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In this article, the problem of semi-parametric inference on the parametersof a multidimensional L\'{e}vy process $L_t$ with independent components basedon the low-frequency observations of the corresponding time-changed L\'{e}vyprocess $L_{\mathcal{T}(t)}$, where $\mathcal{T}$ is a nonnegative,nondecreasing real-valued process independent of $L_t$, is studied. We showthat this problem is closely related to the problem of composite functionestimation that has recently gotten much attention in statistical literature.Under suitable identifiability conditions, we propose a consistent estimate forthe L\'{e}vy density of $L_t$ and derive the uniform as well as the pointwiseconvergence rates of the estimate proposed. Moreover, we prove that the ratesobtained are optimal in a minimax sense over suitable classes of time-changedL\'{e}vy models. Finally, we present a simulation study showing the performanceof our estimation algorithm in the case of time-changed Normal Inverse Gaussian(NIG) L\'{e}vy processes.

Statistical inference for time-changed L\'{e}vy processes via composite characteristic function estimation