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P{X<Y} 的最小方差无偏估计量的容许性与极小极大性AI 解读

Suppose that $X_1,\ldots, X_m$ are i.i.d. from a continuous distribution function $F$ and $Y_1,\ldots, Y_n$ are i.i.d. from a continuous distribution function $G; X$'s and $Y$'s are independent. A minimum variance unbiased estimator of $P\{X < Y\}$ is the Mann-Whitney statistic. We show that the Mann-Whitney statistic is admissible under a class of weighted squared error losses and is minimax under a proper weighted squared error loss.

Kolmogorov–Smirnov 损失下最佳不变分布函数估计量的极小极大性AI 解读

For the invariant decision problem of estimating a continuous distribution function with the Kolmogorov-Smirnov loss, it is proved that the best invariant estimator is minimax.

不变估计中经验分布函数的极小极大性AI 解读

Consider the problem of continuous invariant estimation of a distribution function with the weighted Cramer-von Mises loss. The minimaxity of the empirical distribution function, which is also the best invariant estimator, is proved for any sample size. This solves a long-standing conjecture.

连续不变决策问题中经验分布函数的不可容许性AI 解读

Consider the classical invariant decision problem of estimating an unknown continuous distribution function $F,$ with the loss function $L(F, a) = \int(F(t) - a(t))^2\lbrack F(t) \rbrack^\alpha \lbrack 1 - F(t) \rbrack^\beta dF(t),$ and a random sample of size $n$ from $F.$ It is proved that the best invariant estimator is inadmissible when: 1. $ n > 0, - 1 < \alpha, \beta \leq 0 \text{and} -1 \leq \alpha + \beta.$ 2. $ n > 0, -1 < \alpha = \beta \leq - \frac{1}{2}.$ 3. $ n > 1, (\mathrm{i}) \alpha = -1 \text{and} \beta = 0, \text{or} (\mathrm{ii}) \alpha = 0 \text{and} \beta = -1.$…