Nonparametric Functional Smoothing Techniques

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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.