Summary
The limiting joint distribution of the location and size of the maximum deviation between the historgram and the underlying density is derived. For smooth densities, the location and size of the maximum are asymptotically independent. The size has a limiting double-exponential distribution and the location has a limiting normal distribution.
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Bickel, P.J., Rosenblatt, M.: On some global measures of the deviations of density function estimates. Ann. Statist. 1, 1071–1095 (1973)
Diaconis, P., Freedman, D.: The distribution of the mode of an empirical historgram. Stanford Technical Report No. 105 (1979)
Freedman, D.: Another note on the Borel-Cantelli lemma and the strong law, with the Poisson approximation as a by-product. Ann. Probab. 1, 920–925 (1973)
Freedman, D.: On the maximum of scaled multinomial variables. Pacific J. Math. [To appear (1981)]
Freedman, D., Diaconis, P.: On the maximum difference between the empirical and expected histograms for sums. Pacific J. Math. [To appear (1981a)]
Freedman, D., Diaconis, P.: On the maximum difference between the empirical and expected histograms for sums: Part II. Pacific J. Math. [To appear (1981b)]
Freedman, D., Diaconis, P.: On the histogram as a density estimator: L z theory. Z. Wahrscheinlichkeitstheorie verw. Gebiete [To appear (1981c)]
Freedman, D., Diaconis, P.: On the mode of an empirical historgram for sums. Pacific J. Math. [To appear (1981d)]
Hasminsky, R.: On the lower bounds for risks of nonparametric estimates of density functions in the uniform metrics. Probability Theory and its Applications 23, 824–828 (1978)
Reiss, R.D.: Approximate distributions of the maximum deviation of histograms. Preprint in statistics. Univ. Köln (1976)
Revesz, P.: On empirical density function. Periodica Math. Hungarica 2, 85–110 (1972)
Serfling, R.: Properties and applications of metrics on nonparametric density estimators. Johns Hopkins University Technical Report No. 432 (1980)
Smirnov, N.V.: Approximation of distribution laws of random variables by empirical data. Uspehi. Mat. Nauk. 10, 179–206 (in Russian) (1944)
Tumanjan, S.H.: On the maximal deviation of the empirical density of a distribution. Nauk. Trudy. Erevansk. Univ. 48, 3–48 (1955)
Woodroofe, M.: On the maximum deviation of the sample density. Ann. Math. Statist., 38, 475–481 (1967)
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Research partially supported by NSF grant MCS-80-02535
Research partially supported by NSF grant MCS-77-16974
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Freedman, D., Diaconis, P. On the maximum deviation between the histogram and the underlying density. Z. Wahrscheinlichkeitstheorie verw. Gebiete 58, 139–167 (1981). https://doi.org/10.1007/BF00531558
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DOI: https://doi.org/10.1007/BF00531558