Hermite series estimates of a probability density and its derivatives |
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Authors: | Wκodzimierz Greblicki Mirosκaw Pawlak |
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Institution: | Institute of Engineering Cybernetics, Technical University of Wrocκaw, Wrocκaw, Poland |
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Abstract: | The following estimate of the pth derivative of a probability density function is examined: , where hk is the kth Hermite function and Σi = 1nhk(p)(Xi) is calculated from a sequence X1,…, Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n?α) and O(n?α log n), respectively, where . Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n?β) and , respectively, where . |
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Keywords: | 62G05 Hermite series orthogonal series density estimate |
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