ISSN: 2332-2071 Volume 6 Number 1 2018 Mathematics and Statistics http://www.hrpub.org Horizon Research Publishing, USA http://www.hrpub.org Mathematics and Statistics Mathematics and Statistics is an international peer-reviewed journal that publishes original and high-quality research papers in all areas of mathematics and statistics. As an important academic exchange platform, scientists and researchers can know the most up-to-date academic trends and seek valuable primary sources for reference. The subject areas include, but are not limited to the following fields: Algebra, Analysis, Applied mathematics, Approximation theory, Combinatorics, Computational statistics, Computing in Mathematics, Design of experiments, Discrete mathematics, Dynamical systems, Geometry and Topology, Logic and Foundations of mathematics, Number theory, Numerical analysis, Probability theory, Quantity, Recreational mathematics, Sample Survey, Statistical modelling, Statistical theory. 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Mathematics and Statistics Editor-in-Chief Prof. Dshalalow Jewgeni Florida Inst. of Technology, USA Members of Editorial Board Jiafeng Lu Zhejiang Normal University, China Nadeem-ur Rehman Aligarh Muslim University, India Debaraj Sen Concordia University, Canada Mauro Spreafico University of São Paulo, Brazil Veli Shakhmurov Okan University, Turkey Antonio Maria Scarfone Institute of Complex Systems - National Research Council, Italy Liang-yun Zhang Nanjing Agricultural University, China Ilgar Jabbarov Ganja state university, Azerbaijan Mohammad Syed Pukhta Sher-e-Kashmir University of Agricultural Sciences and Technology, India Vadim Kryakvin Southern Federal University, Russia Rakhshanda Dzhabarzadeh National Academy of Science of Azerbaijan, Azerbaijan Sergey Sudoplatov Sobolev Institute of Mathematics, Russia Birol Altın Gazi University, Turkey Araz Aliev Baku State University, Azerbaijan Francisco Gallego Lupianez Universidad Complutense de Madrid, Spain Hui Zhang St. Jude Children's Research Hospital, USA Yusif Abilov Odlar Yurdu University, Azerbaijan Evgeny Maleko Magnitogorsk State Technical University, Russia İmdat İşcan Giresun University, Turkey Emanuele Galligani University of Modena and Reggio Emillia, Italy Mahammad Nurmammadov Baku State University, Azerbaijan Horizon Research Publishing http://www.hrpub.org ISSN: 2332-2071 Table of Contents Mathematics and Statistics Volume 6 Number 1 2018 New Gradient Methods for Bandwidth Selection in Bivariate Kernel Density Estimation (https://www.doi.org/10.13189/ms.2018.060101) Siloko, I. U., Ishiekwene, C. C., Oyegue, F. O. ............................................................................................................... 1 Exponential Dichotomy and Bifurcation Conditions of Solutions of the Hamiltonian Operators Boundary Value Problems in the Hilbert Space (https://www.doi.org/10.13189/ms.2018.060102) Pokutnyi Oleksandr ......................................................................................................................................................... 9 Mathematics and Statistics 6(1): 1-8, 2018 http://www.hrpub.org DOI: 10.13189/ms.2018.060101 New Gradient Methods for Bandwidth Selection in Bivariate Kernel Density Estimation Siloko, I. U.1,*, Ishiekwene, C. C.2, Oyegue, F. O.2 1Department of Mathematical Sciences, Edwin Clark University, Nigeria 2Department of Mathematics, University of Benin, Nigeria Copyright©2018 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract The bivariate kernel density estimator is smoothing parameter selectors by some researchers [3, 4]. fundamental in data smoothing methods especially for data The importance of the bivariate kernel density estimator exploration and visualization purposes due to its ease of cannot be overemphasized because it occupies a unique graphical interpretation of results. The crucial factor which position of bridging the univariate kernel density determines its performance is the bandwidth. We present estimator and other higher dimensional kernel estimators new methods for bandwidth selection in bivariate kernel [5]. The usefulness of the bivariate kernel density density estimation based on the principle of gradient estimator is mainly in its simplicity of presentation of method and compare the result with the biased probability density estimates, either as surface plots or cross-validation method. The results show that the new contour plots. It also helps in understanding other higher methods are reliable and they provide improved methods dimensional kernel estimators [2]. In bivariate kernel density estimation, , is taken to be the two random for a choice of smoothing parameter. The asymptotic mean variables with a joint probability density function ( , ). integrated squared error is used as the measure of The random variables , , = , ,…, are the set of performance of the new methods. observations and is the sample size. The bivariate Keywords Bandwidth, Bivariate Kernel Density kernel density estimate of ( , ) is of the form Estimator, Biased Cross-validation, Gradient Method, ( , ) = , Asymptotic Mean Integration Squared Error (1.1) − − where �> 0 and ∑>= 0 �are the � smoothing ( , ) parameters in the and axes and is a bivariate kernel function [5, 6]. The bivariate kernel 1. Introduction density estimator in (1.1) can be written as [7] Kernel density estimators are widely used nonparametric ( , ) = (1.2) estimation techniques due to their simple forms and − − = smoothness. Kernel density estimation is the construction Bivariate� bandwidth ∑ selection � is� a difficult � � problem of a probability density estimates from a given sample with which may be simplified by imposing constraints on and . For example, and may be restricted to be few assumptions about the underlying probability density the diagonal elements of the bandwidth matrix and the function and the kernel function. Kernel density estimation is a popular tool for visualising the distribution of data [1]. advantages of imposing restrictions on and has These estimates depend on a bandwidth also known as the been investigated [8]. One of the popular methods of smoothing parameter which controls the smoothness and a bandwidth selection that is data based is the biased kernel function which plays the role of a weighting cross-validation method that considers the asymptotic function [2]. Bandwidth selection is a key issue in kernel mean integrated squared error [9]. The bivariate biased methods and has attracted the attention of researchers over cross-validation method is based on minimizing an the years. It is still an active research area in kernel density estimate of the asymptotic mean integrated squared error estimation. Progress has been made recently on data based and is of the form [10] 2 New Gradient Methods for Bandwidth Selection in Bivariate Kernel Density Estimation , = + × + + + ( ) ( ) ( ) (1.3) = ≠ where = � , � = and −is the standard∑ normal∑