Extension of Likelihood Ratio Analysis Method to Skorokhod $M_1$ Topology (with Application to Poissonian Smooth Change-Point Model)

We extend the Ibragimov-Khasminskii likelihood ratio analysis method, in the one-dimensional parameter case, to a framework based on the Skorokhod $M\_1$ topology. The proposed framework applies to a broad class of statistical models, including those for which the normalized likelihood ratio processes are continuous while the limiting likelihood ratio process is discontinuous, a setting outside the scope of classical approaches based on the uniform or Skorokhod $J\_1$ topologies. We derive sufficient conditions for the weak convergence of likelihood ratio processes in the space of c{à}dl{à}g functions on $\RR$ vanishing at $\pm\infty$, endowed with the Skorokhod $M\_1$ topology, and show how this convergence yields the asymptotic behavior of the maximum likelihood estimator. We also introduce new techniques for controlling the $M\_1$ modulus of continuity. Although these techniques are particularly well suited to models in which all jumps and steep continuous transitions are in the same direction, they are of independent interest and may prove useful beyond this setting. As an application, we study a smooth change-point model for inhomogeneous Poisson processes in the fast regime, where the transition interval shrinks faster than $1/n$. We establish that in this case the maximum likelihood estimator has the same asymptotic behavior (consistency, rate of convergence, limiting distribution and convergence of moments) as in the corresponding ''pure'' change-point model.