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次高斯均值估计量AI 解读

We discuss the possibilities and limitations of estimating the mean of a real-valued random variable from independent and identically distributed observations from a nonasymptotic point of view. In particular, we define estimators with a sub-Gaussian behavior even for certain heavy-tailed distributions. We also prove various impossibility results for mean estimators.

组合结构下的假设检验问题AI 解读

We study a class of hypothesis testing problems in which, upon observing the realization of an n-dimensional Gaussian vector, one has to decide whether the vector was drawn from a standard normal distribution or, alternatively, whether there is a subset of the components belonging to a certain given class of sets whose elements have been “contaminated,” that is, have a mean different from zero. We establish some general conditions under which testing is possible and others…

非渐近通用平滑参数、核复杂度与 Yatracos 类AI 解读

We introduce a method to select a smoothing factor for kernel density estimation such that, for all densities in all dimensions, the $L_1$ error of the corresponding kernel estimate is not larger than three times the error of the estimate with the optimal smoothing factor plus a constant times $\sqrt{\log n/n}$, where n is the sample size, and the constant depends only on the complexity of the kernel used in the estimate. The result is nonasymptotic, that is, the bound is…

核密度估计的一种普遍适用平滑参数AI 解读

We define a minimum distance estimate of the smoothing factor for kernel density estimates, based on a methodology first developed by Yatracos. It is shown that if $f_{nh}$ denotes the kernel density estimate on $\mathbb{R}^d$ for an i.i.d. sample of size n drawn from an unknown density f, where h is the smoothing factor, and if $f_n$ is the kernel estimate with the same kernel and with the proposed new data-based smoothing factor, then, under a regularity condition on the…

最近邻回归函数估计的强普适一致性AI 解读

Two results are presented concerning the consistency of the $k$-nearest neighbor regression estimate. We show that all modes of convergence in $L_1$ (in probability, almost sure, complete) are equivalent if the regression variable is bounded. Under the additional conditional $k/\log n \rightarrow \infty$ we also obtain the strong universal consistency of the estimate.

核密度估计的良好确定性平滑序列AI 解读

We use the probabilistic method to show that if $f_{nh}$ is the standard kernel estimate with smoothing factor $h$, then there exists a deterministic sequence $h_n$ such that, for all densities, $\operatornamewithlimits{\lim\inf}_{n\rightarrow\infty} \frac{\mathbf{E} \int |f_{nh_n} - f|}{\inf_h \mathbf{E} \int |f_{nh} - f|} = 1.$

关于超核在密度估计中作用的说明AI 解读

We consider the Akaike-Parzen-Rosenblatt density estimate $f_{nh}$ based upon any superkernel $L$ (i.e., an absolutely integrable function with $\int L = 1$, whose characteristic function is 1 on $\lbrack -1, 1\rbrack)$, and compare it with a kernel estimate $g_{nh}$ based upon an arbitrary kernel $K$. We show that for a given subclass of analytic densities, $\inf_L \sup_K \lim \sup_{n\rightarrow \infty} \frac{\inf_h \mathbb{E} \int |f_{nh} - f|}{\inf_h \mathbb{E} \int |g_{nh} - f |} = 1,$…

极端次序统计量稳定性与极大似然核密度估计收敛的关系AI 解读

Let $f$ be a density on the real line and let $f_n$ be the kernel estimate of $f$ in which the smoothing factor is obtained by maximizing the cross-validated likelihood product according to the method of Duin and Habbema, Hermans and Vandenbroek. Under mild regularity conditions on the kernel and $f$, we show, among other things that $\int|f_n - f| \rightarrow 0$ almost surely if and only if the sample extremes of $f$ are strongly stable.

核密度估计的渐近性能下界AI 解读

We consider an arbitrary sequence of kernel density estimates $f_n$ with kernels $K_n$ possibly depending upon $n$. Under a mild restriction on the sequence $K_n$, we obtain inequalities of the type $E\big(\int|f_n - f|\big) \geq (1 + o(1))\Psi(n, f),$ where $f$ is the density being estimated and $\Psi(n, f)$ is a function of $n$ and $f$ only. The function $\psi$ can be considered as an indicator of the difficulty of estimating $f$ with any kernel estimate.

Efron–Stein不等式在密度估计中的应用AI 解读

The Efron-Stein inequality is applied to prove that the kernel density estimate $f_n$, with an arbitrary nonnegative kernel and an arbitrary smoothing factor, satisfies the inequality $\operatorname{var}(\int|f_n - f|) \leq 4/n$ for all densities $f$. Similar inequalities are obtained for other estimates.

非参数密度估计：L1 视角AI 解读

Differentiation of Integrals Consistency Lower Bounds for Rates of Convergence Rates of Convergence in L1 The Automatic Kernel Estimate: L1 and Pointwise Convergence Estimates Related to the Kernel Estimate and the Histogram Estimate Simulation, Inequalities, and Random Variate Generation The Transformed Kernel Estimate Applications in Discrimination Operations on Density Estimates Estimators Based on Orthogonal Series Index.

变带宽核估计的 L1 一致性札记AI 解读

A sample $X_1, \cdots, X_n$ of i.i.d. $R^d$-valued random vectors with common density $f$ is used to construct the density estimate $f_n(x) = (1/n) \sum^n_{i = 1} H^{-d}_{ni}K((x - X_i)/H_{ni}),$ where $K$ is a given density on $R^d$, and the $H_{ni}$'s are positive functions of $n, i$ and $X_1, \cdots, X_n$ (but not of $x$). The $H_{ni}$'s can be thought of as locally adapted smoothing parameters. We give sufficient conditions for the weak convergence to 0 of $\int |f_n - f|$…

密度估计中的分布无关下界AI 解读

We consider the kernel estimate on the real line, $f_n(x) = (nh)^{-1} \sum^n_{i=1} K((X_i - x)/h),$ where $K$ is a bounded even density with compact support, and $X_1, \cdots, X_n$ are independent random variables with common density $f$. We treat the problem of placing a lower bound on the $L$1 error $J_n = E(\int |f_n - f|)$ which holds for all $f$. In particular, we show that there exist $A(K) \geq (9/125)^{1/5}$ depending only upon $K$, and $B^\ast(f) \geq 1$ depending…

自动核密度估计的一致性AI 解读

We consider the Parzen-Rosenblatt kernel density estimate on $\mathbb{R}^d$ with data-dependent smoothing factor. Sufficient conditions on the asymptotic behavior of the smoothing factor are given under which the estimate is pointwise consistent almost everywhere for all densities $f$ to be estimated. When the smoothing factor is a function only of the sample size $n$, it is shown that these conditions are also necessary, a generalization of results by Deheuvels. The…

核密度估计在L₁意义下弱收敛、强收敛与完全收敛的等价性AI 解读

Let $f$ be a density on $R^d$, and let $f_n$ be the kernel estimate of $f$, $f_n(x) = (nh^d)^{-1} \sum^n_{i=1} K((x - X_i)/h)$ where $h = h_n$ is a sequence of positive numbers, and $K$ is an absolutely integrable function with $\int K(x) dx = 1$. Let $J_n = \int |f_n(x) - f(x)| dx$. We show that when $\lim_nh = 0$ and $\lim_nnh^d = \infty$, then for every $\varepsilon > 0$ there exist constants $r, n_0 > 0$ such that $P(J_n \geq \varepsilon) \leq \exp(-rn), n \geq n_0$.…

非参数判别的渐近误分类概率AI 解读

Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times \{0, 1\}$, and let $\hat{Y}$ be the $k$-nearest neighbor estimate of $Y$ from $X$ and the $(X_i, Y_i)$'s. We show that for all distributions of $(X, Y)$, the limit of $L_n = P(\hat{Y} \neq Y)$ exists and satisfies $\lim_{n\rightarrow\infty} L_n \leq (1 + a_k)R^\ast, a_k = \frac{\alpha \sqrt k}{k - 3.25}\big(1 + \frac{\beta}{\sqrt{k-3}}\big), k \text{odd}, k \geq 5,$…

非参数回归函数估计的几乎处处收敛AI 解读

Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times R$, and let $E(|Y|^p) < \infty$ for some $p \geq 1$. We wish to estimate the regression function $m(x) = E(Y \mid X = x)$ by $m_n(x)$, a function of $x$ and $(X_1, Y_1), \cdots, (X_n, Y_n)$. For large classes of kernel estimates and nearest neighbor estimates, sufficient conditions are given for $E\{|m_n(x) - m(x)|^p\} \rightarrow 0$ as $n \rightarrow \infty$,…