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二次型检验的效率与最优性AI 解读

Let $Z_1, Z_2, \cdots$ be i.i.d. standard normal variables. Results are obtained which relate to the tail behavior as $x \rightarrow \infty$ of distributions of the form $F(x) = P\{\sum^\infty_{k=1}\lambda_k\lbrack(Z_k + a_k)^2 - 1\rbrack \leqslant x\}$. For test statistics which have such limiting distributions $F$, asymptotic relative efficiency measures are discussed. One of these is the limiting approximate Bahadur efficiency. Applications are to tests of fit and tests…

U 统计量与拟合优度检验的大样本理论AI 解读

Let $X_{ni}, i = 1, \cdots, n$ be i.i.d. random variables on an arbitrary measurable space $(\mathscr{X}, B)$. Suppose $\mathscr{L}(X_{ni}) = Q_{n1}, i = 1, \cdots, n$ and let $P_0$ be a fixed probability measure on $(\mathscr{X}, B)$. We consider limiting distribution theory for $U$-statistics $T_n = n^{-1} \sum_{i \neq j} Q(X_{ni}, X_{nj})$ (1) under conditions which imply the product measures $Q_n = Q_{n1} \times \cdots \times Q_{n1}, n$ times, are contiguous to the…