One of the main objectives of this dissertation is to derive efficient nonparametric estimatorsfor an unknown density f . It is well known that the ordinary kernel densityestimator has, despite of several good properties, some drawbacks. For example, it suffersfrom boundary bias and it also exhibits spurious bumps in the tails. Various solutionsto overcome these defects are presented in this study, which include the application of atransformation kernel density estimator. The latter estimator (if implemented correctly)is pursued as a simultaneous solution for both boundary bias and spurious bumps in thetails. The estimator also has, among others, the ability to detect and estimate densitymodes more effectively.To apply the transformation kernel density estimator an effective transformation of thedata is required. To achieve this objective, an extensive discussion of parametric transformationsintroduced and studied in the literature is presented firstly, emphasizing thepractical feasibility of these transformations. Secondly, known methods of estimating theparameters associated with these transformations are discussed (e.g. profile maximumlikelihood), and two new estimation techniques, referred to as the minimum residual andminimum distance methods, are introduced. Furthermore, new procedures are developedto select a parametric transformation that is suitable for application to a given set ofdata. Finally, utilizing the above techniques, the desired optimal transformation to anytarget distribution (e.g. the normal distribution) is introduced, which has the propertythat it can also be iterated. A polynomial approximation of the optimal transformationfunction is presented. It is shown that the performance of this transformation exceedsthat of any transformation available in the literature.In the context of transformation kernel density estimation, we present a comprehensiveliterature study of current methods available and then introduce the new semi-parametrictransformation estimation procedure based on the optimal transformation of data to normality.However, application of the optimal transformation in this context requires specialattention. In order to create a density estimator that addresses both boundary bias andspurious bumps in the tails simultaneously in an automatic way, a generalized bandwidthadaptation procedure is developed, which is applied in conjunction with a newly developedconstant shift procedure.Furthermore, the optimal transformation function is based on a kernel distribution functionestimator. A new data-based smoothing parameter (bandwidth selector) is invented,and it is shown that this selector has better performance than a well established bandwidthselector proposed in the literature.To evaluate the performance of the newly proposed semi-parametric transformation estimationprocedure, a simulation study is presented based on densities that consist of awide range of forms. Some of the main results derived in the Monte Carlo simulationstudy include that:- the proposed optimal transformation function can take on all the possible shapes of a parametric transformation as well as any combination of these shapes, which result in high p-values when testing normality of the transformed data.- the new minimum residual and minimum distance techniques contribute to better transformations to normality, when a parametric transformation is applicable.- the newly proposed semi-parametric transformation kernel density estimator perform well for unimodal, low and high kurtosis densities. Moreover, it estimates densities with much curvature (e.g. modes and valleys) more effectively than existing procedures in the literature.- the new transformation density estimator does not exhibit spurious bumps in the tail regions.- boundary bias is addressed automatically.In conclusion, practical examples based on real-life data are presented.
