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78
Generalized Likelihood Ratio Statistics And Wilks Phenomenon
, 2000
"... this paper. We introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks phenomenon is unveiled. We demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are ..."
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Cited by 78 (22 self)
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this paper. We introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks phenomenon is unveiled. We demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow
Consistent Specification Testing With Nuisance Parameters Present Only Under The Alternative
, 1995
"... . The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly ex ..."
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Cited by 55 (10 self)
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. The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. How and why the nuisance parameter approach works and how it can be extended bears closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banachvalued random elements and applying the Banach Central Limit Theorem and Law of Iterated Logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted. 1. Introduction In testing whether or not a parametric statistical model is correctly specified, there are a number of apparently distinct approaches one might take. T...
Jump and Sharp Cusp Detection By Wavelets
 Recent Research on the Nitinol Alloys and Their Potential Application in Ocean Engineering,” Ocean Engineering
, 1995
"... this paper we consider only jump and sharp cusp detection in one dimension. There is a great amount of statistical literature on changepoints (Basseville, 1988; Basseville & Nikiforov, 1993). Wahba (1984) and Engle, Granger, Rice &Weiss (1986) were the first to estimate curves with discontinuities ..."
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Cited by 32 (2 self)
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this paper we consider only jump and sharp cusp detection in one dimension. There is a great amount of statistical literature on changepoints (Basseville, 1988; Basseville & Nikiforov, 1993). Wahba (1984) and Engle, Granger, Rice &Weiss (1986) were the first to estimate curves with discontinuities in derivatives, assuming the locations of the jumps are known. McDonald & Owen (1986) proposed an algorithm to compute estimates of regression functions when discontinuities are present. Yin (1988) used onesided moving averages to find the locations of jumps in a function. Lombard (1988) described jump detection by Fourier analysis. Cline & Hart (1991) considered detecting jumps in derivatives. Muller (1992) estimated the location of a jump and its jump size by boundary kernels. Eubank & Speckman (1994) used a semiparametric approach to detect the discontinuities in derivatives of regression functions. Hall & Titterington (1992) studied edgepreserving and peakpreserving by smoothing. Grossmann (1986) and Mallat & Hwang (1992) used wavelet transformation to detect singularities and edges in computer images. The paper is organized as follows. Sections 2 and 3 introduce the white noise model and wavelet transformation, respectively. Testing hypotheses and estimation are considered in Sections 4 and 5. Section 6 discusses implementation of the detection in practice. Simulation results and an application to a real example are reported in this section. Concluding remarks are given in Section 7. Proofs are collected in the Appendix. 2. THE WHITE NOISE MODEL
Inference of Trends in Time Series
 J. the Royal Statistical Society: Series B (Statistical Methodology
, 2007
"... Summary. We consider statistical inference of trends in mean nonstationary models. A test statistic is proposed for the existence of structural breaks in trends. On the basis of a strong invariance principle of stationary processes, we construct simultaneous confidence bands with asymptotically cor ..."
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Cited by 14 (4 self)
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Summary. We consider statistical inference of trends in mean nonstationary models. A test statistic is proposed for the existence of structural breaks in trends. On the basis of a strong invariance principle of stationary processes, we construct simultaneous confidence bands with asymptotically correct nominal coverage probabilities. The results are applied to global warming temperature data and Nile river flow data. Our confidence band of the trend of the global warming temperature series supports the claim that the trend is increasing over the last 150 years.
Polynomial spline confidence bands for regression curves
, 2007
"... Abstract: Asymptotically exact and conservative confidence bands are obtained for a nonparametric regression function, using piecewise constant and piecewise linear spline estimation, respectively. Compared to the pointwise confidence interval of Huang (2003), the confidence bands are inflated by a ..."
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Cited by 10 (6 self)
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Abstract: Asymptotically exact and conservative confidence bands are obtained for a nonparametric regression function, using piecewise constant and piecewise linear spline estimation, respectively. Compared to the pointwise confidence interval of Huang (2003), the confidence bands are inflated by a factor proportional to {log (n)} 1/2, with the same width order as the NadarayaWatson bands of Härdle (1989), and the local polynomial bands of Xia (1998) and Claeskens and Van Keilegom (2003). Simulation experiments corroborate the asymptotic theory. The linear spline band has been used to identify an appropriate polynomial trend for fossil data.
Confidence bands in nonparametric time series regression
, 2008
"... We consider nonparametric estimation of mean regression and conditional variance (or volatility) functions in nonlinear stochastic regression models. Simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct. The imposed dependence structure ..."
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Cited by 10 (3 self)
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We consider nonparametric estimation of mean regression and conditional variance (or volatility) functions in nonlinear stochastic regression models. Simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct. The imposed dependence structure allows applications in many linear and nonlinear autoregressive processes. The results are applied to the S&P 500 Index data. 1. Introduction. There
Testing Monotonicity Of Regression
, 1998
"... this article, we study this problem and construct asymptotically valid tests. Our test statistics are suitable functionals of a stochastic process which may be viewed as a local version of Kendall's tau statistic and have simple natural interpretations. The process involved is a degreetwo Uprocess, ..."
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Cited by 9 (0 self)
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this article, we study this problem and construct asymptotically valid tests. Our test statistics are suitable functionals of a stochastic process which may be viewed as a local version of Kendall's tau statistic and have simple natural interpretations. The process involved is a degreetwo Uprocess, as in Nolan and Pollard (1987). The asymptotic behaviour of the test statistics are studied in three major steps: Approximation of the Uprocess by the empirical process defined by the H'ajek projection, strong approximation of the empirical process by a Gaussian process and finally the extreme value theory for stationary Gaussian processes. The paper is organized as follows. In Section 2, we introduce two different types of test statistics. We also formally describe the model and the hypothesis and explain the notation and regularity conditions in this section. In Section 3, we investigate the asymptotic behaviour of the Uprocess and establish the Gaussian process approximation. Section 4 is devoted to the study of the limiting distribution of the first test statistics using the extreme value theory for stationary Gaussian processes and the results of Section 3. In Section 5, we show that this test is consistent against all alternatives and also determine the minimal rate so that alternatives further apart than this rate can be effectively tested. The second test statistic is studied in Section 6. Technical proofs are presented in Section 7 and the appendix. 2. The Test Statistics
2008): “On Deconvolution with Repeated Measurements
 Annals of Statistics
"... In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without it. However, if additional data are available, then it is p ..."
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Cited by 9 (2 self)
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In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without it. However, if additional data are available, then it is possible to estimate consistently the unknown error density. Data are seldom available directly on the transformation, but repeated, or replicated, measurements increasingly are becoming available. Such data consist of “intrinsic ” values that are measured several times, with errors that are generally independent. Working in this setting we treat the nonparametric deconvolution problems of density estimation with observation errors, and regression with errors in variables. We show that, even if the number of repeated measurements is quite small, it is possible for modified kernel estimators to achieve the same level of performance they would if the error distribution were known. Indeed, density and regression estimators can be constructed from replicated data so that they have the same firstorder properties as conventional estimators in the knownerror case, without any replication, but with sample size equal to the sum of the numbers of replicates. Practical methods for constructing estimators with these properties are suggested, involving empirical rules for smoothingparameter choice. 1. Introduction. Statistical
Goodnessoffit tests via phidivergences
, 2006
"... A unified family of goodnessoffit tests based on φdivergences is introduced and studied. The new family of test statistics Sn(s) includes both the supremum version of the Anderson–Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47–59] as special cas ..."
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Cited by 8 (1 self)
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A unified family of goodnessoffit tests based on φdivergences is introduced and studied. The new family of test statistics Sn(s) includes both the supremum version of the Anderson–Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47–59] as special cases (s = 2 and s = 1, resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk