# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
## 2 Nonlinearity Detection Hypothesis Testing
### Tested Nonlinearity Hypothesis Tests
#### 2.2.1 Non-Parametric Fourier Domain Tests  
Non-parametric tests of the Fourier domain are mainly based on the third-order moment \(M_{3}^{X}(n_{1},n_{2})\) and the bispectrum \(B(\omega_{1},\omega_{2})\) defined as:  
\[\begin{split} M_{3}^{X}(n_{1},n_{2})=E\{X(i)\cdot X(i+n_{1})\cdot X (i+n_{2})\}\\ B(\omega_{1},\omega_{2})=\sum_{n_{1}=-\infty}^{+\infty}\sum_{n_{2}=- \infty}^{+\infty}M_{3}^{X}(n_{1},n_{2})e^{-j(\omega_{1}\cdot n_{1}+\omega_{2} \cdot n_{2})}=E\{\tilde{X}(\omega_{1})\cdot\tilde{X}(\omega_{2})\cdot\tilde{X} ^{*}(\omega_{1}+\omega_{2})\}\\ \text{and}\quad\tilde{X}(\omega)=\sum_{n=-\infty}^{+\infty}X(n) e^{-j\cdot\omega\cdot n}\end{split} \tag{5}\]  
\((\omega_{1},\omega_{2})\) is called a bifrequency, the skewness being derived from the third-order moment at lag \((0,0)\). Under hypothesis \(\mathcal{H}_{0}\), the bispectrum verifies:  
\[B(\omega_{1},\omega_{2})=\gamma_{3}\cdot H(\omega_{1})\cdot H(\omega_{2}) \cdot H^{*}(\omega_{1}+\omega_{2}) \tag{6}\]  
Several tests have been proposed to test whether the modeling of the estimated bispectrum as the triple product of a linear transfer function is valid or not. [PERSON] et al. [22] and [PERSON] [20] proposed two tests for nonlinearity detection based on the bicoherence (normalized bispectrum). However, these tests have been shown to not be very robust due to the bicoherence estimate variance. The problem of bispectrum estimates (and the phase unrolling for the biceptrum for the indices presented below) have been tackled in [24; 46]. Nonredundant bispectrum support is depicted by the triangle in Figure 1, the symmetry relationships being given the aforementioned papers. Two indices are based upon the bispectrum:  
* The first selected index was proposed by [PERSON] and [PERSON] in [23] and uses the bicepstrum (defined as the inverse Fourier transform of the bispectrum logarithm), which has the property of being null outside the lines \(n_{1}=n_{2},n_{2}=0,n_{1}=0\) (see Equation (6)). In [24], [PERSON] and [PERSON] derive an HT based on the test of the nullity of the bicepstrum components outside these three lines. This index is denoted \(q_{1}\) in the latter part of the same paper.
* In the same paper, they propose another based on two relationships involving, respectively, the bispectrum phase (at different bifrequencies) and the bispectrum magnitude logarithm (also at different bifrequencies). \[\begin{split}\psi(\omega_{1},\omega_{2})-\psi(\omega_{1}-\Delta \omega,\omega_{2}+\Delta\omega)+\psi(\Delta\omega,\omega_{1}-\Delta\omega)- \psi(\omega_{1}-\Delta\omega,\omega_{2}+\Delta\omega)=0\\ \log(|B(\omega_{1},\omega_{2})|)-\log(|B(\omega_{1},\omega_{2}+ \Delta\omega)|)+\log(|B(\omega_{1}+\Delta\omega,\omega_{2}-\Delta\omega)|)\\ -\log(|B(\omega_{1}+2\Delta\omega,\omega_{2})|)+\log(|B(\omega_{1} +2\Delta\omega,\omega_{2}-\Delta\omega)|)-\log(|B(\omega_{1}+\Delta\omega, \omega_{2}-\Delta\omega)|)=0\end{split}\] (7) where \(\psi(\omega_{1},\omega_{2})\) is the bispectrum phase and \(\Delta\omega\) an arbitrary lag. These properties being verified for a linear TS, we then form two sets of variables from the relationship of Equation (7). The test estimates the two variables (for all the possible bifrequencies) linking the bispectrum phase and the log-magnitude in Equation (7). After normalization (in order to have a unit variance for both variables [24]), index \(q_{2}\) is given by the interquartile of the concatenated variables.