Pseudo phi-divergence test statistics and multidimensional Ito processes

We consider parametric hypotheses testing for multidimensional It\^o processes, possibly with jumps, observed at discrete time. To this aim, we propose the whole class of pseudo $\phi$-divergence test statistics, which include as a special case the well-known likelihood ratio test but also many other test statistics as well as new ones. Although the final goal is to apply these test procedures to multidimensional It\^o processes, we formulate the problem in the very general setting of regular statistical experiments and then particularize the results to our model of interest. In this general framework we prove that, contrary to what happens to true $\phi$-divergence test statistics, the limiting distribution of the pseudo $\phi$-divergence test statistic is characterized by the function $\phi$ which defines the divergence itself. In the case of contiguous alternatives, it is also possible to study in detail the power function of the test. Although all tests in this class are asymptotically equivalent, we show by Monte Carlo analysis that, in small sample case, the performance of the test strictly depends on the choice of the function $\phi$. In particular, we see that even in the i.i.d. case, the power function of the generalized likelihood ratio test ($\phi=\log$) is strictly dominated by other pseudo $\phi$-divergences test statistics.