# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
## 2 Nonlinearity Detection Hypothesis Testing  
As many financial TS, HF price returns exhibit non-symmetrical statistics [15; 26], a fairly common confusion is to relate the non-symmetry of the statistics to nonlinearity (see the tutorial [27] for the generation of a non-symmetrical distribution induced by the phase coupling due to nonlinear systems). We must point out that the linearity does not imply the gaussianity. When the input noise of the linear system is non-symmetrical, the output TS is also non-symmetrical, although linear. On the other hand, some nonlinear systems can exhibit statistics close to the normal ones (for instance, if we sum a large number of nonlinear terms, then the central limit theorem ensures that we have asymptotical normal statistics). The statistical \"linear\" modeling of a TS means that the estimated statistics of this TS can be modeled as those of the output of a linear system, the input data of which are not available (these conditions are usually assumed for predicting and modeling systems). Thus, the null and the alternative hypothesis can be written as:  
* \(\mathcal{H}_{0}\): The signal can be modeled as the output of a linear system driven by an iid (independent and identically distributed) non-symmetric random excitation.
* \(\mathcal{H}_{1}\): The signal cannot be modeled as the output of a linear system driven by an iid non symmetric random excitation and we decide that the system is nonlinear (driven by a normal excitation). In this case, the input is assumed to be Gaussian, since considering other input statistics only involves an additional nonlinear transform.  
For instance, \(\mathcal{H}_{0}\) has been identified as a possible factorial model in [8], when non-normal residuals are observed. On the other side, \(\mathcal{H}_{1}\) is suggested for the same factorial model in [17] when transforming normal statistics of the risk factors into non-normal statistics of the HF returns. In what follows, with a slight abuse of language, we use the term \"linear\" and \"nonlinear\" TS for the sake of simplicity. Moreover, the HT have been designed (or redesigned) in order to be right sided. In other words, a positive index \(q\) is derived from the observed TS. If \(q\) is below a threshold \(T\), \(\mathcal{H}_{0}\) is accepted and then the decision, denoted \(u\), is equal to 0 (or 1 otherwise). Similarly, the decision can also be made from the probability:  
\[p_{q}=\int_{0}^{q}f_{q}(z|\mathcal{H}_{0})dz \tag{2}\]  
where \(f_{q}(z|\mathcal{H}_{0})\) (resp \(f_{q}(z|\mathcal{H}_{1})\) is the index pdf under hypothesis \(\mathcal{H}_{0}\) (resp \(\mathcal{H}_{1}\)) if \(p_{q}<\int_{0}^{T}f_{q}(z|\mathcal{H}_{0})dz\), then \(u=0\). Thus, three quantities can be merged from several HT as detailed in Section 2.2.  
1. The estimated index denoted \(q_{j}\) of the \(j\)th HT.
2. The probability \(p_{j}\), derived from the estimated index value \(q_{j}\) and the pdf of the estimated probability under \(\mathcal{H}_{0}\) (see Equation (2)), denoted \(f_{j}(p_{j}|\mathcal{H}_{0})\).
3. The decision \(u_{j}\), provided by \(p_{j}\) after fixing the threshold on the probability (i.e., fixing the \(p\)-value), this threshold/\(p\)-value being possibly different for each HT.  
The first approach is the more straightforward, but induces merged quantities with very different ranges of values. The last two have the advantage of merging similar quantities(i.e., a probability in the range \([0,1]\) or binary quantities, the decision \(u_{j}\)), but they require perfectly knowing the statistical index under \(\mathcal{H}_{0}\) to derive these two quantities. A final decision based on each HT decision adds a hyperparameter, the threshold \(T/p\)-value, which can be HT dependent. In our case, the fusion algorithm depends on these two last quantities, leading to two fusion topologies, as seen in Section 3.1. In the next subsection, we inspect the robustness under \(\mathcal{H}_{0}\).