Publication Date


Document Type


First Advisor

Polansky, Alan M.

Degree Name

M.S. (Master of Science)

Legacy Department

Department of Mathematical Sciences


Distribution (Probability theory); Kernel functions; Asymptotic expansions; Smoothing (Statistics)


Many current statistical methods make use of the empirical distribution function to nonparametrically estimate an unknown population distribution function. This method is easy to implement and is pointwise unbiased and consistent. Unfortunately, the empirical distribution function is a step function with discontinuous jumps at each of the observed sample values. Such an estimate often violates an assumption that the underlying population is continuous. Hence, a continuous estimate of the distribution function is often desired. Kernel estimators are useful in providing a continuous estimate of the distribution function. The method relies on the selection of a kernel function and a smoothing parameter for good performance. It is well known that the choice of kernel function does not have a significant impact on the asymptotic performance of the kernel estimator. However, the choice of the smoothing parameter is crucial. Optimal selection of the smoothing parameter for finite samples is difficult. Hence we make this choice based on a truncated asymptotic expansion. The purpose of this thesis is to investigate the error inherent in this asymptotic approximation.


Includes bibliographical references (leaf [21])


19, [2] pages




Northern Illinois University

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