On bootstrapping l 2-statistics in density testing. Stat Probab Lett

Sonderforschungsbereich 373, Humboldt-Universität zu Berlin, Spandauer Straße 1, D-10178 Berlin, Germany
Statistics [?] Probability Letters (Impact Factor: 0.6). 11/2000; 50(2):137-147. DOI: 10.1016/S0167-7152(00)00091-2


We consider non-parametric tests for checking parametric hypotheses about the stationary density of weakly dependent observations. The test statistic is based on the L2-distance between a non-parametric and a smoothed version of a parametric estimate of the stationary density. Since this statistic behaves asymptotically as in the case of independent observations an i.i.d.-type bootstrap to determine the critical value for the test is proposed.

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Available from: Efstathios Paparoditis, Jul 08, 2014
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    • "In particular, Bickel and Rosenblatt (1973) proposed a test based on the L 2 -distance between a non-parametric kernel density estimator and a smoothed version of a parametric fit. Their method was extended by Neumann and Paparoditis (2000) and Bachmann and Dette (2005) to test parametric hypotheses about the marginal distribution of stationary processes. Tests for the parametric form of the density in deconvolution problems have been considered in Holzmann et al. (2007) and Butucea (2007). "
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    ABSTRACT: We consider the problem of testing parametric assumptions in an inverse regression model with a convolution-type operator. An L^2-type goodness-of-fit test is proposed which compares the distance between a parametric and a nonparametric estimate of the regression function. Asymptotic normality of the corresponding test statistic is shown under the null hypothesis and under a general nonparametric alternative with different rates of convergence in both cases. The feasibility of the proposed test is demonstrated by means of a small simulation study. In particular, the power of the test against certain types of alternative is investigated.
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    • "For simulation study they use h = 1/3n −1/5 and h = (ˆ σ 2 /n) 1/5 , respectively. Finally, we also would like to mention that in the case of weakly dependent absolutely regular or β-mixing processes in [4] a fixed bandwidth h = 0.03 has been chosen in the simulation study. "
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    ABSTRACT: The aim of this paper is to analyze the Bickel–Rosenblatt test for simple hypothesis in case of weakly dependent data. Although the test has nice theoretical properties, it is not clear how to implement it in practice. Choosing different band-width sequences first we analyze percentage rejections of the test statistic under H0 by some empirical simulation analysis. This can serve as an approximate rule for choosing the bandwidth in case of simple hypothesis for practical implementation of the test. In the recent paper [12] a version of Neyman goodness-of-fit test was established for weakly dependent data in the case of simple hypotheses. In this paper we also aim to compare and discuss the applicability of these tests for both independent and dependent observations.
    Preview · Article · Jun 2012 · Mathematical Modelling and Analysis
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    • "Surveys of those results are given by Jose, Lishamol and Sreekumar [17] as well as by Block, Langberg and Stoffer [4]. Goodness-of-fit tests for dependent data based on the L 2 -distance between the non-parametric estimate with vanishing bandwidth and a smoothed version of a parametric estimate of the density were considered by Fan and Ullah [13] as well as by Neumann and Paparoditis [20]. Ghosh and Huang [15] showed that L 2 -tests based on statistics with fixed kernels have more power against Pitman alternatives than kernel-based L 2 -tests with vanishing bandwidth. "
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    ABSTRACT: This article proposes two consistent hypothesis tests of L 2 -type for weakly dependent ob-servations based on the empirical characteristic function. We consider a symmetry test and a goodness-of-fit test for the marginal distribution of a time series. Since the asymptotic distributions of the test statistics depend on unknown parameters in a complicated way, we suggest to apply certain parametric bootstrap methods in order to determine critical values of the tests.
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