The problem of estimating a bivariate cumulative distribution function F under the weighted squared error loss and the weighted Cramer–von Mises loss is considered. No restrictions are imposed on the unknown function F . Estimators, which are minimax among procedures being affine transformation of the bivariate empirical distribution function, are found. Then it is proved that these procedures are minimax among all decision rules. The result for the weighted squared error loss is generalized to the case where F is assumed to be a continuous cumulative distribution function. Extensions to higher dimensions are briefly discussed.

中文翻译：

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二元累积分布函数的极大极小估计

考虑了在加权平方误差损失和加权 Cramer-von Mises 损失下估计二元累积分布函数 F 的问题。对未知函数 F 没有任何限制。估计量是二元经验分布函数的仿射变换过程中的极小极大值。然后证明这些过程在所有决策规则中都是极小极大的。加权平方误差损失的结果推广到假设 F 是连续累积分布函数的情况。对更高维度的扩展进行了简要讨论。