Asymptotic theorems for kernel U-quantiles
Let \(X_1, X_2, \ldots\) be independent and identically distributed random variables and let \(h(x_1, \ldots, x_m)\) a real-valued function that is symmetric in its arguments. Let \[ H(y) = P(h(X_1, \ldots, X_n) \leq y)\;\text{and}\;H_n(y) = \sum I(h(X_{i_1},\ldots, X_{i_m}) \leq y)/n_{(m)}, \] where the sum is over the \(n_{(m)} = n(n-1)\ldots(n-m)\) \(m\)-tuples \((i_1, \ldots,i_m)\) of distinct elements drawn from \(\{1,2, \ldots, n\}\). A natural estimator for the parameter \(\xi_p = \inf \{x: H(x)\geq p\}\) is the \(U\)-quantile \(H_n^{-1}(p)\). This paper studies alternative smoothed estimators for \(\xi_p\) which are kernel \(U\)-quantiles. Sufficient condtions are given to establish a strong Bahadur representation theorem for the kernel \(U\)-quantile and an invariance principle for the estimator. The asymptotic normality for estimators based on randomly indexed sequences of random variables is also established.
scientific article; zbMATH DE number 6166972