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凹回归中的一致性AI 解读

For each $t$ in some subinterval $T$ of the real line let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is concave. Let $t_1, t_2, \cdots$ be a sequence of points in $T$ and let $Y_1, Y_2, \cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. We consider estimators $m_n(t) = m_n(t; Y_1, \cdots, Y_n)$ which are concave in $t$ and which minimize $\sum^n_{i=1} \lbrack m_n(t_i; Y_1, \cdots, Y_n) - Y_i\rbrack^2$…

单调回归的一致性研究AI 解读

For each $t$ in some subset $T$ of $N$-dimensional Euclidean space let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is non-decreasing in each of the coordinates of $t$. Let $t_1, t_2,\cdots$ be a sequence of points in $T$ and let $Y_1, Y_2,\cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. Estimators $\hat{m}_n(t; Y_1,\cdots, Y_n)$ of $m(t)$ which are monotone in each coordinate of $t$ and…

两个随机序变量分布的最大似然估计AI 解读

The maximum likelihood estimates and Ĝ of two distribution functions F and G are found, subject to the restrictions that for all x and that and Ĝ are of the discrete type. Random samples x 1, ···, x m and y 1, ···, y n are taken from the respective distributions and these m+n values are ordered according to magnitude, with x's before y's in cases of equality. These n+m values are divided into a number of “strings” as follows. The first string ends after the y value that is…

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