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Divergences test statistics for discretely observed diffusion processes. (English) Zbl 1184.62144

Summary: We propose the use of \(\phi\)-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process
\[ dX_t = b(X_t,\alpha )dt +\sigma(X_t,\beta), \quad \alpha \in R^p,\;\beta \in R^q,\;p,q > 1, \]
from discrete observations \(\{X_{t_i},i=0,\dots ,n\}\) with \(t_i=i\Delta _n, i=0,1,\dots ,n\), under the asymptotic scheme \(\Delta _n\rightarrow 0, n\Delta _n\rightarrow \infty \) and \(n\Delta _n^2 \to 0\). The class of \(\phi\)-divergences is wide and includes several special members like Kullback-Leibler, Rényi, power and \(\alpha \)-divergences. We derive the asymptotic distribution of the test statistics based on the estimated \(\phi\)-divergences. The asymptotic distribution depends on the regularity of the function \(\phi\) and in general differs from the standard \(\chi ^{2}\) distribution as in the i.i.d. case. Numerical analysis is used to show the small sample properties of the test statistics in terms of estimated level and power of the test.

MSC:

62M02 Markov processes: hypothesis testing
62B10 Statistical aspects of information-theoretic topics
62E20 Asymptotic distribution theory in statistics

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References:

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