On the convergence rates of extreme generalized order statistics (Q5943418)

Properties of generalized order statistics are studied. Let \(n \in N\), \(k>0\), \(m_1, \ldots, m_{n-1} \in R\), \(M_r = \sum_{j=r}^{n-1} m_j\), \(1\leq r \leq n-1\), be parameters such that \(\gamma_r =k +n -r + M_r >0\) for all \(r \in \{ 1,\ldots, n-1\}\) and let \(\tilde m = (m_1, \ldots, m_{n-1})\) and \(F\) be an arbitrary distribution function. If the random variables \(U(r, n, \tilde m, k)\), \(r=1, \ldots,n\), have joint density function of the form \[ f(u_1, \ldots, u_n) =k \biggl(\prod_{j=1}^{n-1} \gamma_j\biggr) \biggl(\prod_{i=1}^{n-1} (1 -u_i)^{m_i}\biggr) (1-u_n)^{k-1} \] for \(0\leq u_1 \leq \ldots \leq u_n <1\) of \(R^n\), then they are called uniform generalized order statistics. The random variables \(\chi(r, n, \tilde m, k) = F^{-1}(U(r, n, \tilde m, k))\), \(r=1, \ldots,n\), are called generalized order statistics based on \(F\). Ordinary order statistics are included in the definition by choosing the parameters \(m_1 = \ldots = m_{n-1} =0\) and \(k=1.\) The authors prove limit theorems for generalized order statistics. They find the rate of convergence of the \(n\)-th generalized order statistics and the convergence rate for \(k\)-record values.