Kernel method is one of the most common nonparametric density estimation and recently B-spline is used for estimation of a probability density function. These two methods in some how depend on selecting a smoothing parameter that has an important effect on precision of the estimators. In this paper, we consider Kernel and B-spline methods of density estimation and smoothing parameter selection for these two methods. Then, the accuracy of the obtained estimators is compared by their mean square errors. Also, the effect of the number and dispersion of data on precision of estimators are studied. The results show that for a symmetric probability density, if the dispersion of data increases, the precision of both estimators decreases, while, for an asymmetric probability density function, the precision of the estimators increases for dispersion data.

Background: Breast cancer is the most common cancer in women. An accurate and reliable system for early diagnosis of benign or malignant tumors seems necessary. We can design new methods using the results of FNA and data mining and machine learning techniques for early diagnosis of breast cancer which able detection of breast cancer with high accuracy. The aim of this study was to diagnosis of breast cancer using non-parametric Kernel density estimation.Methods: In this study, 699 samples of benign and malignancy with 9 characteristics from WBCD and 569 samples of benign and malignancy with 30 characteristics from WDBC were used. Then, a model based on non-parametric Kernel density estimation was proposed for classification of WBCD and WDBC data.Results: The results of non-parametric methods showed that Gaussian Kernel method based on Euclidean distance with accuracy %97.93 has the highest accuracy on WDBC data and Gaussian Kernel based on Euclidean distance and k-nearest neighbor methods with accuracy %98.17 has the highest accuracy compared with other methods on WBCD data for breast cancer disease.Conclusion: The result of this study showed that non-parametric Kernel density estimation based classification can be used for breast cancer diagnosis with high accuracy.

In this paper, we prove the strong uniform consistency and asymptotic normality of the Kernel density estimator proposed by Jones for length-biased data. The approach is based on the invariance principle for the empirical processes proved by Horvath. All simulations are drawn for different cases to demonstrate both, consistency and asymptotic normality and the method is illustrated by real automobile brake pads data.

Understanding the underlying structure of vehicle crashes is essential for improving safety on the roads. Past research has found that accident tend to cluster both spatially and temporally. This paper applied spatial, temporal and spatio-temporal techniques to investigate patterns of Road Accident in Karaj-Qazvin highway-Iran between 2009 and 2012, at different characters. Spider graphs were adapted to identify temporal patterns of vehicle crashes at different situation. The spatial structures of vehicle crashes were analyzed using Fuzzy Kernel density and then result was compared with Kernel density Estimation. Comap was then used to demonstrate the spatio-temporal interaction effect on vehicle crashes and spatial pattern of accident at different characters. Characters that influence at spatial and temporal pattern of accident are: Time, Week day, Type of accident (Fatal, Injury and Non-Injury), Type of Collision (Vehicle with Vehicle, Vehicle with Pedestrian, Vehicle with Objects and Overturn), weather situation (Clean, Rain and Snow), Human Behavior (Hurry, Fatigue and Sleepy and Low Inattention). The results show significant differences in spatiotemporal patterns of Road Accident for various crash causes. The techniques used here have the potential to help decision makers in developing effective road safety strategies.

Distance-based clustering methods categorize samples by optimizing a global criterion, finding ellipsoid clusters with roughly equal sizes. In contrast, density-based clustering techniques form clusters with arbitrary shapes and sizes by optimizing a local criterion. Most of these methods have several hyper-parameters, and their performance is highly dependent on the hyper-parameter setup. Recently, a Gaussian density Distance (GDD) approach was proposed to optimize local criteria in terms of distance and density properties of samples. GDD can find clusters with different shapes and sizes without any free parameters. However, it may fail to discover the appropriate clusters due to the interfering of clustered samples in estimating the density and distance properties of remaining unclustered samples. Here, we introduce Adaptive GDD (AGDD), which eliminates the inappropriate effect of clustered samples by adaptively updating the parameters during clustering. It is stable and can identify clusters with various shapes, sizes, and densities without adding extra parameters. The distance metrics calculating the dissimilarity between samples can affect the clustering performance. The effect of different distance measurements is also analyzed on the method. The experimental results conducted on several well-known datasets show the effectiveness of the proposed AGDD method compared to the other well-known clustering methods.

Kernel density estimators are the basic tools for density estimation in non-parametric statistics. The k-nearest neighbor Kernel estimators represent a special form of Kernel density estimators, in which the bandwidth is varied depending on the location of the sample points. In this paper, we initially introduce the k-nearest neighbor Kernel density estimator in the random left-truncation model, and then prove some of its asymptotic behaviors, such as strong uniform consistency and asymptotic normality. In particular, we show that the proposed estimator has truncation-free variance. Simulations are presented to illustrate the results and show how the estimator behaves for finite samples. Moreover, the proposed estimator is used to estimate the density function of a real data set.

Traditional method in flood frequency analysis is parametric approach. This method lacks the ability to describe multimodal and Asymmetric densities. In order to overcome this problem, the nonparametric models can be used. Two methods of nonparametric approach are: fixed and variable Kernel density. In fixed Kernel density method, the probability density function can be estimated by selecting a Kernel function and optimal bandwidth and in variable Kernel density method the probability density function can be estimated by selecting a Kernel function and bandwidth at each observation point. Cross validation and Rule of thumb are common methods for estimating the optimum bandwidth. In this paper, besides mentioned methods Plug in bandwidth method is used and nonparametric flood frequency analysis is performed using annual maximum flood data of the Dez river. Finally results were compared with parametric method. According to RMSE, it is concluded that plug in bandwidth is the most accurate method for estimating optimum bandwidth. As well as Nonparametric method based on variable Kernel density is more accurate than fixed Kernel density and both types of these models are more accurate than LP3 distribution.