This survey intends to collect the developments on Goodness-of-Fit for regression models during the last 20 years, from the very first origins with the proposals based on the idea of the tests for density and distribution, until the most recent advances for complex data and models. Far from being exhaustive, the contents in this paper are focused on two main classes of tests statistics: smoothing-based tests (kernel-based) and tests based on empirical regression processes, although other tests based on Maximum Likelihood ideas will be also considered. Starting from the simplest case of testing a parametric family for the regression curves, the contributions in this field provide also testing procedures in semiparametric, nonparametric, and functional models, dealing also with more complex settings, as those ones involving dependent or incomplete data.

https://www.eco.uc3m.es/temp/Gonzalez%203.pdf

An updated review of Goodness-of-Fit tests for regression models

The 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 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. Copyright 2011, Oxford University Press.

/pdf/testing-parametric-assumptions-of-trends-of-a-nonstationary-2su2ckegp7.pdf

Testing parametric assumptions of trends of a nonstationary time series

We discuss estimation and testing of hypotheses in a partial linear regression model, that is, a regression model where the regression function is the sum of a linear and a nonparametric component. We focus on the case where the covariables and the random noise do not necessarily have summable autocovariance functions, and the estimators and test statistics are based on kernel smoothing. We obtain the bias, variance and asymptotic distribution of both estimators for the parametric and nonparametric parts, as well as the asymptotic distributions of the statistics used, both under the null hypothesis and local alternatives. We thus generalize the results of Speckman and of Beran and Ghosh to the case of general structures for the autocovariance function and complete the results of Gonzalez-Manteiga and Vilar-Fernandez to the case of a partial linear regression model. Simulations and a real data example provide promising results for our tests.

/pdf/estimation-and-testing-in-a-partial-linear-regression-model-3ozsicd65h.pdf

Estimation and testing in a partial linear regression model under long-memory dependence

In this paper, we propose a new testing procedure for detecting smooth (non)monotonic trends embedded into a linear noise that possibly does not degenerate to a finite-dimensional representation or into a conditionally heteroscedastic (autoregressive conditionally heteroscedastic/generalized autoregressive conditionally heteroscedastic (ARCH/GARCH)) noise. The proposed nonparametric trend test is local regression-based, and we develop a flexible and computationally efficient hybrid bootstrap procedure to approximate its finite sample behavior. Because the proposed trend test does not assume prior knowledge on the dependence structure and probability distribution of the observed process, the new testing procedure is fully data-driven and robust to misspecification of dependence structure and distributional assumptions, which is of particular importance for noisy environmental measurements. Moreover, because the proposed methodology allows to test for monotonic versus non-monotonic trends and hence, to assess existence of extremums in the hypothesized trend function, the developed approach may be also employed for preliminary detection of regime shifts and change points in the observed environmental data series. Our simulation studies indicate competitive performance of the proposed nonparametric procedure for detection of (non)monotonic trends against conventional trend tests. Copyright © 2013 John Wiley & Sons, Ltd.

On detecting non‐monotonic trends in environmental time series: a fusion of local regression and bootstrap

In this paper we describe some general methods for constructing goodness of fit tests in nonparametric regression models. Our main concern is the development of statisticial methodology for the assessment (validation) of specific parametric models M as they arise in various fields of applications. The fundamental idea which underlies all these methods is the investigation of certain goodness of fit statistics (which may depend on the particular problem and may be driven by different criteria) under the assumption that a specified model (which has to be validated) holds true as well as under a broad range of scenaria, where this assumption is violated. This is motivated by the fact that outcomes of tests for the classical hypothesis: "The model M holds true" (and their associated p values) bear various methodological flaws. Hence, our suggestion is always to accompany such a test by an analysis of the type II error, which is in goodness of fit problems often the more serious one. We give a careful description of the methodological aspects, the required asymptotic theory, and illustrate the main principles in the problem of testing model assumptions such as a specific parametric form or homoscedasticity in nonparametric regression models.

/pdf/some-methodological-aspects-of-validation-of-models-in-4uwhxpykda.pdf

W. Manteiga Gonzalez

Papers

Testing linear regression models using non-parametric regression estimators when errors are non-independent