Nonparametric Functional Smoothing Techniques

人口中一组测量数据的相对频率分布模型称为概率密度函数，它描述了在不同间隔内观察到的数据的比例，并回答了这样的问题：“加拿大人中有多少人生活在贫困线以下？”或者“装配线上生产的产品有多大比例低于规定的质量门槛？”这是一条数学曲线，如钟形高斯曲线，然后用于计算人口的各种其他特征，如平均值、中位数和百分位数。通常，被称为参数模型的标准模型在计算人口测量的复杂函数时是有用的，这些模型仅依赖于几个常量，例如在高斯模型的情况下的平均值和标准差。例如，在钟形曲线上，我们估计95%的观测值将落在与平均值的两个标准差以内。如果模型不合适，这种基于模型的推理将是无效的。 在这种情况下，可以使用具有关于基础分布的宽松(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，逼近直方图的连续曲线可以用作非参数密度估计器。这项建议属于非参数平滑的一般领域，以期探索在各种应用领域中可能具有重要意义的重要应用。 我开发和研究了非参数平滑方法，当对称核可能不合适时(例如处理包括加权数据、删失数据和相关数据在内的非标准数据)，非参数平滑方法可以替代传统的核平滑。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，这些分布可用于估计样本中最大观察值以外的生存概率，以及总体的其他重要特征，如以给定年龄为条件的预期寿命。这项建议的基本目标是探索将适用于上述非标准数据情况的方法用于其他重要问题，如识别极值和离群值、依赖于基本概率密度函数的聚类和分类。 这些发展的意义在于它们的应用，例如，在使用更新函数的估计器的保证分析中，以及在分类和聚类中，参数密度的作用被它们的半参数/非参数对应所取代。这项建议将进一步用于培训本科生和研究生在几个应用领域的非参数曲线光顺知识和应用。

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.