A LOCATION SHIFT PROBLEM IN NONPARAMETRIC DENSITY ESTIMATION
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概要
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Let $ X_1, X_2, \cdots, X_n $ be i.i.d. random variables having a probability density function $ f(x) $ and $ f_n(x) $ be a nonparametric density estimator of $ f(x) $. We investigate the property of a location shift random variable $ a_n $ which minimizes integrated squared error $ \mathrm{ISE}_n(a) $: $ \mathrm{ISE}_n(a) = \int_{-x}^{x}{\mid f_n(x) - f(x - a) \mid^2}dx $. The asymptotic normality and the order of strong convergence of the $ \mathrm{r.v.}a_n $ and those of $ \mathrm{ISE}_n(a_n) $ are studied. We also give some numerical examples and some simulations which show the effectiveness of using the $ a_n $ when one estimates $ f(x) $ by $ f_n(x) $.
- Research Association of Statistical Sciencesの論文
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