The problem of nonparametric estimation of probability density function is considered. The performance of kernel estimators based on various common kernels and a new kernel K (see (14)) with both fixed and adaptive smoothing bandwidth is compared in terms of the symmetric mean absolute percentage error using the Monte Carlo method. The kernel K is everywhere positive but has lighter tails than the Gaussian density. Gaussian mixture models from a collection introduced by Marron and Wand (1992) are taken for Monte Carlo simulations. The adaptive kernel method outperforms the smoothing with a fixed bandwidth in the majority of models. The kernel K shows better performance for Gaussian mixtures with considerably overlapping components and multiple peaks (double claw distribution).

Ultrasonic and digital dermatoscopy diagnostic methods are used in order to estimate the changes of structure, as well as to non-invasively measure the changes of parameters of lesions of human tissue. These days, it is very actual to perform the quantitative analysis of medical data, which allows to achieve the reliable early-stage diagnosis of lesions and help to save more lives. The proposed automatic statistical post-processing method based on integration of ultrasonic and digital dermatoscopy measurements is intended to estimate the parameters of malignant tumours, measure spatial dimensions (e.g. thickness) and shape, and perform faster diagnostics by increasing the accuracy of tumours differentiation. It leads to optimization of time-consuming analysis procedures of medical images and could be used as a reliable decision support tool in the field of dermatology.

Ultrasonic diagnostic methods are used to estimate the structural changes and to measure parameters of lesions of the human tissue. Nowadays, the special algorithms of medical data analysis are able to perform diagnosis and monitor the progress of treatment, efficiency of treatment methods, also to estimate the health status and to make prognosis of the diseases evolution. The aim of the presented research is to check the goodness of fit test for thicknesses of the skin tumours measured in two different ways (ultrasound examination and histological analysis) and to compare the compatibility of likely density of histological thicknesses distribution of the skin tumours and density of Normal distribution. As a result, the study has showed that thicknesses of the skin tumours measured by ultrasonic method are strongly similar to histological values, which means that the density of ultrasonic thicknesses distribution and density of Normal distribution are closely interconnected. Therefore, the obtained results show the sufficient level of reliability in the case of application of non-invasive ultrasonic thickness measurement comparing with reference invasive technique based on biopsy and histological thickness evaluation.

This paper presents nonparametric statistical estimation of distribution density. The Monte Carlo  method is used to show the effects of kernel function for multimodal kernel density estimation. Here it is shown that the novel kernel function is effective for asymmetrical heavy tails distributions.