Multivariate recursive kernel density estimation with adjustable discounting of old data

The advent of the information technology era changes the data paradigm and produces complex data. The evolution of the data paradigm challenges the methodology of data analysis. Recently streaming data, which means that massive data comes continuously without end, are available in many areas. Especially streaming data sets are more complex because the underlying structure changes with time. Multivariate recursive kernel density estimators are suggested in order to estimate the density of the streaming data effectively with respect to computation and storage. The asymptotic bias and variance are developed and strong convergence is shown using almost sure invariance principle and the law of iterated logarithm. Multivariate recursive kernel density estimator with exponential smoother is proposed in order to capture the changing distributional pattern of the streaming data. Asymptotic bias and variance are investigated for the independent and identically distributed case and for the stationary case. Multivariate recursive kernel density estimator with exponential smoother is applied to the Internet traffic data to see how it captures the changing distributional pattern with various exponential smoothing parameters.