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估计收敛率、Kolmogorov熵与回归中的降维原则AI 解读

$L_1$-optimal minimum distance estimators are provided for a projection pursuit regression type function with smooth functional components that are either additive or multiplicative, in the presence of or without interactions. The obtained rates of convergence of the estimate to the true parameter depend on Kolmogorov's entropy of the assumed model and confirm Stone's heuristic dimensionality reduction principle. Rates of convergence are also obtained for the error in…

一个回归型估计问题AI 解读

Let $X_1, \cdots, X_n$ be random vectors that take values in a compact set in $R^d, d = 1, 2$. Let $Y_1, \cdots, Y_n$ be random variables (the responses) which conditionally on $X_1 = x_1, \cdots, X_n = x_n$ are independent with densities $f(y \mid x_i, \theta(x_i)), i = 1, \cdots, n$. Assuming that $\theta$ lies in a sup-norm compact space $\Theta$ of real-valued functions, an $L_1$-consistent estimator (of $\theta$) is constructed via empirical measures. The rate of…

非参数回归类问题中的估计误差下界AI 解读

Let $(X_1, Y_1), \cdots, (X_n, Y_n)$ be a sample, denote the conditional density of $Y_i\mid X_i = x_i$ as $f(y\mid x_i, \theta(x_i))$ and $\theta$ an element of a metric space $(\Theta, d)$. A lower bound is provided for the $d$-error in estimating $\theta$. The order of the bound depends on the local behavior of the Kullback information of the conditional density. As an application, we consider the case where $\Theta$ is the space of $q$-smooth functions on $\lbrack 0, 1 \rbrack^d$…

最小距离估计量的收敛速率与 Kolmogorov 熵AI 解读

Let $(\mathscr{X, A})$ be a space with a $\sigma$-field, $M = \{P_s; s \in \Theta\}$ be a family of probability measures on $\mathscr{A}$ with $\Theta$ arbitrary, $X_1, \cdots, X_n$ i.i.d. observations on $P_\theta.$ Define $\mu_n(A) = (1/n) \sum^n_{i = 1} I_A(X_i),$ the empirical measure indexed by $A \in \mathscr{A}.$ Assume $\Theta$ is totally bounded when metrized by the $L_1$ distance between measures. Robust minimum distance estimators $\hat{\theta}_n$ are constructed…