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11
Nonparametric inference for additive models
 J. Amer. Statist. Assoc
, 2005
"... Additive models with backfitting algorithms are popular multivariate nonparametric fitting techniques. However, the inferences of the models have not been very well developed, due partially to the complexity of the backfitting estimators. There are few tools available to answer some important and fr ..."
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Cited by 31 (2 self)
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Additive models with backfitting algorithms are popular multivariate nonparametric fitting techniques. However, the inferences of the models have not been very well developed, due partially to the complexity of the backfitting estimators. There are few tools available to answer some important and frequently asked questions, such as whether a specific additive component is significant or admits a certain parametric form. In an attempt to address these issues, we extend the generalized likelihood ratio (GLR) tests to additive models, using the backfitting estimator. We demonstrate that under the null models, the newly proposed GLR statistics follow asymptotically rescaled chisquared distributions, with the scaling constants and the degrees of freedom independent of the nuisance parameters. This demonstrates that the Wilks phenomenon continues to hold under a variety of smoothing techniques and more relaxed models with unspecified error distributions. We further prove that the GLR tests are asymptotically optimal in terms of rates of convergence for nonparametric hypothesis testing. In addition, for testing a parametric additive model, we propose a bias corrected method to improve the performance of the GLR. The biascorrected test is shown to share the Wilks type of property. Simulations are conducted to demonstrate the Wilks phenomenon and the power of the proposed tests. A real example is used to illustrate the performance of the testing approach.
Generalized Likelihood Ratio Tests for Additive Models
"... Additive models with back tting algorithms are popular multivariate nonparametric tting techniques. However, the inferences of the models have not been much developed due partially to the complexity of the back tting estimators. There are few tools available to answer some important and freque ..."
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Cited by 4 (3 self)
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Additive models with back tting algorithms are popular multivariate nonparametric tting techniques. However, the inferences of the models have not been much developed due partially to the complexity of the back tting estimators. There are few tools available to answer some important and frequentlyasked questions, such as whether a speci c additive component is signi cant or admits a certain parametric form. In an attempt to address these issues, we extend the generalized likelihood ratio tests to additive models, using the back tting estimator. We demonstrate that under the null models the newly proposed generalized likelihood ratio statistics follow asymptotically rescaled distributions, with the scale constants and the degrees of freedom being independent of the nuisance parameters. This demonstrates that the Wilks phenomenon continues to hold under a variety of smoothing techniques and more relaxed models with unspeci ed error distributions. We further prove that the generalized likelihood ratio tests are asymptotically optimal in terms of rates of convergence for nonparametric hypothesis testing.
Feature Matching in Time Series Modelling
, 2010
"... Abstract: Using a time series model to mimic an observed time series has a long history. However, with regard to this objective, conventional estimation methods for discretetime dynamical models are frequently found to be wanting. In fact, they are characteristically misguided in at least two respe ..."
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Cited by 3 (0 self)
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Abstract: Using a time series model to mimic an observed time series has a long history. However, with regard to this objective, conventional estimation methods for discretetime dynamical models are frequently found to be wanting. In fact, they are characteristically misguided in at least two respects: (i) assuming that there is a true model; (ii) evaluating the efficacy of the estimation as if the postulated model is true. There are numerous examples of models, when fitted by conventional methods, that fail to capture some of the most basic global features of the data, such as cycles with good matching periods, singularities of spectral density functions (especially at the origin) and others. We argue that the shortcomings need not always be due to the model formulation but the inadequacy of the conventional fitting methods. After all, all models are wrong, but some are useful if they are fitted properly. The practical issue becomes one of how to best fit the model to data. Thus, in the absence of a true model, we prefer an alternative approach to conventional model fitting that typically involves onestepahead prediction
Score based goodnessoffit tests for time series. submitted for publication
, 2010
"... Abstract: This paper studies a class of tests useful for testing goodness of fit of a wide variety of time series models. These tests are based on a class of empirical processes marked by certain scores. Major advantages of these tests are that they are easy to implement, require only weak conditio ..."
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Cited by 1 (0 self)
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Abstract: This paper studies a class of tests useful for testing goodness of fit of a wide variety of time series models. These tests are based on a class of empirical processes marked by certain scores. Major advantages of these tests are that they are easy to implement, require only weak conditions that are usually satisfied in practical applications, the relevant critical values are readily available without bootstrap, and are more powerful than the LjungBox test, the LiMak test and the KoulStute test in all the cases we have tried. A comparison with the FanZhang test is included. We also extend the class of tests to include scorelike statistics.
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"... TEST manuscript No. (will be inserted by the editor) Nonparametric inference with generalized likelihood ratio tests ⋆ ..."
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TEST manuscript No. (will be inserted by the editor) Nonparametric inference with generalized likelihood ratio tests ⋆
Supplementary materials for this article are available at
"... We develop a specification test for the transition density of a discretely sampled continuoustime jumpdiffusion process, based on a comparison of a nonparametric estimate of the transition density or distribution function with their corresponding parametric counterparts assumed by the null hypothe ..."
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We develop a specification test for the transition density of a discretely sampled continuoustime jumpdiffusion process, based on a comparison of a nonparametric estimate of the transition density or distribution function with their corresponding parametric counterparts assumed by the null hypothesis. As a special case, our method applies to pure diffusions. We provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. We investigate the finitesample properties through simulations and compare them with those of other tests. This article has supplementary material online.
Generalized Likelihood Ratio Tests for FunctionalCoecient Partially Linear Regression Model
"... Key words: Ecient estimate; Functionalcoecient partially linear regression model; Generalized likelihood ..."
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Key words: Ecient estimate; Functionalcoecient partially linear regression model; Generalized likelihood
doi:http://dx.doi.org/10.5705/ss.2009.090 SCORE BASED GOODNESSOFFIT TESTS FOR TIME SERIES
"... Abstract: This paper studies a class of tests useful for testing goodness of fit of a wide variety of time series models. These tests are based on a class of empirical processes marked by certain scores. Major advantages of these tests are that they are easy to implement, require only weak conditio ..."
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Abstract: This paper studies a class of tests useful for testing goodness of fit of a wide variety of time series models. These tests are based on a class of empirical processes marked by certain scores. Major advantages of these tests are that they are easy to implement, require only weak conditions that are usually satisfied in practical applications, the relevant critical values are readily available without bootstrap, and are more powerful than the LjungBox test, the LiMak test and the KoulStute test in all the cases we have tried. A comparison with the FanZhang test is included. We also extend the class of tests to include scorelike statistics. Key words and phrases: Empirical process, goodnessoffit test, nonlinear time series, score, time series models. 1.
Local Whittle Likelihood Estimators and Tests for nonGaussian Linear Processes
"... In this paper, we propose a local Whittle likelihood estimator for spectral densities of nonGaussian linear processes and a local Whittle likelihood ratio test statistic for the problem of testing whether the spectral density of a nonGaussian stationary linear process belongs to a parametric famil ..."
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In this paper, we propose a local Whittle likelihood estimator for spectral densities of nonGaussian linear processes and a local Whittle likelihood ratio test statistic for the problem of testing whether the spectral density of a nonGaussian stationary linear process belongs to a parametric family or not. Introducing a local Whittle likelihood of a spectral density fθ(λ) around λ, we propose a local estimator θ ̂ = θ̂(λ) of θ which minimizes the local Whittle likelihood around λ, and use fθ̂(λ)(λ) as an estimator of the true spectral density. For the testing problem, we use a local Whittle likelihood ratio test statistic based on the local Whittle likelihood estimator. The asymptotics of these statistics are elucidated. It is shown that their asymototic distributions do not depend on nonGaussianity of the processes. Because our models include nonlinear stationary time series models, we can apply the results to stationary GARCH processes. Advantages of the proposed estimator and test are demonstrated by a few simulated numerical examples.