Multinomial Density Estimation - A Projection Pursuit Approach and a Wavelet Approach

这项研究涉及新的估计，一个投影的发展， 追求估计和小波估计，细胞概率 高维有序列联表 一个高维的偶然性 表有两个独特的特点：它通常是非常稀疏的，它有一个大的 边界细胞数。采用投影寻踪方法研究了 高维数据的低维特征 预测。 投影数据不再稀疏，投影 追踪法不受边界效应的影响，因为该方法 而不是基于当地的平均水平。当引入平滑方法时， 人们通常假设列联表具有潜在的密度， 具有一定平滑度的函数。 两者的顺序都是 核函数和平滑参数的最佳选择取决于底层的平滑度。 密度函数 这种假设似乎是不自然的。 一个人可能没有前科 关于未知密度的平滑度的知识。 小波方法可以 适应未知密度的平滑性，避免了 多少钱才能顺利。 因此，人们不需要假设存在 一个基本的密度函数 列联表或交叉表经常出现在生物医学科学、社会科学和教育研究中。 例如，研究公众对 五个不同的政策问题（医疗保健，税收，. . .)以五分制形式 （强烈同意，同意，中立，不同意，强烈不同意），a 需要一个有3 125个单元格的五维交叉表格。 通常，当单元格的数量很大时，表格通常是稀疏的， 也就是说，有很多空的细胞。 大而稀疏的表格为统计分析带来了许多挑战。 这项研究开发了新的工具来分析这种类型的 分类数据

This research involves the development of new estimators, a projection pursuit estimator and a wavelet estimator, for cell probabilities of high-dimensional ordered contingency tables. A high-dimensional contingency table has two unique features: it is usually very sparse, and it has a large number of boundary cells. The projection pursuit method is used to study the features of high-dimensional data by looking at its low-dimensional projections. The projected data is no longer sparse and the projection pursuit method does not suffer from boundary effect since the method is not based on local averages. When a smoothing method is introduced, one usually assumes that the contingency table has an underlying density function which has a certain degree of smoothness. Both the order of a kernel function and the optimal choice of a smoothing parameter depend on the smoothness of the underlying density function. This assumption seems unnatural. One may not have a prior knowledge about the smoothness of an unknown density. The wavelet method can adapt to the smoothness of an unknown density, and it avoids the problem of how much to smooth. Therefore, one does not need to assume the existence of an underlying density function. Contingency or cross tabulation tables occur very frequently in biomedical science, social science, and educational research. For example, to study the public's opinions on five different policy issues (health care, tax, . . .) on a five point scale (strongly agree, agree, neutral, disagree, strongly disagree), a five-dimensional cross tabulation table with 3,125 cells would be needed. Frequently when the number of cells is large, the table is usually sparse, i.e., has many empty cells. Large and sparse tables provide many challenges for statistical analysis. The research develops new tools for analyzing this type of categorical data.