Test of bivariate independence based on angular probability integral transform with emphasis on circular-circular and circular-linear data

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 ( 0 , 2 π ] with common starting direction equal to zero and are angles in radians measured all counterclockwise or clockwise. In this sense, if Θ is a circular random variable with starting direction zero and measured in radians counterclockwise,

and

for k an integer number. The density function for an absolutely continuous circular random variable satisfies being a periodic function such that f Θ ( θ ) ≥ 0 , f Θ ( θ + 2 k π ) = f Θ ( θ ) , and ∫ 0 2 π f Θ ( θ ) d θ = 1 . For two random variables, X 1 and X 2 , bivariate (pairwise) independence tests have null hypothesis, H 0 : F X 1 , X 2 ( x 1 , x 2 ) = F X 1 ( x 1 ) F X 2 ( x 2 ) , where F X 1 , X 2 ( x 1 , x 2 ) = P { X 1 ≤ x 1 , X 2 ≤ x 2 } is the bivariate joint distribution function and F X 1 ( x 1 ) = P { X 1 ≤ x 1 } and F X 2 ( x 2 ) = P { X 2 ≤ x 2 } are the corresponding marginal distribution functions. A mutual independence test for a set of random variables, X 1 , X 2 , … , X d considers the null hypothesis H 0 : F X 1 , X 2 , … , X d ( x 1 , x 2 , … , x d ) = ∏ k = 1 d F X k ( x k ) , where F X 1 , X 2 , … , X d ( x 1 , x 2 , … , x d ) is the joint distribution and F X k ( x k ) for k = 1 , 2 , … , d are the marginal univariate distribution functions. Finally, groupwise independence tests consider the independence between disjoint subsets of random variables. If the parametric functional form of F X 1 , X 2 , … , X d and F X k for k = 1 , 2 , … , d are specified, a likelihood ratio test of independence from a sample of random vectors X ̲ i = ( X i 1 , X i 2 , … , X i d ) ⊤ of size n ( i = 1 , 2 , … , n ), can be constructed. The most commonly used test for independence is the chi-squared test of independence for contingency tables, which is not adequate when dealing with absolutely continuous random variables. Alternatively to the likelihood ratio independence test, other tests for independence were developed by considering nonparametric (distribution free) methods, rank tests [24,31], and measures of dependence (association) derived from the empirical copula process [6,19–21,43,44]. The empirical copula, C n , for a vector of d absolutely continuous linear random variables, X ̲ ⊤ = ( X 1 , X 2 , … , X d ) ⊤ , and a sample of size n is defined as follows:

where I ( ) is an indicator function, which is equal to one if the condition in its argument is satisfied and zero otherwise, and F ˆ 1 , F ˆ 2 , … , F ˆ d are the empirical distribution functions of the random variables X 1 , X 2 , … , X d . A test of independence based on the distance between the empirical copula and independence copula for absolutely continuous linear random variables is implemented in the R package c o p u l a [25,32].

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 Cramér-von Mises and Kolmogorov-Smirnov functionals [3,6,7]. For example, Hoeffding [24] considered the Cramér-von 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]. Kernel-based 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 X ̲ D 1 and X ̲ D 2 with X ̲ = X ̲ D 1 ⋃ D 2 ⊤ = ( X ̲ D 1 , X ̲ D 2 ) ⊤ considers the following test statistic for a sample of size n ,

where Σ ˆ D 1 ⋃ D 2 = ∑ j = 1 n ( x ̲ j − x ̲ ¯ ) ( x ̲ j − x ̲ ¯ ) ⊤ is the estimated covariance matrix of the complete vector of observations x ̲ = x ̲ D 1 ⋃ D 2 , which is partitioned into Σ ˆ D 1 and Σ ˆ D 2 with Σ ˆ D 1 = ∑ j = 1 n ( x ̲ D 1 , j − x ̲ ¯ D 1 ) ( x ̲ D 1 , j − x ̲ ¯ D 1 ) ⊤ and Σ ˆ D 2 = ∑ j = 1 n ( x ̲ D 2 , j − x ̲ ¯ D 2 ) ( x ̲ D 2 , j − x ̲ ¯ D 2 ) ⊤ . The statistic W then measures the extent of the distance between the determinant of Σ ˆ D 1 ⋃ D 2 and the product of the determinants of Σ ˆ D 2 and Σ ˆ D 2 . The equality relationship is satisfied in the multivariate Gaussian population under the null hypothesis of independence between X ̲ D 1 and X ̲ D 2 . Asymptotically and under regularity conditions, − n ln ( W ) follows a chi-squared distribution with ♯ D 1 ♯ D 2 degrees of freedom, where ♯ D 1 and ♯ D 2 are the cardinalities of sets D 1 and D 2 , respectively.

For the circular-circular (angular-angular) and circular-linear (angular-linear) 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 distribution-free 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 ( 0 , 2 π ], and an additional result of the theory of circular statistics ([17,26,35,38,47]). The test was evaluated using flexible nonnegative trigonometric sums (NNTS) distributions [10,11]. Although the proposed test of independence is a general one, it is especially suitable to test for the independence in the circular-circular and circular-linear cases. Thus, a measure of dependence was developed.

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 linear-linear, circular-linear, and circular-circular 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, X 1 and X 2 , with corresponding cumulative distribution functions F X 1 ( x 1 ) and F X 2 ( x 2 ) , can be expressed as follows:

where C ( u , v ) is the copula (distribution) function that corresponds to a bivariate joint distribution of two identically distributed uniform random variables on the interval [ 0 , 1 ] , U = F X 1 ( X 1 ) and V = F X 2 ( X 2 ) . A linear-linear copula function must be an increasing function satisfying that C ( u , 1 ) = u , C ( 1 , v ) = v , C ( 0 , v ) = C ( u , 0 ) = 0 [36]. By differentiation of equation (3), one obtains the joint density function of X 1 and X 2 , f X 1 , X 2 ( x 1 , x 2 ) , as follows:

where f X 1 ( x 1 ) and f X 2 ( x 2 ) are the marginal density functions of X 1 and X 2 , respectively. The function c ( u , v ) is the copula density function defined as c ( u , v ) = ∂ 2 C ( u , v ) ∂ u ∂ v . Johnson and Wehrly [28] and Wehrly and Johnson [48] proposed a large family of joint density functions for circular-circular ( Θ 1 and Θ 2 ) and circular-linear ( Θ and T ) random vectors in the following way:

and

Fernández-Durán [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 g must be the density function of an angular (circular) random variable on the interval ( 0 , 2 π ] . For the case of circular-circular and circular-linear bivariate copulas, the function c has also to be periodic in their circular arguments to satisfy the periodicity of the density function of a circular random variable, and this is the reason that in the Johnson and Wehrly model the joining function g is the density of a circular random variable.

Johnson and Wehrly derived bivariate circular–circular and circular–linear models by considering conditional arguments. When function g corresponds to a uniform circular density on the circle, g ( θ ) = 1 2 π , the joint density of the Johnson and Wehrly model corresponds to the independence case in which the joint density of the circular-circular (circular-linear) model is the product of the marginal univariate densities.

This property of the Johnson and Wehrly model motivated our independence test by approximating the circular density function g with a density function from NNTS family of densities [10,11], which includes uniform circular density as a particular case. In addition, by using the NNTS family of circular densities, it is possible to generate joining densities g that are in closer proximity to the circular uniform density as desired; this is explained below. Pewsey and Kato [39] developed a goodness-of-fit test for the circular-circular model of Wehrly and Johnson [48] in equation (5). Their test considers the independence between each circular random variable, Θ 1 and Θ 2 , and the argument of the joining function g , Ω = 2 π { F Θ 2 ( θ 2 ) ± F Θ 1 ( θ 1 ) } ( mod 2 π ) . In a similar way to this article, their independence test is implemented by testing for the toroidal uniformity of the two bivariate random vectors, ( 2 π F Θ 1 ( θ 1 ) , 2 π F Ω ( ω ) ) ⊤ and ( 2 π F Θ 2 ( θ 2 ) , 2 π F Ω ( ω ) ) ⊤ .

The circular density function based on NNTS for a circular (angular) random variable Θ ∈ ( 0 , 2 π ] [10] is defined as follows:

where i = − 1 , c k are complex numbers c k = c r k + i c c k for k = 0 , … , M and c ¯ k = c r k − i c c k is the conjugate of c k . To have a valid density function, f Θ ( θ ; M , c ̲ ) , which integrates to one,

where ∣ ∣ c k ∣ ∣ 2 = c r k 2 + c c k 2 . The parameter space of the vector of parameters, c ̲ = ( c 0 , c 1 , … , c M ) ⊤ , is a subset of the surface of a hypersphere since c ̲ and − c ̲ gives the same NNTS density and the conjugate of c ̲ written in reverse order also gives the same NNTS density as c ̲ . Then, for identifiability of the parameter vector c ̲ and given equation (9), the following constraints are imposed: c c 0 = 0 and c r 0 ≥ 0 , i.e., c 0 is a nonnegative real number and c 0 2 ≥ ∣ ∣ c M ∣ ∣ 2 . There is a total of 2 M free parameters c , and M is an additional parameter that determines the total number of terms in the sum defining the density and it is related to the maximum number of modes that the density can have. Note that an NNTS model with M = M 1 is nested on NNTS models with M = M 2 such that M 2 > M 1 . The circular uniform density on ( 0 , 2 π ] corresponds to an NNTS density with M = 0 , f Θ ( θ ; M = 0 , c ̲ ) = 1 2 π , and it is nested in all NNTS models with M > 0 or, for an NNTS model with M > 0 , and as c 0 approaches the value of 1, the NNTS density converges to the circular uniform density or, equivalently, for an NNTS density with c ̲ = ( 1 , 0 , 0 , … , 0 ) ⊤ , that is, with c 0 = 1 and the other elements of c ̲ equal to zero corresponds to the circular uniform density. It is this property that will be used to evaluate the power for the proposed independence test by generating samples from NNTS alternative densities with values of c 0 as close to one as desired. Fernández-Durán and Gregorio-Domínguez [12] developed an efficient algorithm based on optimization on manifolds to obtain the maximum likelihood estimates of the c parameters. This algorithm is included with other routines for the analysis of circular data based on NNTS models in the free R [42] package CircNNTSR [13].

3 Proposed test for bivariate independence

For absolutely continuous independent and identically distributed (i.i.d.) circular uniform random variables, U 1 , U 2 , … , U d ∼ U ( 0 , 2 π ] , consider ∑ k = 1 d U k ( mod 2 π ) ∼ U ( 0 , 2 π ] , that is, the sum modulus 2 π of i.i.d. circular uniform random variables is also circular and uniformly distributed. This is a consequence of the fact that the characteristic function of a circular uniform random variable, ψ ( t ) = E ( e i t Θ ) , is equal to one when t = 0 and equal to zero elsewhere ( t ≠ 0 ). Then, if the sum modulus 2 π of circular uniform random variables is not circular uniformly distributed, this implies that the circular uniform random variables are not independent [35, p. 35]. The proposed test of independence is based on this result by considering the angular probability integral transform (APIT) of arbitrary (linear or circular) absolutely continuous random variables. Let X 1 , X 2 , … , X d be d arbitrary absolutely continuous random variables. The angular probability transform of X k is defined as the angular (circular) random variable APIT ( X k ) = 2 π F X k ( X k ) , which is uniformly distributed on the unit circle. By considering the null hypothesis of mutual joint independence, APIT ( X 1 ) , APIT ( X 2 ) , … , APIT ( X d ) are i.i.d. U ( 0 , 2 π ] , then ∑ k = 1 d ± APIT ( X k ) ( mod 2 π ) ∼ U ( 0 , 2 π ] is also circular and uniformly distributed. The proposed test for bivariate independence is based on testing for the circular uniformity of ( APIT ( X 1 ) + APIT ( X 2 ) ) ( mod 2 π ) ( ( APIT ( X 1 ) − APIT ( X 2 ) ) ( mod 2 π ) ) for absolutely continuous (circular or linear) random variables X 1 and X 2 . Testing for bivariate independence is equivalent to testing for uniformity of the sum (difference) modulus 2 π of the APITs. To test for circular uniformity of the sum (difference) modulus 2 π of the angular integral transforms, we considered the tests of Rayleigh and Pycke [41]. The Rayleigh test considers an alternative unimodal circular density and has a test statistic for a sample θ 1 , θ 2 , … , θ n , which is defined as follows [35]:

where R ¯ is the sample mean resultant length. The statistic T RT asymptotically has a chi-squared distribution with two degrees of freedom. The p-values of the Rayleigh test used in this article correspond to the approximation given by Fisher [17, p. 70] that includes a correction for small sample sizes. The Pycke test considers an alternative multimodal density, and its test statistic for a sample θ 1 , θ 2 , … , θ n is defined as follows:

The critical values of the Pycke test are obtained via simulation.

The steps of the proposed independence test for two absolutely continuous random variables, X 1 and X 2 , are as follows. First, the empirical distributions F ˆ X 1 and F ˆ X 2 were calculated from the observed values of each random variable. Second, the APITs were calculated by multiplying the pseudo-observations by 2 π ( 2 π F ˆ X 1 and 2 π F ˆ X 2 ). Third, the sum (difference) modulus 2 π of the two APITs was calculated. Finally, the Rayleigh or Pycke test was applied to the vector of the observed values of the sum (difference) modulus 2 π of the APITs. In the case of a positive association between the random variables, the proposed independence test must be applied to the difference modulus 2 π of the APITs, and in the case of a negative association, it must be applied to the sum modulus 2 π of the APITs. The case in which there is no prior indication regarding whether the association is positive or negative, one can calculate the two test statistics, one for the sum modulus 2 π and other for the difference modulus 2 π of the APITs, and modify the minimum of the two p -values in accordance to some multiple testing correction such as Bonferroni procedure [4,22]. In practice, the user can plot the histogram of the sum (difference) modulus 2 π of APITs and decide on the use of the Rayleigh or Pycke test in terms of the observed number of modes. In case of doubt, the Pycke test should be preferred. Other omnibus tests for circular uniformity can be considered as those of Ajne et al. [35].

Derived from the fitting of an NNTS model with M = 1 to the sum (difference) modulus 2 π of APITs, the following measure of dependence can be defined as follows:

where the correction term M + 1 M = 2 comes from the fact that the NNTS density, in the case M = 1 , with the highest concentration around zero has a parameter vector c ̲ in which the squared norm of each of its components is equal to 1 M + 1 = 1 2 . This implies that 1 2 ≤ c 0 2 ≤ 1 and λ ˆ c 0 , for M = 1 , takes values in the interval [ 0 , 1 ] with values close to zero, implying low dependence (independence) and values closer to one, further implying high dependence between the considered random variables. For independent random variables, the theoretical value of λ ˆ c 0 is equal to zero. The measure of dependence λ ˆ c 0 is particularly useful in the circular-circular and circular-linear cases.

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 α (10, 5, and 1%), the powers of the tests were obtained by generating 100 samples of the specified sample size from the bivariate density, by calculating the values of the test statistics for each of the 100 samples, and determining the number of times that the test statistics considered a value that rejected the null hypothesis of independence at the given value of α . Thus, the reported powers of the tests considered values in the range of 0–100 and can be interpreted in terms of percentages. The Rayleigh test of circular uniformity was performed using the circular R package [1]. The Pycke test of circular uniformity was performed using the CircMLE R package [18,33]. The R package c o p u l a was used to calculate the empirical copula test, and the measure of dependence λ ˆ c 0 was obtained by fitting an NNTS model with M = 1 using the R package CircNNTSR [13].

4.1 Circular-linear models

Table 1 includes the powers of the proposed ART and APT, and the WT and ECT when simulating samples from the circular-linear model of Johnson and Wehrly with the circular marginal density being an NNTS density with M = 3 , which is plotted in the first plot of Figure 1; three different linear marginals (exponential, Gaussian, and Cauchy); and a joining circular density that corresponds to an NNTS density with M = 3 with five different values of the parameter c 0 (0.7, 0.8, 0.9, 0.99, and 0.9999) to account for different degrees of association between the circular and linear random variables, as depicted in the last plot of Figure 1, which includes the plots of the angular joining functions for the five different values of parameter c 0 . The case c 0 = 0.9999 corresponds to an almost circular uniform density and to the null hypothesis of independence (refer the last plot of Figure 1). In general, the proposed ART and APT demonstrated significantly larger powers when compared to the ECT, which was only similar for large sample sizes and low values of c 0 (0.7, 0.8, and 0.9), further representing highly dependent circular and linear random variables. The power of the WT was considerably lower when compared to that of the ART, APT, and ECT.

Table 1

NNTS angular joining density circular-linear power study

Marginals	SS	c 0	λ ˆ c 0 = 2 ( 1 ‒ c ˆ 0 2 )	α = 10 %				α = 5 %				α = 1 %
				RT	PT	WT	ECT	RT	PT	WT	ECT	RT	PT	WT	ECT
Θ ∼ NNTS ( M = 3 )	20	0.7	0.61	91	89	29	43	89	88	22	32	68	64	8	3
X ∼ Exp ( 1 )	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
Θ ∼ NNTS ( M = 3 )	20	0.7	0.62	89	86	13	36	79	80	7	25	55	52	2	4
X ∼ N ( 0 , 1 )	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
Θ ∼ NNTS ( M = 3 )	20	0.7	0.52	81	75	17	28	73	66	13	14	46	46	6	6
X ∼ Cauchy ( 0 , 1 )	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 circular-linear density function constructed from an NNTS angular joining density with M = 3 , an NNTS marginal density function with M = 3 (Figure 1) and, three different linear marginals (exponential, Gaussian, and Cauchy). The NNTS angular joining density is defined with five different values of the parameter c 0 (0.7, 0.8, 0.9, 0.99, and 0.9999). The case with c 0 = 1 corresponds to the null independence model.

Figure 1

Circular-circular copula model: The first two plots show the marginal NNTS circular densities ( M 1 = 3 and M 2 = 2 ) and the last third plot show the angular (circular) joining density for different values of the parameter c 0 (0.7, 0.8, 0.9, 0.99, and 0.9999). The case c 0 = 1 corresponds to the circular uniform density (null independence model).