## Kernel estimators of density function of directional data.(English)Zbl 0669.62015

The authors considered the estimation of a density function f(x) based on n independent observations $$X_ 1,X_ 2,...,X_ n$$ on X, where X is a unit vector random variable taking values on a k-dimensional sphere $$\Omega$$ with probability density f(x).
The proposed estimator is of the form $f_ n(x)=(nh^{k-1})^{- 1}C(h)\sum K[(1-x'X_ i)/h^ 2],\quad x\in \Omega,$ where K is a kernel function defined on $$R_+$$. Conditions are imposed on K and F to prove pointwise strong consistency, and strong $$L_ 1$$-norm consistency of $$f_ n$$ as an estimator of f.
Reviewer: A.K.Hosni

### MSC:

 62G05 Nonparametric estimation
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### References:

 [1] Batschelet, E., Recent statistical methods for orientation, Amer. inst. biol. sci. bull., (1971) [2] Batschelet, E., () [3] Devroye, L., The equivalence of weak, strong and complete convergence in L1 for kernel density estimates, Ann. statist., 11, 896-904, (1983) · Zbl 0521.62033 [4] Bertrand-Retali, R., Convergence uniforme d’une estimateur de la densité par la méthod du noyau, Rev. roumaine math. pures appl., 23, 361-385, (1978) · Zbl 0375.62080 [5] Devroye, L.P.; Wagner, T.J., The strong uniform consistency of kernel density estimates, (), 59-77 · Zbl 0431.62024 [6] Hoeffding, W., Probability inequalities for sums of bounded random variables, J. amer. statist. assoc., 58, 13-30, (1963) · Zbl 0127.10602 [7] Mardia, K.V., () [8] Pukkila, T.; Rao, C.R., (), Technical Report 86-09 [9] Rao, J.S., Nonparametric methods in directional data analysis, (), 757-770 [10] Vapnik, V.N.; Chervonenkis, A.Ya., On the uniform convergence of relative frequencies of events to their probabilities, Theory probab. appl., 16, 264-280, (1971), (English) · Zbl 0247.60005 [11] Watson, G.S., () [12] Wenocur, R.S.; Dudley, R.M., Some special vapnik-chervonenkis classes, Discrete math., 33, 313318, (1981) · Zbl 0459.60008 [13] Zhao, L.C., (), Technical Report 85-30
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