# zbMATH — the first resource for mathematics

Sequential confidence bands for densities. (English) Zbl 0897.62093
Summary: 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 P. J. Bickel and M. Rosenblatt [ibid. 1, 1071-1095 (1973; Zbl 0275.62033)]. Its implementation in the sequential setting cannot be obtained using Anscombe’s theorem [F. Anscombe, Proc. Camb. Philos. Soc. 48, 600-607 (1952; Zbl 0047.13401)], 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.

##### MSC:
 62L12 Sequential estimation 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference
Full Text:
##### References:
 [1] ANSCOMBE, F. 1952. Large sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 48 600 607. Z. · Zbl 0047.13401 [2] 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. · Zbl 0275.62033 [3] CARROLL, R. J. 1976. On sequential density estimation. Z. Wahrsch. Verw. Gebiete 36 137 151. Z. · Zbl 0321.62091 [4] 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. · Zbl 0441.60075 [5] 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. · Zbl 0142.15601 [6] CSORGO, M. and REVESZ, P. 1979. How big are the increments of a Wiener process? Ann. \" \' Ṕrobab. 7 731 737. Z. · Zbl 0412.60038 [7] ISOGAI, E. 1981. Stopping rules for sequential density estimation. Bulletin of Mathematical Statistics 19 53 67. Z. · Zbl 0474.62078 [8] ISOGAI, E. 1987. The convergence rate of fixed-width sequential confidence intervals for a probability density function. Sequential Anal. 6 55 69. Z. · Zbl 0622.62084 [9] ISOGAI, E. 1988. A note on sequential density estimation. Sequential Anal. 7 11 21. Z. · Zbl 0639.62079 [10] IZENMAN, A. 1991. Recent developments in nonparametric density estimation. J. Amer. Statist. Assoc. 86 205 224. Z. JSTOR: · Zbl 0734.62040 [11] KOMLOS, J., MAJOR, P. and TUSNADY, G. 1975. An approximation of partial sums of independent \' ŔV’s and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111 131. Z. · Zbl 0308.60029 [12] MARTINSEK, A. T. 1983. Second order approximation to the risk of a sequential procedure. Ann. Statist. 11 827 836. Z. · Zbl 0521.62067 [13] MARTINSEK, A. T. 1993. Fixed width confidence bands for density functions. Sequential Anal. 12 169 177. Z. · Zbl 0773.62060 [14] PARZEN, E. 1962. On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065 1076. Z. · Zbl 0116.11302 [15] 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 ánd Wiksell, Stockholm. Z. · Zbl 0095.13005 [16] ROSENBLATT, M. 1956. Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 832 837. Z. · Zbl 0073.14602 [17] STUTE, W. 1982. The oscillation behavior of empirical processes. Ann. Probab. 10 86 107. Z. · Zbl 0489.60038 [18] STUTE, W. 1983. Sequential fixed-width confidence intervals for a nonparametric density function. Z. Wahrsch. Verw. Gebiete 62 113 123. Z. · Zbl 0488.62032 [19] WOODROOFE, M. 1982. Nonlinear Renewal Theory in Sequential Analy sis. SIAM, Philadelphia. Z. · Zbl 0487.62062 [20] XU, Y. and MARTINSEK, A. T. 1994. Sequential confidence bands for densities. Technical report, Dept. Statistics, Univ. Illinois. Z. · Zbl 0897.62093 [21] ZHENG, Z. 1988. Strong uniform consistency for density estimator from randomly censored data. Chinese Ann. Math. Ser. B 9 167 175. · Zbl 0651.62030 [22] CHAMPAIGN, ILLINOIS 61820
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.