Abstract
Let X1, X2, ..., Xn be i.i.d. random variables with common unknown density function f. We are interested in estimating the unknown density f with bounded Mean Integrated Absolute Error (MIAE). Devroye and Győrfi (1985, Nonparametric Density Estimation: The L1 View, Wiley, New York) obtained asymptotic bounds for the MIAE in estimating f by a kernel estimate fn. Using these bounds one can identify an appropriate sample size such that an asymptotic upper bound for the MIAE is smaller than some pre-assigned quantity w > 0. But this sample size depends on the unknown density f. Hence there is no fixed sample size that can be used to solve the problem of bounding the MIAE. In this work we propose stopping rules and two-stage procedures for bounding the L1 distance. We show that these procedures are asymptotically optimal in a certain sense as w → 0, i.e., as one requires increasingly better fit.
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REFERENCES
Bowyer, A. (1980). Experiments and computer modelling in stick-slip, Ph.D. Thesis, Department of Actuarial Science and Statistics, University of London.
Cao, R., Cuevas, A. and Manteiga, W. G. (1994). A comparative study of several smoothing methods in density estimation, Comput. Statist. Data Anal., 17, 153–176.
Carroll, R. J. (1976). On sequential density estimation, Zeit. Wahrscheinlichkeitsth., 36, 137–151.
Devroye, L. (1983). The equivalence of weak, strong and complete convergence in L 1 for kernel density estimates, Ann. Statist., 11, 896–904.
Devroye, L. (1988). Asymptotic performance bounds for the kernel estimate, Ann. Statist., 16, 1162–1179.
Devroye, L. (1991). Exponential inequalities in nonparametric estimation, Nonparametric Functional Estimation and Related Topics (ed. G. Roussas), 31–44, Kluwer.
Devroye, L. and Cyőrfi, L. (1985). Nonparametric Density Estimation: The L 1 View, Wiley, New York.
Devroye, L. and Wand, M. P. (1993). On the effect of density shape on the performance of its kernel estimate, Statistics, 24, 215–233.
Efroimovich, S. Y. (1989). Sequential nonparametric estimation of a density, Theory Probab. Appl., 34, 228–239.
Hall, P. and Wand, M. P. (1988). Minimizing L 1 distance in nonparametric density estimation, J. Mullivar. Anal., 26, 59–88.
Isogai, E. (1987). The convergence rate of fixed-width sequential confidence intervals for a probability density function, Sequential Anal., 6, 55–69.
Isogai, E. (1988). A note on sequential density estimation, Sequential Anal., 7, 11–21.
Koronacki, J. and Wertz, D. (1988). A global stopping rule for recursive density estimators, J. Statist. Plann. Inference, 20, 946–954.
Martinsek, A. T. (1992). Using stopping rules to bound the mean integrated squared error in density estimation, Ann. Statist., 20, 797–806.
Nadaraya, E. A. (1974). On the integral mean square error of some nonparametric estimates for the density function, Theory Probab. Appl., 19, 133–141.
Neutra, R. R., Fienberg, S. E., Greenland, S. and Friedman, E. A. (1978). The effect of fetal monitoring on neonatal death rates, New England Journal of Medicine, 299, 324–326.
Park, B. U. and Turlach, B. A. (1992). Practical performance of several data driven bandwidth selectors, Comput. Statist., 7, 251–270.
Pinelis, I. (1990). To Devroye's estimates for the distribution of density estimators, Tech. Rep. (personal communication).
Prakasa Rao, B. L. S. (1983). Nonparametric Functional Estimation, Academic Press, Orlando.
Rice, J. (1975). Statiotical methods of use in analyzing sequences of carthquakes, Geophysical Journal of the Royal Astronomical Society, 42, 671–683.
Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27, 832–837.
Rosenblatt, M. (1971). Curve estimates, Ann. Math. Statist., 42, 1815–1842.
Sheather, S. J. (1992). The performance of six popular bandwidth selection methods on some real data sets, Comput. Statist., 7, 225–250.
Silverman, B. W. (1978). Weak and strong uniform consistency of the kernel estimate of a density and its derivatives, Ann. Statist., 6, 177–184.
Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, Chapman and Hall, London.
Stute, W. (1983). Sequential fixed-width confidence intervals for a nonparametric density function, Zeit. Wahrscheinlichkeitsth., 62, 113–123.
Wand, M. P. and Devroye, L. (1993). How easy is a given density to estimate?, Comput. Statist. Data Anal., 16, 311–323.
Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing, Chapman and Hall, London.
Wegman, E. J. and Davies, H. I. (1975). Sequential nonparametric density estimation, IEEE Transactions on Information Theory, 21, 619–628.
Yamato, H. (1971). Sequential estimate of a continuous probability density function and mode, Bull. Math. Statist., 14, 1–12.
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Kundu, S., Martinsek, A.T. Bounding the L1 Distance in Nonparametric Density Estimation. Annals of the Institute of Statistical Mathematics 49, 57–78 (1997). https://doi.org/10.1023/A:1003110605331
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DOI: https://doi.org/10.1023/A:1003110605331