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On estimation of a density and its derivatives

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Letf (p)n be a recursive kernel estimate off (p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of\(\left\| {f_n^{(p)} - f^{(p)} } \right\|_{L_2 } = \left[ {\smallint [f_n^{(p)} (x) - f^{(p)} (x)]^2 dx} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \) and show that the rate of almost sure convergence of\(\left\| {f_n^{(p)} - f^{(p)} } \right\|_{L_2 } \) to zero isO(n −α), α<(r−p)/(2r+1), iff (r),r>p≧0, is a continuousL 2(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of\(\left\| {f_n^{(p)} - f^{(p)} } \right\|_\infty = \mathop {\sup }\limits_x \left| {f_n^{(p)} (x) - f^{(p)} (x)} \right|\) to zero under different conditions onf.

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Authors and Affiliations

  1. State University of New York at Buffalo, Buffalo, USA

    K. F. Cheng

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  1. K. F. Cheng
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This work was supported in part by the Research Foundation of SUNY.

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Cheng, K.F. On estimation of a density and its derivatives. Ann Inst Stat Math 34, 479–489 (1982). https://doi.org/10.1007/BF02481046

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  • DOI: https://doi.org/10.1007/BF02481046

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