In nonparametric statistics, kernel-based methods are usually used for the estimation of density, distribution, quantile, and quantile density functions. However, kernel-type estimators are often unstable due to the choice of the bandwidth parameter and can be seriously biased near boundaries and/or at tails. When quantile density estimation is of interest, the problem may be even worse.
In addition, it is well known that the kernel density estimator converges to the true density of the data when the bandwidth goes to zero, but in practice, the bandwidth is not near zero, otherwise the resulting density curve is under-smoothed, leading to a useless density estimator.
To alleviate the issues of bias or due to the bandwidth selection, three different types of quantile and quantile density estimators are introduced and compared with the existing methods in this thesis.
Moreover, we study a general family of distributions, which is generated by providing a base distribution that is related to the kernel density estimator asymptotically, whether the bandwidth is sufficiently small or not. It includes a reparametrized skew normal distribution and a new class of bimodal distributions as special cases and also hints at a kernel-type density estimator of a single order statistic. Furthermore, tests of normality are constructed based on this kernel related function.