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Dümbgen, Lutz

New goodness-of-fit tests and their application to nonparametric confidence sets. (English) Zbl 0930.62034

Ann. Stat. 26, No. 1, 288-314 (1998).
Summary: Suppose one observes a process \(V\) on the unit interval, where \(dV=f_0 +dW\) with an unknown parameter \(f_0\in L_1 [0,1]\) and standard Brownian motion \(W\). We propose a particular test of one-point hypotheses about \(f_0\) which is based on suitably standardized increments of \(V\). This test is shown to have desirable consistency properties if, for instance, \(f_0\) is restricted to various Hölder classes of functions. The test is mimicked in the context of nonparametric density estimation, nonparametric regression and interval-censored data. Under shape restrictions on the parameter, such as monotonicity or convexity, we obtain confidence sets for \(f_0\) adapting to its unknown smoothness.

Cited in 22 Documents

MSC:

62G07 Density estimation
62M02 Markov processes: hypothesis testing
62G10 Nonparametric hypothesis testing
62G08 Nonparametric regression and quantile regression
62G15 Nonparametric tolerance and confidence regions

Keywords:

adaptivity; conditional median; convexity; distribution-free; interval censoring; modality; monotonicity; signs of residuals; spacings
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[1] Anderson, T. W. (1955). The integral of a sy mmetric unimodal function over a sy mmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 170-176. JSTOR: · Zbl 0066.37402 · doi:10.2307/2032333
[2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[3] Brown, L. D. and Low, M. (1996). Asy mptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398. · Zbl 0867.62022 · doi:10.1214/aos/1032181159
[4] Davies, P. L. (1995). Data features. Statist. Neerlandica 49 185-245. · Zbl 0831.62001 · doi:10.1111/j.1467-9574.1995.tb01464.x
[5] Donoho, D. L. (1988). One-sided inference about functionals of a density. Ann. Statist. 16 1390- 1420. · Zbl 0665.62040 · doi:10.1214/aos/1176351045
[6] Dudley, R. M. (1978). Central limit theorems for empirical measures. Ann. Probab. 6 899-929. · Zbl 0404.60016 · doi:10.1214/aop/1176995384
[7] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimators. Birkhäuser, Basel. · Zbl 0757.62017
[8] Hengartner, N. W. and Stark, P. B. (1995). Finite-sample confidence envelopes for shaperestricted densities. Ann. Statist. 23 525-550. · Zbl 0828.62043 · doi:10.1214/aos/1176324534
[9] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30. JSTOR: · Zbl 0127.10602 · doi:10.2307/2282952
[10] Ingster, Y. I. (1993). Asy mptotically minimax hy pothesis testing for nonparametric alternatives I-III. Math. Methods Statist. 2 85-114, 171-189, 249-268. · Zbl 0798.62057
[11] Khas’minskii, R. Z. (1978). A lower bound on the risks of nonparametric estimates of densities in the uniform metric. Theory Probab. Appl. 23 794-798. · Zbl 0449.62032 · doi:10.1137/1123095
[12] Khas’minskii, R. Z. (1979). Lower bounds for the risk of nonparametric estimates of the mode. In Contributions to Statistics, Jaroslav Hájek Memorial Volume (J. Jureckova, ed.) 91-97. Academia, Prague.
[13] Low, M. G. (1997). Personal communication.
[14] Mammen, E. (1991). Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19 741-759. · Zbl 0737.62039 · doi:10.1214/aos/1176348118
[15] Marron, J. S. and Tsy bakov, A. B. (1995). Visual error criteria for qualitative smoothing. J. Amer. Statist. Assoc. 90 499-507. JSTOR: · Zbl 0826.62026 · doi:10.2307/2291060
[16] M üller, D. W. (1991). Quantile regression. Preprint 624, Univ. Heidelberg, Sonderforschungsbereich 123.
[17] Nussbaum, M. (1996). Asy mptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399-2430. · Zbl 0867.62035 · doi:10.1214/aos/1032181160
[18] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045
[19] Romano, J. P. (1988). On weak convergence and optimality of kernel density estimates of the mode. Ann. Statist. 16 629-647. · Zbl 0658.62053 · doi:10.1214/aos/1176350824
[20] Sawitzki, G. (1996). Extensible statistical software: on a voy age to Oberon. J. Comput. Graph. Statist. 5 263-283.
[21] Shorack, G. R. and Wellner, J. A. (1982). Limit theorems and inequalities for the uniform empirical process indexed by intervals. Ann. Probab. 10 639-652. · Zbl 0497.60026 · doi:10.1214/aop/1176993773
[22] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365
[23] Spokoiny, V. G. (1996). Adaptive hy pothesis testing using wavelets. Ann. Statist. 24 2477-2498. · Zbl 0898.62056 · doi:10.1214/aos/1032181163
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