An asymptotic representation for the likelihood ratio for multidimensional samples with discontinuous desities (Q2752955)

Let \(X_1,\dots, X_n\) be a sample consisting of independent \(d\)-dimensional random vectors having common density \(f(\bar{ x},\theta)\) with respect to the Lebesgue measure in~\({\mathbb R}^d\). Assume that this density is continuous with respect to \(\bar{x} = (x_1, \dots, x_d)\) except a set of points \(K_\theta\) depending on an unknown parameter \(\theta \in \Theta = (a, b)\), where \(a\) and \(b\) are not necessarily finite. Suppose also that \(K_\theta\) is a smooth manifold of dimension \(d - 1\) for all \(\theta \in \Theta,\) and the sets \(\Omega_{\theta}^1\) and \(\Omega_{\theta}^2\) are defined such that first their combination with \(K_\theta\) generates a partition of \({\mathbb R}^d\) and secondly, the density \(f(\bar{x},\theta)\) has at points \(\bar{y} \in K_\theta\) discontinuities of the first kind along directions specified by the sets \(\Omega_{\theta}^1\) and \(\Omega_{\theta}^2\). Let \(\theta\) be the true value and \(l(\bar{x},\delta)=\ln f(\bar{x},\delta)\), \(\delta \in \Theta\), and NEWLINE\[NEWLINE\Delta(\bar{x},\delta)=l(\bar{x},\delta)-l(\bar{x},\theta),\qquad Y_n (u)=\sum _{i\leq n} \Delta \Big(X_i, \theta +u/n\Big),\quad \theta +u/n \in \Theta.NEWLINE\]NEWLINE The authors study the asymptotic behavior of the likelihood ratio process \(Y_n (u)\). In particular, they give an asymptotic expansion of \(Y_n (u)\).