Estimating the Variance In Nonparametric Regression—What is a Reasonable Choice?

In this paper a new test for the parametric form of the variance function in the common nonparametric regression model is proposed which is applicable under very weak assumptions. The new test is based on an empirical process formed from pseudo residuals, for which weak convergence to a Gaussian process can be established. In the special case of testing for homoscedasticity the limiting process is essentially a Brownian bridge, such that critical values are easily available. The new procedure has three main advantages. First, in contrast to many other methods proposed in the literature, it does not depend directly on a smoothing parameter. Secondly, it can detect local alternatives converging to the null hypothesis at a rate n −1/2 . Thirdly, -in contrast to most of the currently available tests -it does not require strong smoothness assumptions regarding the regression and variance function. We also present a simulation study and compare the tests with the procedures which are currently available for this problem and require the same minimal assumptions.

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“…, d r for the calculation of the pseudo residuals. As pointed out by Dette, Munk and Wagner (1998), the sequence (d 0 , . .…”

Section: Testing For Homoscedasticitymentioning

confidence: 93%

A simple test for the parametric form of the variance function in nonparametric regression

Dette

Hetzler

2008

Ann Inst Stat Math

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“…where l denotes the difference operator applied l subsequent times (Rice, 1984;Hall et al, 1990;Seifert et al, 1993;Dette et al, 1998). This estimator was introduced in Vrugt et al (2005) and was shown to work well for daily and hourly discharge data.…”

Section: Case Studies: Hydrologic Modelingmentioning

confidence: 99%

Bridging the gap between GLUE and formal statistical approaches: approximate Bayesian computation

Sadegh

Vrugt

2013

Hydrol. Earth Syst. Sci.

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Abstract. In recent years, a strong debate has emerged in the hydrologic literature regarding how to properly treat nontraditional error residual distributions and quantify parameter and predictive uncertainty. Particularly, there is strong disagreement whether such uncertainty framework should have its roots within a proper statistical (Bayesian) context using Markov chain Monte Carlo (MCMC) simulation techniques, or whether such a framework should be based on a quite different philosophy and implement informal likelihood functions and simplistic search methods to summarize parameter and predictive distributions. This paper is a follow-up of our previous work published in Vrugt and Sadegh (2013) and demonstrates that approximate Bayesian computation (ABC) bridges the gap between formal and informal statistical model-data fitting approaches. The ABC methodology has recently emerged in the fields of biology and population genetics and relaxes the need for an explicit likelihood function in favor of one or multiple different summary statistics that measure the distance of each model simulation to the data. This paper further studies the theoretical and numerical equivalence of formal and informal Bayesian approaches using discharge and forcing data from different watersheds in the United States, in particular generalized likelihood uncertainty estimation (GLUE). We demonstrate that the limits of acceptability approach of GLUE is a special variant of ABC if each discharge observation of the calibration data set is used as a summary diagnostic.

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“…If the errors e i were independent and identically distributed, 185 then g(t) ≡ σ 2 e = e i 2 , the variance of e i . In this case we can apply difference-based variance estimators and there is a huge literature on the estimation of σ 2 e ; see Rice (1984), and Dette et al (1998), among others. In our setting, however, due to the dependence, the difference-based approach is generally invalid.…”

Section: ·6 Estimation Of Variance Functionsmentioning

confidence: 99%

Testing parametric assumptions of trends of a nonstationary time series

Zhang

2011

Biometrika

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SUMMARYThe paper considers testing whether the mean trend of a nonstationary time series is of certain parametric forms. A central limit theorem for the integrated squared error is derived, and with that a hypothesis-testing procedure is proposed. The method is illustrated in a simulation study, and is applied to assess the mean pattern of lifetime-maximum wind speeds of global tropical cyclones from 1981 to 2006. We also revisit the trend pattern in the central England temperature series.

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Estimating the Variance In Nonparametric Regression—What is a Reasonable Choice?

Cited by 120 publications

References 21 publications

A simple test for the parametric form of the variance function in nonparametric regression

A simple test for the parametric form of the variance function in nonparametric regression

Bridging the gap between GLUE and formal statistical approaches: approximate Bayesian computation

Testing parametric assumptions of trends of a nonstationary time series

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