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Central Limit Theorem; global errors; strong approximation; empirical processes; $U$-statistics; Poissonization
The asymptotic behavior of global errors of functional estimates plays a key role in hypothesis testing and confidence interval building. Whereas for pointwise errors asymptotic normality often easily follows from standard Central Limit Theorems, global errors asymptotics involve some additional techniques such as strong approximation, martingale theory and Poissonization. We review these techniques in the framework of density estimation from independent identically distributed random variables, i.e., the context for which they were introduced. This will avoid the mathematical difficulties associated with more complex statistical situations in which these tools have proved to be useful.
[1] Barbour, A. D., Holst, L. and Janson, S.: Poisson Approximation. University Press, Oxford, 1992. MR 1163825
[2] Barron, A. R.: The convergence in information of probability density estimators. (1988), Presented at IEEE ISIT, Kobe, Japan, June 19–24, 1988.
[3] Barron, A. R., Györfi, L. and van der Meulen, E. C.: Distribution estimates consistent in total variation and in two types of information divergence. IEEE Trans. on Information Theory 38 (1992), 1437–1454. DOI 10.1109/18.149496 | MR 1178189
[4] Beirlant, J., Györfi, L. and Lugosi, G.: On the asymptotic normality of the $L_1$ and $L_2$ errors in histogram density estimation. Canadian Journal of Statistics 3 (1994), 309–318. DOI 10.2307/3315594
[5] Beirlant, J. and Mason, D. M.: On the asymptotic normality of $L_p$ norms of empirical functionals. Mathematical Methods of Statistics 4 (1994), 1–15. MR 1324687
[6] Berlinet, A.: Central limit theorems in functional estimation. Bulletin of the International Statistical Institute 56 (1995), 531–548. Zbl 0880.62041
[7] Berlinet, A., Devroye, L. and Györfi, L.: Asymptotic normality of $L_1$ error in density estimation. Statistics 26 (1995), 329–343. DOI 10.1080/02331889508802500 | MR 1365682
[8] Berlinet, A., Györfi, L. and van der Meulen, E. C.: Asymptotic normality of relative entropy in multivariate density estimation. Revue de l’Institut de Statistique de l’Université de Paris 41 (1997), 3–27. MR 1743681
[9] Berlinet, A., Vajda, I. and van der Meulen, E. C.: About the Asymptotic Accuracy of Barron density estimates. Trans. IEEE on Inform. Theory 44 (19981998), 999–1009. DOI 10.1109/18.669143 | MR 1616679
[10] Bickel, P. and Rosenblatt, M.: On some global measures of the deviation of density function estimates. The Annals of Statistics 1 (1973), 1071–1095. DOI 10.1214/aos/1176342558 | MR 0348906
[11] Csörgő, M. and Horváth, L.: Weighted Approximations in Probability and Statistics. Wiley, New York, 1993. MR 1215046
[12] Györfi, L., Liese, F., Vajda, I. and van der Meulen, E. C.: Distribution estimates consistent in $\chi ^2$-divergence. Statistics 32 (1998), 31–58. DOI 10.1080/02331889808802651 | MR 1708074
[13] Hall, P.: Central limit theorem for integrated square error of multivariate nonparametric density estimators. Journal of Multivariate Analysis 14 (1984), 1–16. DOI 10.1016/0047-259X(84)90044-7 | MR 0734096 | Zbl 0528.62028
[14] Heyde, C. C.: Central limit theorem. Encyclopedia of Statistical Sciences 4 (1983), 651–655. MR 0517475
[15] Horváth, L.: On $L_p$ norms of multivariate density estimators. The Annals of Statistics 19 (1991), 1933–1949. DOI 10.1214/aos/1176348379 | MR 1135157
[16] Johnson, N. L. and Kotz, S.: Urn Models and their Application. Wiley, New York, 1977. MR 0488211
[17] Kac, S.: On deviations between theoretical and empirical distributions. Proceedings of The National Academy of Sciences of USA 35 (1949), 252–257. MR 0029490 | Zbl 0033.19303
[18] Morris, C.: Central limit theorems for multinomial sums. The Annals of Statistics 3 (1975), 165–188. DOI 10.1214/aos/1176343006 | MR 0370871 | Zbl 0305.62013
[19] Pollard, D.: Beyond the Heuristic Approach to Kolmogorov-Smirnov Theorems. Essays in Statistical Science. Festschrift for P. A. P. Moran, J. Gani and E. J. Hannan (eds.), Applied Probability Trust, 1982. MR 0633205 | Zbl 0495.60008
[20] Rosenblatt, M.: A quadratic measure of deviation of two-dimensional density estimates and a test of independence. The Annals of Statistics 3 (1975), 1–14. DOI 10.1214/aos/1176342996 | MR 0428579 | Zbl 0325.62030
[21] van der Vaart, A. W. and Wellner, J. A.: Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York, 1996. MR 1385671
[22] Shorack, G. R. and Wellner, J. A.: Empirical Processes with Applications to Statistics. Wiley, New York, 1986. MR 0838963
[23] Steck, G. P.: Limit theorems for conditional distributions. 2 (1957), University of California Publications in Statistics, 237–284. MR 0091552 | Zbl 0077.33104
[24] Wellner, J. A.: Empirical processes in action: a review. International Statistical Review 60 (1992), 247–269. DOI 10.2307/1403678 | Zbl 0757.62028
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