Annals of the Institute of Statistical Mathematics

, Volume 34, Issue 3, pp 479–489

# On estimation of a density and its derivatives

• K. F. Cheng
Article

## Summary

Letf n (p) 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 continuousL2(−∞, ∞) 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.

## AMS 1970 subject classification

Primary 62G05 Secondary 60F15

## Key words and phrases

Recursive kernel density derivatives almost sure convergence rates

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