# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
## 4 Results
### Hedge Fund Nonlinearity  
The decision rule on the nonlinearity has then been applied over two databases provided by TASS and S&P, the first one containing HF recorded from 1992 to 2003 (i.e., 132 samples), the second one 120 samples from 1996 to 2006. As stated in the introduction, the low number of samples makes the statistical decisions fairly difficult [5; 9; 63]. However, the statistical properties of HF returns are known to vary over time [8; 9; 63]. Then, largest return time series can violate the stationarity assumption needed for our tests (see Section 2). Only fully recorded TS have been considered in the results presented below. This induces a survivor bias, theoretically not relevant in our case, since we are not interested in the relative performance, but we discuss this point again below. Another bias can be relevant \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**N** & **\(\mathbf{N}\)** & **NN** & **NNL** & **NNL** & **ML** & **ML** & **OD** & **OD** & **DSO** & **DSO** & **ME** & **ME** & **GM** & **GM** \\ \hline
128 & 90 & 196 & 381 & 406 & 203 & 337 & 218 & 274 & 187 & 180 & 410 & 500 & 193 & 251 \\ \hline
256 & 96 & 194 & 338 & 410 & 206 & 337 & 222 & 312 & 160 & 147 & 415 & 500 & 169 & 273 \\ \hline
512 & 110 & 204 & 338 & 397 & 198 & 306 & 224 & 310 & 165 & 147 & 422 & 500 & 159 & 269 \\ \hline
1024 & 111 & 210 & 339 & 390 & 176 & 288 & 217 & 291 & 192 & 156 & 359 & 500 & 147 & 247 \\ \hline
2048 & 119 & 225 & 338 & 381 & 147 & 251 & 182 & 278 & 165 & 147 & 317 & 500 & 131 & 250 \\ \hline
4096 & 125 & 227 & 338 & 366 & 102 & 185 & 134 & 249 & 158 & 158 & 205 & 427 & 120 & 219 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(\bar{R}\) (multiplied by \(10^{3}\)) for all models and all NLERs for the different fusion methods, learning base (left) and test base (right). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**N** & \(\mathbf{q_{1}}\) & \(\mathbf{q_{2}}\) & \(\mathbf{q_{3}}\) & \(\mathbf{q_{4}}\) & \(\mathbf{q_{5}}\) & \(\mathbf{q_{6}}\) & \(\mathbf{q_{7}}\) & \(\mathbf{q_{8}}\) & \(\mathbf{q_{9}}\) & \(\mathbf{q_{10}}\) & \(\mathbf{q_{11}}\) & \(\mathbf{q_{12}}\) \\ \hline
128 & 357 & 466 & 426 & 458 & 409 & 405 & 483 & 473 & 207 & 361 & 429 & 270 \\ \hline
256 & 330 & 470 & 405 & 478 & 359 & 364 & 480 & 473 & 181 & 343 & 424 & 224 \\ \hline
512 & 292 & 465 & 384 & 491 & 306 & 317 & 472 & 461 & 168 & 337 & 419 & 193 \\ \hline
1024 & 260 & 465 & 353 & 474 & 251 & 275 & 471 & 456 & 161 & 323 & 410 & 218 \\ \hline
2048 & 228 & 470 & 327 & 459 & 202 & 238 & 477 & 454 & 160 & 302 & 400 & 216 \\ \hline
4096 & 205 & 472 & 298 & 443 & 156 & 205 & 479 & 452 & 153 & 263 & 391 & 192 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean \(\bar{R}\) (multiplied by \(10^{3}\)) for all models and all NLERs. in the nonlinearity detection case. Since the HF prices are declarative, managers may smooth their returns [5] to reduce high returns for augmenting these returns when low or negative. This smoothing has been observed through serial correlation of the returns. Such \"saturation\" effects can induce nonlinearities. Results on the two HF bases are given in Table 5. Two main conclusions can be drawn from this table. * The first one is that the number of TS classified as nonlinear is much greater in the TASS base than in the S&P base. For instance, 53.9% are classified as nonlinear for the first HF base, but only 36.3% for the second one. * According to Table 5, we can observe four situations: 1. The HFs mostly classified as linear in the two bases. For instance, Event-driven, Emerging Market, Fixed Income and Distressed Securities. 2. The HFs mostly classified as nonlinear in the two bases (for the reason exposed in the previous point, an equivalent number of linear/nonlinear HFs in the S& P base leads to a conclusion of nonlinearity detection). These nonlinear styles are Equity Hedge, Equity Non Hedge and Managed Future. 3. The Hedge funds equivalently classified as nonlinear or linear in both bases.