Moderate and Cramér-type large deviation theorems for M-estimators
Let \(X_ 1,...,X_ n\) be i.i.d. r.v.'s. Let \(\{T_ n\}\) be a sequence of M-estimators of a parameter \(t_ 0\). For various type M-estimators the authors obtained sufficient conditions which assure either that for each \(\{\xi_ n\}\) with \(\lim \epsilon_ n=0\) and \(\lim n \epsilon_ n=\infty\), \[ -\lim_{n\to \infty}(n \epsilon^ 2_ n)^{-1} \log P(| T_ n-t_ 0| >\epsilon_ n)=(2\sigma^ 2)^{-1} \] or that uniformly in the range \(-c\leq x\leq o(n^{1/6})\) \((c\geq 0)\) \[ P((T_ n-t_ 0)n^{1/2} \sigma^{-1}>x)=\{1-\Phi (x)\}(1+o(1))\quad (n\to \infty). \]
scientific article; zbMATH DE number 4041050