# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
## 3 HT Fusion Algorithms
### Fusion Methods  
In [30], we tested fusion rules. In what follows, we consider having selected some indices to perform the fusion (see Table 2). Obliviously, the selection is based on the HT robustness according to the estimated values of the criterion of Equation (3) over the learning base. 1. Neural Network on probability and Entropy/Log-Probability (NN and NNL). The first two methods are based on Neural Networks (NNs). Details of the architecture are given in [30]. We use the learning base to train the NN and estimate the results on the validation base. The first NN has for input the probability \(p_{j}\) for all the indices of Section 2, assuming that the NN discard the non-pertinent input (i.e., non robust HT, but also shared information in order to robustify the final decision), the output being 0 or 1 according to the hypothesis. For the same kind of NN, the same process is developed over the Log-Probability and then the NN input is the entropy. The idea being to ease the elimination of the shared information but possibly losing the robustness information borne by the probabilities. Maximum Likehood (ML). Obviously, since we have a labeled (linear/nonlinear) database, we can estimate the joint multivariate density pdf of all the probabilities and make the decision: \[u=0\quad\text{if}\quad\frac{f(p_{i}\cdots p_{j}|\mathcal{H}_{0})}{f(p_{i}\cdots p _{j}|\mathcal{H}_{1})}\geq\lambda,\quad u=1\quad\text{either}\] (11) In this case, the \(f(p_{i}\cdots p_{j}|\mathcal{H}_{0})\) (resp. \(f(p_{i}\cdots p_{j}|\mathcal{H}_{1})\)) is the joint probability density of the \(p_{j}\), under \(\mathcal{H}_{0}\) (resp. \(\mathcal{H}_{1}\)). The threshold \(\lambda\) is fixed according to value \(\alpha\). This joint probability could be estimated either on all the HT or by only considering the more robust ones, as previously stated. 3. Optimized Decision (OD). This approach [59] first concatenates the decision of each HT into an array \(\vec{U}\), the final decision being made as: \[u=0\quad\text{if}\quad\frac{P(\vec{U}|\mathcal{H}_{0})}{P(\vec{U}|\mathcal{H} _{1})}>1,\quad u=1\quad\text{either}\] (12) \[P(\vec{U}|\mathcal{H}_{0})\text{ (resp. }P(\vec{U}|\mathcal{H}_{1})\text{) is the probability of }\vec{U}\text{ under }\mathcal{H}_{0}\text{ (resp. }\mathcal{H}_{1}\text{). Thus, }\vec{U}\text{ has }2^{n}\text{ possible values, }n\text{ being the number of selected HT. One advantage of the approach is that it is set to different TIEP \(\alpha\) for each decision, but it needs an operational research of all the \(p\)-values to take into account, the possible correlation between the decisions, this optimization being performed over a predefined bounded grid.
4. Decision with Security Offset (DSO). This approach needs first to rank the HT according to the performance over the learning based. The idea is to make this ranking using first the most robust HT, except when its decision is not sure, i.e., when the pdf at the estimated value is close to: \[\frac{f_{j}(p_{j}|\mathcal{H}_{0})}{f_{j}(p_{j}|\mathcal{H}_{1})}> t_{max}^{j}\quad\text{then}\quad u=0,\] \[t_{max}^{j}>\frac{f_{j}(p_{j}|\mathcal{H}_{0})}{f_{j}(p_{j}| \mathcal{H}_{1})}>t_{min}^{j}\quad\text{then no decision,}\] (13) \[\frac{f_{j}(p_{j}|\mathcal{H}_{0})}{f_{j}(p_{j}|\mathcal{H}_{1})}< t_{min}^{j}\quad\text{then}\quad u=1\] When there is not a decision, the second-most robust test is tested. The process is repeated, in case of failure, until the less robust HT. If none of the HT can provide a \"robust\" decision, in the meaning of Equation (13), then the final decision is given by the more robust (i.e., first) HT. \(t_{min}^{j}\) and \(t_{max}^{j}\) are estimated over the learning basis, by an exhaustive search, in order to minimize \(\bar{R}\). 5. Maximum of Entropy (ME). One of the few papers [25; 60], taking into account the MI between the indices in the final decision.