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Let $(X, Y)$ be an $\mathbb{R}^d \times \mathbb{R}$-valued random vector and let $(X_1, Y_1), \cdots, (X_n, Y_n)$ be a random sample drawn from its distribution. We study the consistency properties of the kernel estimate $m_n(x)$ of the regression function $m(x) = E\{Y\mid X = x\}$ that is defined by $m_n(x) = \Sigma^n_{i=1} Y_ik((X_i - x)/h_n)/\Sigma^n_{i=1}k((X_i - x)/h_n)$ where $k$ is a bounded nonnegative function on $\mathbb{R}^d$ with compact support and $\{h_n\}$ is…

核密度估计的L1收敛AI 解读

Let $X_1, \cdots, X_n$ be a sequence of independent random vectors taking values in $\mathbf{\mathbb{R}}^d$ with a common probability density $f$. If $f_n(x) = (1/h) h^{-d}_n\sum^n_{i = 1}K((x - X_i)/h_n)$ is the kernel estimate of $f$ from $X_1, \cdots, X_n$ then conditions on $K$ and $\{h_n\}$ are given which insure that $\int|f_n(x) - f(x)|dx \rightarrow_n 0$ in probability or with probability one. No continuity conditions are imposed on $f$.

局部分类规则的有限样本分布无关性能界AI 解读

In the discrimination problem the random variable $\theta$, known to take values in $\{1,\cdots, M\}$, is estimated from the random vector $X$. All that is known about the joint distribution of $(X, \theta)$ is that which can be inferred from a sample $(X_1, \theta_1),\cdots, (X_n, \theta_n)$ of size $n$ drawn from that distribution. A discrimination rule is any procedure which determines a decision $\hat{\theta}$ for $\theta$ from $X$ and $(X_1, \theta_1),\cdots, (X_n, \theta_n)$…