On large deviation theorem for data-driven Neyman's statistic
Let \(U_1, U_2,...\) be i.i.d. random variables uniformly distributed on the unit interval. Let \(\Phi_0, \Phi_2,...\), \(\Phi_0\equiv 1\), be a complete orthogonal system in \(L_2[0,1]\). Let \((m_n)\) be a a sequence of natural nambers such that \[ (A2)\qquad m_m\to \infty\;\text{as} n\to \infty,\;\text{and} m_n= o\Bigl((n/log n))^{1/(2\omega-2)}\Bigr)\;\text{for some} \omega\geq 0. \] Then define for every \(n\): \[ S2=\min\{k\leq m_n: |\overline{\Phi}|_k^2-k(log n/n)\geq \max_{1\leq j\leq m_n}\{|\overline{\Phi}|_j^2-j(log n/n)\}\}, \] \[ \text{where}\qquad |\overline{\Phi}|_k^2=n^{-2}\sum_{j=1}^{k} \Bigl(\sum_{i=1}^{n}\Phi_j(U_i)\Bigr)^2. \] Then the data-driven Neyman's statistic is \[ T_{S2}=n|\overline{\Phi}|_{S2}^2=n^{-1}\sum_{j=1}^{S2} \Bigl(\sum_{i=1}^{n}\Phi_j(U_i)\Bigr)^2. \] In fact \(T_{S2}\) is completely defined by the choice of a basis \(\Phi\) and a control sequence \((m_n)\). The main theorem of the paper is \textbf{Theorem 1}. If \((A_2)\) is satisfied and \(\Phi\) is a Legendre basis or the cosine basis, then for every bounded sequence \((x_n)\) of positive numbers such that \(m_n^{2\omega +1}x_n^2\to \infty\) as \(n\to \infty\) we have \[ (nx_n^2)^{-1}\log\mathbb{P}(T_{S2}\geq nx_n^2)\to 0,\;\text{as} n\to \infty. \] The proof of Theorem 1 is based on a version of the lower bound inequality given by \textit{A.A. Mogulskij} [Sib. Math. J. 37, No. 4, 782-787 (1996); translation from Sib. Math. Zh. 37, No. 4, 889-894 (1996; Zbl 0878.60022)].
scientific article; zbMATH DE number 1448749