A Review on Dimension-Reduction Based Tests For Regressions

Curse of dimensionality is a big obstacle for constructing efficient goodness-of-fit tests for regression models with large or moderate number of covariates. To alleviate this difficulty, numerous efforts have been devoted in the last two decades. This review intends to collect and comment on the developments in this aspect. To make the paper self-contained, basic ideas on goodness-of-fit testing for regression models are also briefly reviewed, and the main classes of methods and their advantages and disadvantages are presented. Further, the difficulty caused by the dimensionality (number of covariates) is then discussed. The relevant dimension reduction methodologies are presented. Further, as a dedication to Stute’s 70th birthday, we also include a section to summarize his great contributions other than the results in dimension reduction-based tests.

[1]  Winfried Stute,et al.  Bootstrap Approximations in Model Checks for Regression , 1998 .

[2]  Quang Vuong,et al.  NONPARAMETRIC SIGNIFICANCE TESTING , 2000, Econometric Theory.

[3]  Wenceslao González Manteiga,et al.  Significance testing in nonparametric regression based on the bootstrap , 2001 .

[4]  Holger Dette,et al.  A power comparison between nonparametric regression tests , 2004 .

[5]  Testing for Differences Between Conditional Means in a Time Series Context , 2004 .

[6]  Wei Biao Wu,et al.  Testing parametric assumptions of trends of a nonstationary time series , 2011 .

[7]  H. Koul,et al.  Minimum distance regression model checking with Berkson measurement errors , 2009, 0902.4827.

[8]  Hira L. Koul,et al.  Nonparametric model checks for time series , 1999 .

[9]  H. Tong,et al.  Article: 2 , 2002, European Financial Services Law.

[10]  H. Bierens.  Consistent model specification tests , 1982 .

[11]  A Significance Test for Covariates in Nonparametric Regression , 2014, 1403.7063.

[12]  W. Stute,et al.  Nonparametric model checks in censored regression , 2000 .

[13]  Li-Xing Zhu,et al.  MODEL CHECKING OF DIMENSION-REDUCTION TYPE FOR REGRESSION , 2003 .

[14]  Jianqing Fan,et al.  Nonparametric inference with generalized likelihood ratio tests , 2007 .

[15]  V. Patilea,et al.  Nonparametric model checks of single-index assumptions , 2014, 1410.4934.

[16]  E. Khmaladze,et al.  Martingale Approach in the Theory of Goodness-of-Fit Tests , 1982 .

[17]  W. Stute.  Asymptotic Normality of Nearest Neighbor Regression Function Estimates , 1984 .

[18]  Herman J. Bierens,et al.  A consistent conditional moment test of functional form , 1990 .

[19]  Qi Li,et al.  CONSISTENT MODEL SPECIFICATION TESTS , 2000, Econometric Theory.

[20]  Marie Husková,et al.  Goodness-of-Fit Tests for Parametric Regression Models Based on Empirical Characteristic Functions , 2009, Kybernetika.

[21]  Marie Hušková,et al.  Tests for the error distribution in nonparametric possibly heteroscedastic regression models , 2010 .

[22]  Lixing Zhu,et al.  A projection-based adaptive-to-model test for regressions , 2016, 1604.01874.

[23]  V. Patilea,et al.  Powerful nonparametric checks for quantile regression , 2014, 1404.0216.

[24]  E. Khmaladze,et al.  Goodness-of-fit problem for errors in nonparametric regression: Distribution free approach , 2009, 0909.0170.

[25]  Lixing Zhu,et al.  Model checks for regression: an innovation process approach , 1998 .

[26]  L. J. Wei,et al.  A Lack-of-Fit Test for the Mean Function in a Generalized Linear Model , 1991 .

[27]  H. Koul,et al.  An Adaptive-to-Model Test for Parametric Single-Index Errors-in-Variables Models , 2016, Statistica Sinica.

[28]  Lixing Zhu,et al.  Heteroscedasticity testing for regression models: A dimension reduction-based model adaptive approach , 2015, Comput. Stat. Data Anal..

[29]  W. Stute,et al.  Nonparametric checks for single-index models , 2005, math/0507416.

[30]  Hua Liang,et al.  Integrated conditional moment test for partially linear single index models incorporating dimension-reduction , 2014 .

[31]  Lixing Zhu,et al.  Model checking for parametric single‐index models: a dimension reduction model‐adaptive approach , 2016 .

[32]  Lixing Zhu,et al.  Model Checks for Generalized Linear Models , 2002 .

[33]  Valentin Patilea,et al.  One for All and All for One: Regression Checks With Many Regressors , 2012 .

[34]  H. Koul,et al.  Minimum distance regression model checking , 2004 .

[35]  J. Zheng,et al.  A consistent test of functional form via nonparametric estimation techniques , 1996 .

[36]  Wenceslao González-Manteiga,et al.  An updated review of Goodness-of-Fit tests for regression models , 2013, TEST.

[37]  Holger Dette,et al.  A new test for the parametric form of the variance function in non‐parametric regression , 2007 .

[38]  Winfried Stute,et al.  Nonparametric comparison of regression functions , 2010, J. Multivar. Anal..

[39]  Léopold Simar,et al.  Nonparametric stochastic frontiers: A local maximum likelihood approach , 2007 .

[40]  Winfried Stute,et al.  Nonparametric model checks for regression , 1997 .

[41]  Lixing Zhu,et al.  Consistently determining the number of factors in multivariate volatility modelling , 2015 .

[42]  Lixing Zhu,et al.  Sufficient dimension reduction through discretization-expectation estimation , 2010 .

[43]  Lixing Zhu,et al.  An adaptive-to-model test for partially parametric single-index models , 2015, Stat. Comput..

[44]  Yingcun Xia,et al.  Model checking in regression via dimension reduction , 2009 .

[45]  E. Mammen,et al.  Comparing Nonparametric Versus Parametric Regression Fits , 1993 .

[46]  Valentin Patilea,et al.  Breaking the curse of dimensionality in nonparametric testing , 2008 .

[47]  Jianqing Fan,et al.  Generalized likelihood ratio statistics and Wilks phenomenon , 2001 .

[48]  Zhou Yu,et al.  On Partial Sufficient Dimension Reduction With Applications to Partially Linear Multi-Index Models , 2013 .

[49]  W. González-Manteiga,et al.  Rejoinder on: An updated review of Goodness-of-Fit tests for regression models , 2013, TEST.

[50]  Ingrid Keilegom,et al.  Goodness-of-fit tests in parametric regression based on the estimation of the error distribution , 2008 .

[51]  Yiguo Sun,et al.  A consistent nonparametric test of parametric regression functional form in fixed effects panel data models , 2014 .

[52]  W. Stute,et al.  NN goodness-of-fit tests for linear models , 1996 .

[53]  Lixing Zhu,et al.  Model diagnosis for parametric regression in high-dimensional spaces , 2008 .

[54]  Zhu Lixing,et al.  Dimension-reduction type test for linearity of a stochastic regression model , 1998 .

[55]  W. Stute,et al.  Model checks of higher order time series , 2006 .

[56]  Holger Dette,et al.  A consistent test for the functional form of a regression based on a difference of variance estimators , 1999 .

[57]  Hira L. Koul,et al.  Model diagnosis for setar time series , 2005 .

[58]  Jeffrey D. Hart,et al.  Nonparametric Smoothing and Lack-Of-Fit Tests , 1997 .

[59]  W. Stute,et al.  Bootstrap based goodness-of-fit-tests , 1993 .

[60]  Yanqin Fan,et al.  Consistent model specification tests : Omitted variables and semiparametric functional forms , 1996 .

[61]  J. Escanciano.  A CONSISTENT DIAGNOSTIC TEST FOR REGRESSION MODELS USING PROJECTIONS , 2006, Econometric Theory.

[62]  Wenceslao González-Manteiga,et al.  A lack-of-fit test for quantile regression models with high-dimensional covariates , 2015, Comput. Stat. Data Anal..

[63]  Lixing Zhu,et al.  Enhancements of Non‐parametric Generalized Likelihood Ratio Test: Bias Correction and Dimension Reduction , 2015, 1507.06090.