Narrow Results

Motivated by Jin–Ke–Wang’s work (J. Jin, Z. T. Ke and W. Wang, Ann. Statist. 45(5) (2017) 2151–2189), this paper studies estimation of misclassification rate in the Asymptotic Rare and Weak (ARW) model. In contrast to Jin–Ke–Wang’s theorem, we measure the performance of the estimator by the misclassification rate instead of Hamming distance, and extend the Gaussian noise to sub-Gaussian’s. The probability estimation with convergence rate is first given under some conditions. Then we prove that condition necessary as well. A direct corollary of our estimation can be compared with Jin–Ke–Wang’s theorem. It turns out that our statistical limit coincides with theirs.

In a graph $G=(V,E)$, a non-empty set $S\subseteq V(G)$ is said to be an open packing set if no two vertices of $S$ have a common neighbor in $G.$ Let $v\in V$ and let $OS(v)$ denote the maximum cardinality of an open packing set in $G$ which contains $v$. Then $OS(G)=min\{OS(v):v\in V\}$ is called the open packing saturation number of $G$. In this paper, we initiate a study on this parameter.