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

The probability integral transform of a continuous random variable <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>X</m:mi></m:math> X with distribution function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>F</m:mi></m:mrow><m:mrow><m:mi>X</m:mi></m:mrow></m:msub></m:math> {F}X is a uniformly distributed random variable <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>U</m:mi><m:mo>=</m:mo><m:msub><m:mrow><m:mi>F</m:mi></m:mrow><m:mrow><m:mi>X</m:mi></m:mrow></m:msub><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>X</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:math> U={F}X(X) . We define the angular probability integral transform (APIT) as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>θ</m:mi></m:mrow><m:mrow><m:mi>U</m:mi></m:mrow></m:msub><m:mo>=</m:mo><m:mn>2</m:mn><m:mi>π</m:mi><m:mi>U</m:mi><m:mo>=</m:mo><m:mn>2</m:mn><m:mi>π</m:mi><m:msub><m:mrow><m:mi>F</m:mi></m:mrow><m:mrow><m:mi>X</m:mi></m:mrow></m:msub><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>X</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:math> {θ }U=2π U=2π {F}X(X) , which corresponds to a uniformly distributed angle on the unit circle. For circular (angular) random variables, the sum modulus 2 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>π</m:mi></m:math> π of absolutely continuous independent circular uniform random variables is a circular uniform random variable, that is, the circular uniform distribution is closed under summation modulus 2 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>π</m:mi></m:math> π , and it is a stable continuous distribution on the unit circle. If we consider the sum (difference) of the APITs of two random variables, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub></m:math> {X}₁ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>X</m:mi></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow></m:msub></m:math> {X}₂ , and test for the circular uniformity of their sum (difference) modulus 2 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>π</m:mi></m:math> π , this is equivalent to test of independence of the original variables. In this study, we used a flexible family of nonnegative trigonometric sums (NNTS) circular distributions, which include the uniform circular distribution as a member of the family, to evaluate the power of the proposed independence test by generating samples from NNTS alternative distributions that could be at a closer proximity with respect to the circular uniform null distribution.