# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
## 2 Nonlinearity Detection Hypothesis Testing
### Robustness  
* The HT robustness involves correctly defining a performance metric for the HT (such a comparison has already been completed, see [24; 28; 29], but with non rigorous definition of the two hypotheses, as seen below). As usual, two types of error are possible, the TIEP (Type I Error Probability), denoted \(\alpha=\int_{T}^{+\infty}f_{q}(z|\mathcal{H}_{0})dz\) that is the probability of rejecting the null hypothesis although right; and the Type II Error Probability (TIIEP, i.e., the error of accepting the null hypothesis although false, denoted \(\beta(\alpha)\)). The usual HT decision making involves fixing \(\alpha\), thus deriving threshold \(T\) using the index pdf function under \(\mathcal{H}_{0}\) and then comparing the index to this value, leading to the acceptance or rejection of \(\mathcal{H}_{0}\). Admitting that the two errors have the same importance in the decision process, the mean Bayesian risk of the HT (see [25] chapter one) is equal to: \[\bar{R}=\frac{1}{2}\int_{0}^{1}(\alpha+\beta(\alpha))d\alpha.\] (3) This criterion is the parameter to minimize as the metric of performance in what follows. * In order to rigorously estimate the HT robustness and derive the parameters needed for the HT fusion algorithm, TS under the two hypotheses have to be simulated. However, the (linear/nonlinear) TS sets have to be carefully designed. The nonlinear TS set can be derived from nonlinear models. In this paper and in [30], we consider three kinds of second-order nonlinear system: \[X(n)=\sum_{i=0}^{q}b_{i}\cdot e(n-i)+\sum_{k=0}^{r}\sum_{l=0}^{r}d _{kl}\cdot e(n-k)\cdot e(n-l)\] (4a) \[\sum_{j=0}^{p}a_{j}\cdot X(n-j)=\sum_{i=0}^{q}b_{i}\cdot e(n-i)+ \sum_{k=1}^{r_{1}}\sum_{l=0}^{r_{2}}c_{kl}\cdot X(n-k)\cdot e(n-l)\] (4b) \[\sum_{j=0}^{p}a_{j}\cdot X(n-j)+\sum_{k=0}^{r}\sum_{l=0}^{r}c_{kl} \cdot X(n-k)\cdot X(n-l)=e(n)\] (4c) As usual, in TS modeling, the excitation \[e(n)\] is assumed to be iid white normal noise, while \[X(n)\] is assumed to convey the statistical properties of the observed TS. Volterra models ( 4a ) are easily derived by nonlinear expansion for differentiable nonlinear transfer function and the ACF finite length model, unlike the other two (when the summation upper bounds are finite). There are necessary conditions for the stability of QARMA ( 4c ), Quadratic Autoregressive Moving Average Model, see [31; 32; 33]) or bilinear ( 4b ), see [34; 35; 36] for instance), extensively used in econometric TS modeling. A first inspection of the polynomial models of Equations ( 4a )-(4c ) shows that these models contain two kernels, a first linear kernel and a quadratic one. Thus, for nonlinear systems, the excitation variance induces different weights of each (linear/quadratic) kernel and thus fairly different outputs. For instance, a weak variance induces a strong weight of the linear kernel output and thus an almost linear TS, making the nonlinearity detection fairly difficult. For this reason, the metric performance (Equation (3)) also depends on the NonLinear Energy Ratio (NLER, that is the energy from the quadratic kernel over the total energy). In what follows, we have considered NLER values of 0.25, 0.5 and 0.75 (i.e., weakly, fairly, strongly nonlinear systems). * The system simulating the linear system (hypotheses \(\mathcal{H}_{0}\)) also has to be carefully designed. In fact, for each generated output of nonlinear systems, the corresponding output linear system requires: 1. Having the same second order moment, i.e., the same ACF and then the same Power Spectrum Density (PSD). In fact, the second order moment has an effect on the variance of the estimated indices. Thus, after estimating the theoretical ACF/PDF by choosing one of the three models, initializing the model parameters and the excitation variance (see above), we derive the theoretical ACF of the output; this ACF is used to calculate the transfer function of the linear system.