Nonparametric inference with generalized likelihood ratio tests

Abstract The advance of technology facilitates the collection of statistical data. Flexible and refined statistical models are widely sought in a large array of statistical problems. The question arises frequently whether or not a family of parametric or nonparametric models fit adequately the given data. In this paper we give a selective overview on nonparametric inferences using generalized likelihood ratio (GLR) statistics. We introduce generalized likelihood ratio statistics to test various null hypotheses against nonparametric alternatives. The trade-off between the flexibility of alternative models and the power of the statistical tests is emphasized. Well-established Wilks’ phenomena are discussed for a variety of semi- and non-parametric models, which sheds light on other research using GLR tests. A number of open topics worthy of further study are given in a discussion section.

[1]  Jianqing Fan,et al.  Nonparametric Transition-Based Tests for Jump Diffusions , 2005 .

[2]  R. Fisher,et al.  On the Mathematical Foundations of Theoretical Statistics , 1922 .

[3]  M. Nussbaum,et al.  Asymptotic equivalence for nonparametric regression , 2002 .

[4]  Stanley R. Johnson,et al.  Varying Coefficient Models , 1984 .

[5]  Lianfen Qian,et al.  Nonparametric Curve Estimation: Methods, Theory, and Applications , 1999, Technometrics.

[6]  N. Hjort,et al.  Nonparametric Density Estimation with a Parametric Start , 1995 .

[7]  Jianqing Fan.  On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems , 1991 .

[8]  G. Wahba.  Automatic Smoothing of the Log Periodogram , 1980 .

[9]  Efstathios Paparoditis,et al.  Spectral Density Based Goodness-of-Fit Tests for Time Series Models , 2000 .

[10]  Campbell R. Harvey,et al.  An Empirical Comparison of Alternative Models of the Short-Term Interest Rate , 1992 .

[11]  Joel L. Horowitz,et al.  An Adaptive, Rate-Optimal Test of a Parametric Model Against a Nonparametric Alternative , 1999 .

[12]  Jianqing Fan,et al.  Goodness-of-Fit Tests for Parametric Regression Models , 2001 .

[13]  T. Hettmansperger,et al.  Robust analysis of variance based upon a likelihood ratio criterion , 1980 .

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

[15]  H. T. Davis,et al.  Estimation of the Innovation Variance of a Stationary Time Series , 1968 .

[16]  Zongwu Cai,et al.  Adaptive varying‐coefficient linear models , 2000 .

[17]  V. LaRiccia,et al.  Asymptotic Comparison of Cramer-von Mises and Nonparametric Function Estimation Techniques for Testing Goodness-of-Fit , 1992 .

[18]  Jianqing Fan,et al.  Functional-Coefficient Regression Models for Nonlinear Time Series , 2000 .

[19]  V. Spokoiny,et al.  Minimax Nonparametric Hypothesis Testing: The Case of an Inhomogeneous Alternative , 1999 .

[20]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

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

[22]  Tadeusz Inglot,et al.  Asymptotic optimality of data-driven Neyman's tests for uniformity , 1996 .

[23]  Chunming Zhang.  Calibrating the Degrees of Freedom for Automatic Data Smoothing and Effective Curve Checking , 2003 .

[24]  Jianqing Fan,et al.  Efficient Estimation and Inferences for Varying-Coefficient Models , 2000 .

[25]  Young K. Truong,et al.  RATE OF CONVERGENCE FOR LOGSPLINE SPECTRAL DENSITY ESTIMATION , 1995 .

[26]  John W. Tukey,et al.  POWER SPECTRAL METHODS OF ANALYSIS AND THEIR APPLICATION TO PROBLEMS IN AIRPLANE DYNAMICS , 1959 .

[27]  P. Hall,et al.  Optimal Rates of Convergence for Deconvolving a Density , 1988 .

[28]  R. Tibshirani,et al.  Linear Smoothers and Additive Models , 1989 .

[29]  R. Tibshirani,et al.  Generalized Additive Models , 1991 .

[30]  Robert Tibshirani,et al.  An Introduction to the Bootstrap , 1994 .

[31]  Qi Li,et al.  Semiparametric Smooth Coefficient Models , 2002 .

[32]  J. Shao,et al.  The jackknife and bootstrap , 1996 .

[33]  R. Carroll,et al.  Semiparametric Regression for Clustered Data Using Generalized Estimating Equations , 2001 .

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

[35]  ADAPTIVE HYPOTHESIS TESTING USING WAVELETS BY V. G. SPOKOINY Weierstrass Institute for Applied Analysis and Stochastics and Institute for Information Transmission Problems , 1996 .

[36]  Jianqing Fan,et al.  A Reexamination of Diffusion Estimators With Applications to Financial Model Validation , 2003 .

[37]  Yudi Pawitan,et al.  Nonparametric Spectral Density Estimation Using Penalized Whittle Likelihood , 1994 .

[38]  J. S. Marron,et al.  Variable window width kernel estimates of probability densities , 1988 .

[39]  Spectral density estimation with amplitude modulation and outlier detection , 2004 .

[40]  Li Ping Yang,et al.  Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data , 1998 .

[41]  Jianqing Fan,et al.  Statistical Estimation in Varying-Coefficient Models , 1999 .

[42]  Zhiliang Ying,et al.  Semiparametric and Nonparametric Regression Analysis of Longitudinal Data , 2001 .

[43]  R. Wolpert,et al.  Likelihood Principle , 2022, The SAGE Encyclopedia of Research Design.

[44]  Jianqing Fan,et al.  Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems , 1996 .

[45]  R. L. Eubank,et al.  Testing Goodness-of-Fit in Regression Via Order Selection Criteria , 1992 .

[46]  J. Rice,et al.  Smoothing spline models for the analysis of nested and crossed samples of curves , 1998 .

[47]  A. Birnbaum.  On the Foundations of Statistical Inference , 1962 .

[48]  M. C. Jones,et al.  Spline Smoothing and Nonparametric Regression. , 1989 .

[49]  Jianqing Fan,et al.  Sieve empirical likelihood ratio tests for nonparametric functions , 2004, math/0503667.

[50]  L. Hansen.  Large Sample Properties of Generalized Method of Moments Estimators , 1982 .

[51]  Michael Nussbaum,et al.  Minimax quadratic estimation of a quadratic functional , 1990, J. Complex..

[52]  James Stephen Marron,et al.  Variable window width kernel estimates of probability densities , 1992 .

[53]  R. Tibshirani,et al.  Varying‐Coefficient Models , 1993 .

[54]  Hengyan Li,et al.  Nonparametric Specification Testing for Continuous-Time Models with Applications to Term Structure of Interest Rates , 2005 .

[55]  Jianqing Fan,et al.  Nonparametric Inferences for Additive Models , 2005 .

[56]  Xin-Yuan Song,et al.  Local Polynomial Fitting in Semivarying Coefficient Model , 2002 .

[57]  T. W. Anderson.  Goodness of Fit Tests for Spectral Distributions , 1993 .

[58]  Wolfgang Härdle,et al.  Partially Linear Models , 2000 .

[59]  Chong Gu.  Smoothing Spline Anova Models , 2002 .

[60]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[61]  Wilbert C.M. Kallenberg,et al.  Data-Driven Smooth Tests When the Hypothesis is Composite , 1997 .

[62]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[63]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[64]  V. Spokoiny,et al.  Statistical inference for time-inhomogeneous volatility models , 2004, math/0406430.

[65]  T. Ozaki,et al.  Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model , 1981 .

[66]  A. W. F. Edwards,et al.  A History of Likelihood , 1974 .

[67]  Raymond J. Carroll,et al.  Semiparametric regression for clustered data , 2001 .

[68]  Z. Q. John Lu,et al.  Nonlinear Time Series: Nonparametric and Parametric Methods , 2004, Technometrics.

[69]  Maurice Kendall,et al.  Time Series , 2009, Encyclopedia of Biometrics.

[70]  Richard A. Davis,et al.  Time Series: Theory and Methods (2Nd Edn) , 1993 .

[71]  Leonard J. Savage,et al.  The foundations of statistical inference : a discussion , 1962 .

[72]  Jianqing Fan,et al.  On the estimation of quadratic functionals , 1991 .

[73]  Chunming Zhang.  Adaptive tests of regression functions via multiscale generalized likelihood ratios , 2003 .

[74]  Susan A. Murphy,et al.  Testing for a Time Dependent Coefficient in Cox's Regression Model , 1993 .

[75]  Ruey S. Tsay,et al.  Functional-Coefficient Autoregressive Models , 1993 .

[76]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .

[77]  David Ruppert,et al.  A Fully Automated Bandwidth Selection Method for Fitting Additive Models , 1998 .

[78]  Joel L Horowitz,et al.  An Adaptive, Rate-Optimal Test of Linearity for Median Regression Models , 2002 .

[79]  Jiancheng Jiang,et al.  Two-stage local M-estimation of additive models , 2008 .

[80]  Jean D. Opsomer,et al.  Asymptotic Properties of Backfitting Estimators , 2000 .

[81]  Joel L. Horowitz,et al.  Nonparametric estimation of an additive model with a link function , 2002, math/0508595.

[82]  Jianqing Fan,et al.  Adaptive varying co-efficient linear models , 2003 .

[83]  Jianqing Fan,et al.  Profile likelihood inferences on semiparametric varying-coefficient partially linear models , 2005 .

[84]  P. Hall.  The Bootstrap and Edgeworth Expansion , 1992 .

[85]  Yongmiao Hong,et al.  Nonparametric Specification Testing for Continuous-Time Models with Application to Spot Interest Rates , 2002 .

[86]  Ingrid Van Keilegom,et al.  Empirical likelihood tests for two-sample problems via nonparametric density estimation , 2006 .

[87]  William Alexander,et al.  Nonparametric Smoothing and Lack-of-Fit Tests , 1999, Technometrics.

[88]  Jianqing Fan,et al.  Generalised likelihood ratio tests for spectral density , 2004 .

[89]  D. Ruppert,et al.  Nonparametric estimation via local estimating equations, with applications to nutrition calibration , 1997 .

[90]  C. J. Stone,et al.  Additive Regression and Other Nonparametric Models , 1985 .

[91]  G. Wahba.  Spline models for observational data , 1990 .

[92]  J. Friedman,et al.  Projection Pursuit Regression , 1981 .

[93]  Ingrid K. Glad,et al.  Parametrically Guided Non‐parametric Regression , 1998 .

[94]  Young K. Truong,et al.  LOGSPLINE ESTIMATION OF A POSSIBLY MIXED SPECTRAL DISTRIBUTION , 1995 .

[95]  Jianqing Fan.  Test of Significance Based on Wavelet Thresholding and Neyman's Truncation , 1996 .

[96]  L. Brown,et al.  A constrained risk inequality with applications to nonparametric functional estimation , 1996 .

[97]  Grace Wahba,et al.  Spline Models for Observational Data , 1990 .

[98]  Joel L. Horowitz,et al.  An Adaptive, Rate-Optimal Test of a Parametric Mean-Regression Model Against a Nonparametric Alternative , 2001 .

[99]  R. Royall.  Statistical Evidence: A Likelihood Paradigm , 1997 .

[100]  Adrian Bowman,et al.  On the use of nonparametric regression for model checking , 1989 .

[101]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[102]  K. Dzhaparidze.  Parameter estimation and hypothesis testing in spectral analysis of stationary time series , 1986 .

[103]  Jianqing Fan,et al.  New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis , 2004 .

[104]  Helmut Herwartz,et al.  Time Inhomogeneous Multiple Volatility Modeling , 2003 .

[105]  Jianhua Z. Huang,et al.  Varying‐coefficient models and basis function approximations for the analysis of repeated measurements , 2002 .

[106]  Cun-Hui Zhang.  Fourier Methods for Estimating Mixing Densities and Distributions , 1990 .

[107]  G. Barnard,et al.  Likelihood Inference and Time Series , 1962 .

[108]  S. Portnoy.  Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity , 1988 .

[109]  Xuejun Jiang,et al.  Generalized likelihood ratio tests for the structure of semiparametric additive models , 2007 .

[110]  Wolfgang Härdle,et al.  Direct estimation of low-dimensional components in additive models , 1998 .

[111]  J. Neyman.  »Smooth test» for goodness of fit , 1937 .

[112]  Trevor Hastie,et al.  Statistical Models in S , 1991 .

[113]  D. Brillinger.  Time series - data analysis and theory , 1981, Classics in applied mathematics.

[114]  P. Bickel,et al.  On Some Global Measures of the Deviations of Density Function Estimates , 1973 .

[115]  Gerhard Tutz,et al.  On model diagnostics using varying coefficient models , 1999 .

[116]  Nouna Kettaneh,et al.  Statistical Modeling by Wavelets , 1999, Technometrics.

[117]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[118]  David Ruppert,et al.  Local Estimating Equations , 1998 .