We propose a kernel function for ordered categorical data that overcomes limitations present in ordered kernel functions appearing in the literature on the estimation of probability mass functions for multinomial ordered data. Some limitations arise from assumptions made about the support of the underlying random variable. Furthermore, many existing ordered kernel functions lack a particularly appealing property, namely the ability to deliver discrete uniform probability estimates for some value of the smoothing parameter. We propose an asymmetric empirical support kernel function that adapts to the data at hand and possesses certain desirable features. There are no difficulties arising from zero counts caused by gaps in the data while it encompasses both the empirical proportions and the discrete uniform probabilities at the lower and upper boundaries of the smoothing parameter. We propose likelihood and least-squares cross-validation for smoothing parameter selection and study their asymptotic and finite-sample behaviour.

中文翻译：

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有序数据类型的核平滑概率质量函数

我们提出了一种用于有序分类数据的核函数，它克服了文献中出现的关于多项有序数据的概率质量函数估计的有序核函数中存在的局限性。一些限制源于对基础随机变量的支持所做的假设。此外，许多现有的有序核函数缺乏一个特别吸引人的特性，即为平滑参数的某个值提供离散均匀概率估计的能力。我们提出了一个不对称的经验支持核函数，它适应手头的数据并具有某些理想的特征。由于数据中的差距导致零计数没有任何困难，同时它包含经验比例和平滑参数下边界和上边界的离散均匀概率。我们提出了用于平滑参数选择的似然和最小二乘交叉验证，并研究了它们的渐近和有限样本行为。