Nonparametric Functional Smoothing Techniques

一组人口测量值的相对频率分布模型，被称为概率密度函数，描述了在不同间隔内的观测值的比例，并回答了诸如“加拿大人有多少百分比生活在贫困线以下？”或“装配线上生产的产品有多少百分比低于规定的质量阈值？”这是一条数学曲线，如钟形高斯曲线，然后用于计算总体的各种其他特征，如平均值、中位数和百分位数。通常标准模型，被称为参数模型，只依赖于几个常数，如高斯模型中的平均值和标准差，在计算总体测量的复杂函数时很有用。例如，在钟形曲线上，我们估计95%的观测值将落在离平均值两个标准差的范围内。如果模型不合适，这种基于模型的推断将无效。******在这种情况下，可以使用对底层分布具有宽松（或更少）假设的非参数模型。非参数方法背后的基本思想是对一组离散点的连续逼近，通常称为平滑。因此，近似直方图的连续曲线可以用作非参数密度估计量。本提案是在非参数平滑的一般领域，以期探索在各个应用领域可能具有重要意义的重要应用。******我已经开发和研究了非参数平滑方法，作为传统核平滑的替代方案，当对称核可能不合适（例如处理非标准数据，包括加权数据，审查数据和依赖数据）。这些方法依赖于非对称核和离散分布，如二项分布和泊松分布，这些分布对于估计样本中最大观测值之外的生存概率以及种群的其他重要特征（如给定年龄条件下的预期寿命）很有用。本提案的基本目标是探索使用适用于上述非标准数据情况的方法来解决其他重要问题，例如依赖于潜在概率密度函数的极端值和异常值的识别，聚类和分类。******这些发展的意义在于它们的应用，例如在使用更新函数估计器的保修分析中，以及在分类和聚类中，参数密度的作用被其半/非参数对应物所取代。本课题将进一步用于培养本科生和研究生在多个应用领域的非参数曲线平滑知识和应用。

A model for the relative frequency distribution of a set of measurements on a population, known as a probability density function, describes the proportion of observations falling in various intervals, and answers such questions as "what percentage of Canadians are below the poverty line?" or "what percentage of items produced on an assembly line will be found below the prescribed quality threshold?" This is a mathematical curve, such as the bell-shaped Gaussian curve, that is then used to compute various other characteristics of the population such the average, median and percentiles. Often the standard models, known as the parametric models that depend only on a few constants, such as the mean and standard deviation in the case of the Gaussian model, are useful in computing complex functions of the population measurements. For example, on a bell curve, we estimate that 95% of the observations will fall within two standard deviations from the mean. If the model is not appropriate, such model based inferences will be invalid. ******In such situations, non-parametric model with relaxed (or less) assumptions about the underlying distribution may be employed. The basic idea behind the non-parametric method is a continuous approximation to a set of discrete points, commonly known as smoothing. Thus a continuous curve approximating the histogram may serve as a non-parametric density estimator. This proposal is in the general area of non-parametric smoothing, with a view to explore important applications that may be of importance in various applied fields. ******I have developed and studied non-parametric smoothing methods as an alternative to traditional kernel smoothing when symmetric kernels may not be appropriate (such as when dealing with non-standard data including weighted data, censored data and dependent data). These methods depend on asymmetric kernels and discrete distributions such as the binomial and Poisson distributions that are useful for estimating the survival probability beyond the largest observation in the sample as well as other important characteristics of the population such as the expected life time conditional on a given age. The basic objective of this proposal is to explore the use of methods that are applicable to non-standard data situations as mentioned above for other important problems such as the identification of extremes and outliers, clustering and classification which depend on the underlying probability density function. ******Significance of these developments is in their applications, for example in warranty analysis using the estimator of a renewal function, and in classification and clustering where the role of parametric densities is replaced by their semi/non-parametric counterparts. This proposal will be further used for training of undergraduate and graduate students in the knowledge and applications of non-parametric curve smoothing in several applied fields.