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基于不精确观测的时间序列预测函数AI 解读

Let $T \subseteq I$ be sets of real numbers. Let $\{Y(t): t\in I\}$ be a real time series whose covariance kernel is assumed known and positive definite. The mean is assumed either to be known or to be an unknown member of a known class of functions on $I$. For each fixed $s\in I, Y(s)$ is predicted by a minimum mean square error unbiased linear predictor $\hat{Y}(s)$ based on $\{Y(t): t\in T\}$. If $\hat{y}(s)$ is the evaluation of $\hat{Y}(s)$ given that the sample path…

单调依赖AI 解读

Random variables $X$ and $Y$ are mutually completely dependent if there exists a one-to-one function $g$ for which $P\lbrack Y = g(X)\rbrack = 1.$ An example is presented of a pair of random variables which are mutually completely dependent, but "almost" independent. This example motivates considering a new concept of dependence, called monotone dependence, in which $g$ above is now required to be monotone. Finally, this monotone dependence concept leads to defining and…

时间序列的预测函数与均值估计函数AI 解读

Let $T \subseteqq I$ be sets of real numbers. Let $\{Y(t): t \in I\}$ be a real time series whose mean is an unknown element of a known class of functions on $I$ and whose covariance kernel $k$ is assumed known. For each fixed $s \in I, Y(s)$ is predicted by a minimum mean square error unbiased linear predictor $\hat{Y}(s)$ based on $\{Y(t): t \in T\}$. If $\hat{y}(s)$ is the evaluation of $\hat{Y}(s)$ given a set of observations $\{Y(t) = g(t): t \in T\}$, then the function…

自回归随机过程中的回归实验设计AI 解读

This paper extends some recent results on asymptotically optimal sequences of experimental designs for regression problems in stochastic processes. In the regression model $Y(t) = \beta f(t) + X(t), 0 \leqq t \leqq 1$, the constant $\beta$ is to be estimated based on observations of $Y(t)$ and its first $m - 1$ derivatives at each of a set $T_n$ of $n$ distinct points. The function $f$ is assumed known as is the covariance kernel of $X(t)$, a zero-mean $m$th order…

一类协方差不等式AI 解读

This article considers the class of covariance inequalities of the form cov (X,Y) ≤ var g(X,Y). We present a functional inequality and a partial differential inequality, each of which is a necessary and sufficient condition on g for the covariance inequality to be satisfied for all random variables X, Y jointly taking values in a given region.

一类协方差不等式AI 解读

This article considers the class of covariance inequalities of the form cov (X,Y) ≤ var g(X,Y). We present a functional inequality and a partial differential inequality, each of which is a necessary and sufficient condition on g for the covariance inequality to be satisfied for all random variables X, Y jointly taking values in a given region.