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

## References

1. [1]
Bhattacharya, P. K. (1967). Estimation of a probability density function and its derivatives,Sankhyá, A,29, 373–382.
2. [2]
Carroll, R. J. (1976). On sequential density estimation,Z. Wahrscheinlichkeitsth.,36, 137–151.
3. [3]
Davies, H. I. (1973). Strong consistency of a sequential estimator of a probability density function,Bull. Math. Statist.,15, 49–54.
4. [4]
Davies, H. I. and Wegman, E. J. (1975). Sequential nonparametric density estimation,IEEE Trans. Inf. Theory, IT-21, 619–628.
5. [5]
Deheuvals, P. (1974). Conditions necessaires et suffisantes de convergence ponctuelle presque sûre et uniforme presque sûre des estimateurs de la densité,C. R. Acad. Sci. Paris, A.278, 1217–1220.
6. [6]
Fryer, M. J. (1977). A review of some non-parametric methods of density estimation,J. Inst. Math. Appl.,20, 335–354.
7. [7]
Lamperti, J. (1966).Probability, W. A. Benjamin, Inc. N.Y.
8. [8]
Lin, P. E. (1975). Rates of convergence in empirical Bayes problems: Continuous case,Ann. Statist.,3, 155–164.
9. [9]
Nadaraya, E. A. (1965). On nonparametric estimates of density functions and regression curves,Theory Prob. Appl.,10, 186–190.
10. [10]
Nadaraya, E. A. (1973). On convergence in theL 2-norm of probability density estimates,Theory Prob. Appl.,18, 808–811.
11. [11]
Parzen, E. (1962). On estimation of a probability density function and mode,Ann. Math. Statist.,33, 1065–1076.
12. [12]
Silverman, B. S. (1978). Weak and strong uniform consistency of the kernel estimate of a density and its derivatives,Ann. Statist.,6, 177–184.
13. [13]
Singh, R. S. (1977a). Improvement on some known nonparametric uniformly consistent estimators of derivatives of a density,Ann. Statist.,5, 394–400.
14. [14]
Singh, R. S. (1977b). Applications of estimators of a density and its derivatives to certain statistical problems,J.R. Statist. Soc., B,39, 357–363.
15. [15]
Singh, R. S. (1979). On necessary and sufficient conditions for uniform strong consistency of estimators of a density and its derivatives,J. Multivariate Anal.,9, 157–164.
16. [16]
Singh, R. S. (1981). On the exact asymptotic behavior of estimators of a density and its derivatives,Ann. Statist.,9, 453–456.
17. [17]
Walter, G. G. (1977). Properties of Hermite series estimation of probability density,Ann. Statist.,5, 1258–1264.
18. [18]
Walter, G. G. (1980). Addendum to “Properties of Hermite series estimation of probability density”,Ann. Statist.,8, 454–455.
19. [19]
Wegman, E. J. (1972a). Nonparametric probability density estimation: I. A. summary of available methods,Technometrics,14, 533–546.
20. [20]
Wegman, E. J. (1972b). Nonparametric probability density estimation: II. A comparison of density estimation methods,J. Statist. Comp. Simul.,1, 225–245.
21. [21]
Wegman, E. J. and Davies, H. I. (1979). Remarks on some recursive estimators of a probability density,Ann. Statist.,7, 316–327.
22. [22]
Wolverton, C. T. and Wagner, T. J. (1969). Asymptotically optimal discriminant functions for pattern classification,IEEE Trans. Inf. Theory, IT-15, 258–265.
23. [23]
Yamato, H. (1971). Sequential estimation of a continuous probability density and mode,Bull. Math. Statist.,14, 1–12.