  98  
20  5  0.4  55  48  72  89  46  41  62  77  31  19  38  59  
20  0  0.14  14  10  4  6  4  2  1  3  1  0  0  0  
50  50  1  100  100  100  100  100  100  100  100  100  100  100  100  
50  15  0.99  100  100  100  100  100  100  100  100  100  100  100  100  
50  10  0.9  100  100  100  100  100  100  100  100  100  100  100  100  
50  5  0.34  97  89  95  100  94  84  94  99  81  67  83  99  
50  0  0.05  14  12  3  13  6  7  2  7  1  2  0  1  
100  50  1  100  100  100  100  100  100  100  100  100  100  100  100  
100  15  1  100  100  100  100  100  100  100  100  100  100  100  100  
100  10  0.89  100  100  100  100  100  100  100  100  100  100  100  100  
100  5  0.31  100  100  100  100  100  100  100  100  100  97  100  100  
100  0  0.02  9  9  3  9  4  4  2  4  1  0  2  0  
200  50  1  100  100  100  100  100  100  100  100  100  100  100  100  
200  15  1  100  100  100  100  100  100  100  100  100  100  100  100  
200  10  0.88  100  100  100  100  100  100  100  100  100  100  100  100  
200  5  0.29  100  100  100  100  100  100  100  100  100  100  100  100  
200  0  0.01  8  10  5  8  4  5  2  6  1  1  0  2  

20  50  1  100  100  100  100  100  100  100  100  100  100  100  100 

20  15  0.95  100  100  100  100  100  99  100  100  99  97  100  100 
20  10  0.83  97  95  100  100  96  90  99  99  88  78  99  97  
20  5  0.4  55  48  94  90  46  41  88  72  31  19  69  54  
20  0  0.14  14  10  10  6  4  2  5  3  1  0  1  0  
50  50  1  100  100  100  100  100  100  100  100  100  100  100  100  
50  15  0.99  100  100  100  100  100  100  100  100  100  100  100  100  
50  10  0.9  100  100  100  100  100  100  100  100  100  100  100  100  
50  5  0.34  97  89  99  100  94  84  99  99  81  67  99  99  
50  0  0.05  14  12  13  13  6  7  6  7  1  2  0  1  
100  50  1  100  100  100  100  100  100  100  100  100  100  100  100  
100  15  1  100  100  100  100  100  100  100  100  100  100  100  100  
100  10  0.89  100  100  100  100  100  100  100  100  100  100  100  100  
100  5  0.31  100  100  100  100  100  100  100  100  100  97  100  100  
100  0  0.02  9  9  5  9  4  4  3  4  1  0  0  0  
200  50  1  100  100  100  100  100  100  100  100  100  100  100  100  
200  15  1  100  100  100  100  100  100  100  100  100  100  100  100  
200  10  0.88  100  100  100  100  100  100  100  100  100  100  100  100  
200  5  0.29  100  100  100  100  100  100  100  100  100  100  100  100  
200  0  0.01  8  10  9  8  4  5  6  6  1  1  1  2  

20  50  1  100  100  95  100  100  100  92  100  100  100  91  100 

20  15  0.95  100  100  81  100  100  99  73  100  99  97  62  100 
20  10  0.83  97  95  71  100  96  90  64  99  88  78  47  97  
20  5  0.4  55  48  42  90  46  41  33  72  31  19  19  54  
20  0  0.14  14  10  12  6  4  2  7  3  1  0  2  0  
50  50  1  100  100  98  100  100  100  95  100  100  100  93  100  
50  15  0.99  100  100  81  100  100  100  72  100  100  100  62  100  
50  10  0.9  100  100  66  100  100  100  63  100  100  100  52  100  
50  5  0.34  97  89  43  100  94  84  38  99  81  67  25  99  
50  0  0.05  14  12  12  13  6  7  9  7  1  2  4  1  
100  50  1  100  100  92  100  100  100  89  100  100  100  81  100  
100  15  1  100  100  70  100  100  100  65  100  100  100  49  100  
100  10  0.89  100  100  64  100  100  100  54  100  100  100  35  100  
100  5  0.31  100  100  32  100  100  100  25  100  100  97  20  100  
100  0  0.02  9  9  7  9  4  4  5  4  1  0  5  0  
200  50  1  100  100  86  100  100  100  82  100  100  100  77  100  
200  15  1  100  100  62  100  100  100  55  100  100  100  43  100  
200  10  0.88  100  100  49  100  100  100  42  100  100  100  35  100  
200  5  0.29  100  100  30  100  100  100  26  100  100  100  14  100  
200  0  0.01  8  10  5  9  4  5  4  7  1  1  2  2 
The powers of the proposed test implemented using the ART and APT circular uniformity tests, the WT and the ECT are compared when simulating 100 times samples of sizes 20, 50, 100, and 200 from a linearlinear density function constructed from a Frank copula and three different marginals (exponential, Gaussian, and Cauchy). The Frank copula is defined with five different values of the dependence parameter
4.3.1 Bivariate Gaussian copula
The bivariate Gaussian (normal) copula correspond to a multivariate distribution, which is defined as follows:
where
Table 3 compares the powers of the proposed independence test when using a ART and APT circular uniformity tests with respect to those of the WT and ECT independence tests when using simulated samples from a bivariate linearlinear distribution with a Gaussian copula and three different cases of marginal distributions following Herwatz and Maxand [23]: exponential, Gaussian, and Cauchy. For the Gaussian copula, we considered five different values of the correlation coefficient
4.3.2 Bivariate Frank copula
The Frank bivariate copula is defined as follows:
where
5 Application to real circularcircular and circularlinear data
5.1 Test of bivariate independence
5.1.1 Circularlinear real examples
Figure 2 depicts the scatterplots of the considered real examples. We applied the proposed independence test to the circularlinear data on wind direction (circular variable) and ozone concentration (linear variable) originally analyzed by Johnson and Wehrly [28], and later included them as dataset B.18 in Fisher [17]. A total of 19 measurements were taken at a weather station in Milwaukee at 6 o’clock in the morning every fourth day starting on April 18 and ending on June 29, 1975. The scatterplot of this data is included in the top left plot of Figure 2, which presents the values of the wind direction and ozone concentration, further indicating a possible positive association between the circular and linear variables and considering the periodicity of the circular random variable. By applying the Pycke and Rayleigh circular uniformity tests to the difference modulus 2
A second example analyzed by Fisher [17] is a dataset on the directions and distances travelled by 31 small blue periwinkles after undergoing transplantation from their normal place of living. The topright plot depicted in Figure 2 includes the scatterplot for this dataset, further indicating a possible negative association between the direction and travelled distance. When applying the uniformity test to the sum modulus 2
5.1.2 Circularcircular real examples
The first example in the circularcircular test of independence corresponds to pairs of wind directions measured at a weather monitoring station at Milwaukee. The measurements were taken at 6:00 and 12:00 o’clock for 21 consecutive days and were originally included in Johnson and Wehrly [28]. The bottomleft plot depicted in Figure 2 includes a scatterplot of the pairs of wind directions, which indicates a possible positive association between the two angles. Fisher [17] listed this dataset as the B.21 dataset, and the main conclusion of Fisher [17] was that there exists a strong positive association between the wind directions when applying a hypothesis test based on a circularcircular correlation coefficient. When applying the proposed methodology to the difference modulus 2
The second example corresponds to 233 pairs (
6 Conclusion
By using the result that the sum modulus 2
Acknowledgments
We express our sincere gratitude to the Asociación Mexicana de Cultura, A.C. for their support.

Funding information: No funding was received to assist with the preparation of this manuscript.

Conflict of interest: The authors have no conflict of interest to declare that are relevant to the content of this article.
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