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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
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[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
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