The Annals of Statistics

Sequential confidence bands for densities

Adam T. Martinsek and Yi Xu

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This paper proposes a fully sequential procedure for constructing a fixed-width confidence band for an unknown density on a finite interval and shows the procedure has the desired coverage probability asymptotically as the width of the band approaches zero. The procedure is based on a result of Bickel and Rosenblatt. Its implementation in the sequential setting cannot be obtained using Anscombe's theorem, because the normalized maximal deviations between the kernel estimate and the true density are not uniformly continuous in probability (u.c.i.p.). Instead, we obtain a slightly weaker version of the u.c.i.p. property and a correspondingly stronger convergence property of the stopping rule. These together yield the desired results.

Article information

Ann. Statist., Volume 23, Number 6 (1995), 2218-2240.

First available in Project Euclid: 15 October 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L12: Sequential estimation
Secondary: 62G07: Density estimation 62G20: Asymptotic properties

Density estimation confidence band sequential estimation stopping rule uniform continuity in probability


Xu, Yi; Martinsek, Adam T. Sequential confidence bands for densities. Ann. Statist. 23 (1995), no. 6, 2218--2240. doi:10.1214/aos/1034713654.

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  • ANSCOMBE, F. 1952. Large sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 48 600 607. Z.
  • BICKEL, P. J. and ROSENBLATT, M. 1973. On some global measures of the deviations of the density function estimates. Ann. Statist. 1 1071 1095. Z.
  • CARROLL, R. J. 1976. On sequential density estimation. Z. Wahrsch. Verw. Gebiete 36 137 151. Z.
  • CHOW, Y. S., HSIUNG, C. A. and YU, K. F. 1980. Limit theorems for a positively drifting process and its related first passage times. Bull. Inst. Math. Acad. Sinica 8 141 172. Z.
  • CHOW, Y. S. and ROBBINS, H. 1965. Asy mptotic theory of fixed width confidence intervals for the mean. Ann. Math. Statist. 36 457 462. Z.
  • CSORGO, M. and REVESZ, P. 1979. How big are the increments of a Wiener process? Ann. ¨ ´ ´ Probab. 7 731 737. Z.
  • ISOGAI, E. 1981. Stopping rules for sequential density estimation. Bulletin of Mathematical Statistics 19 53 67. Z.
  • ISOGAI, E. 1987. The convergence rate of fixed-width sequential confidence intervals for a probability density function. Sequential Anal. 6 55 69. Z.
  • ISOGAI, E. 1988. A note on sequential density estimation. Sequential Anal. 7 11 21. Z.
  • IZENMAN, A. 1991. Recent developments in nonparametric density estimation. J. Amer. Statist. Assoc. 86 205 224. Z.
  • KOMLOS, J., MAJOR, P. and TUSNADY, G. 1975. An approximation of partial sums of independent ´ ´ RV's and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111 131. Z.
  • MARTINSEK, A. T. 1983. Second order approximation to the risk of a sequential procedure. Ann. Statist. 11 827 836. Z.
  • MARTINSEK, A. T. 1993. Fixed width confidence bands for density functions. Sequential Anal. 12 169 177. Z.
  • PARZEN, E. 1962. On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065 1076. Z.
  • ROBBINS, H. 1959. Sequential estimation of the mean of a normal population. In Probability Z. and Statistics. The Harald Cramer Volume U. Grenander, ed. 235 245. Almquist ´ and Wiksell, Stockholm. Z.
  • ROSENBLATT, M. 1956. Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 832 837. Z.
  • STUTE, W. 1982. The oscillation behavior of empirical processes. Ann. Probab. 10 86 107. Z.
  • STUTE, W. 1983. Sequential fixed-width confidence intervals for a nonparametric density function. Z. Wahrsch. Verw. Gebiete 62 113 123. Z.
  • WOODROOFE, M. 1982. Nonlinear Renewal Theory in Sequential Analy sis. SIAM, Philadelphia. Z.
  • XU, Y. and MARTINSEK, A. T. 1994. Sequential confidence bands for densities. Technical report, Dept. Statistics, Univ. Illinois. Z.
  • ZHENG, Z. 1988. Strong uniform consistency for density estimator from randomly censored data. Chinese Ann. Math. Ser. B 9 167 175.