Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Nonparametric inference with generalized likelihood ratio tests

  • Invited Paper
  • Published:
TEST Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aït-Sahalia Y, Fan J, Peng H (2005) Nonparametric transition-based tests for Jump-diffusions. Unpublished manuscript

  • Anderson TW (1993) Goodness of fit tests for spectral distributions. Ann Stat 21:830–847

    MATH  Google Scholar 

  • Azzalini A, Bowman AN, Härdle W (1989) On the use of nonparametric regression for model checking. Biometrika 76:1–11

    MATH  MathSciNet  Google Scholar 

  • Barnard GA, Jenkins GM, Winsten CB (1962) Likelihood inference and time series. J R Stat Soc Ser A 125:321–372

    Google Scholar 

  • Berger JO, Wolpert RL (1988) The likelihood principle, 2nd edn. Institute of Mathematical Statistics, Haywood

    Google Scholar 

  • Bickel PJ, Ritov Y (1988) Estimating integrated squared density derivatives: sharp order of convergence estimates. Sankhyā Ser A 50:381–393

  • Bickel PJ, Rosenblatt M (1973) On some global measures of the deviation of density function estimates. Ann Stat 1:1071–1095

    MATH  MathSciNet  Google Scholar 

  • Birnbaum A (1962) On the foundations of statistical inference (with discussion). J Am Stat Assoc 57:269–326

    MathSciNet  Google Scholar 

  • Brillinger DR (1981) Time series. Data analysis and theory, 2nd edn. Holden–day series in time series analysis. Holden–Day, Oakland

    MATH  Google Scholar 

  • Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2nd edn. Springer, New York

    Google Scholar 

  • Brown LD, Low M (1996) A constrained risk inequality with applications to nonparametric functional estimation. Ann Stat 24:2524–2535

    MATH  MathSciNet  Google Scholar 

  • Brumback B, Rice JA (1998) Smoothing spline models for the analysis of nested and crossed samples of curves (with discussion). J Am Stat Assoc 93:961–994

    MATH  MathSciNet  Google Scholar 

  • Buja A, Hastie TJ, Tibshirani RJ (1989) Linear smoothers and additive models. Ann Stat 17:453–555

    MATH  MathSciNet  Google Scholar 

  • Cai Z, Fan J, Li R (2000a) Efficient estimation and inferences for varying-coefficient models. J Am Stat Assoc 95:888–902

    MATH  MathSciNet  Google Scholar 

  • Cai Z, Fan J, Yao Q (2000b) Functional-coeficient regression models for nonlinear time series. J Am Stat Assoc 95:941–956

    MATH  MathSciNet  Google Scholar 

  • Carrol RJ, Ruppert D, Welsh AH (1998) Nonparametric estimation via local estimating equations. J Am Stat Assoc 93:214–227

    Google Scholar 

  • Chan KC, Karolyi AG, Longstaff FA, Sanders AB (1992) An empirical comparison of alternative models of the short-term interest rate. J Finance 47:1209–1227

    Google Scholar 

  • Chen R, Tsay RJ (1993) Functional-coefficient autoregressive models. J Am Stat Assoc 88:298–308

    MATH  MathSciNet  Google Scholar 

  • Cleveland WS, Grosse E, Shyu WM (1991) Local regression models. In: Chambers, JM, Hastie, TJ (eds) Statistical models in S. Chapman & Hall computer science series. CRC Press, Boca Raton, pp 309–376

    Google Scholar 

  • Cox JC, Ingersoll JE, Ross SA (1985) A theory of the term structure of interest rates. Econometrica 53:385–467

    MathSciNet  Google Scholar 

  • Davis HT, Jones RH (1968) Estimation of the innovation variance of a stationary time series. J Am Stat Assoc 63:141–149

    MATH  MathSciNet  Google Scholar 

  • Donoho DL, Nussbaum M (1990) Minimax quadratic estimation of a quadratic functional. J Complex 6:290–323

    MATH  MathSciNet  Google Scholar 

  • Dzhaparidze K (1986) Parameter estimation and hypothesis testing on spectral analysis of stationary time series. Springer, New York

    Google Scholar 

  • Edwards AWF (1972) Likelihood, 1st edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Edwards AWF (1974) The history of likelihood. Int Stat Rev 42:9–15

    Article  Google Scholar 

  • Efromovich S (1999) Nonparametric curve estimation: methods, theory and applications. Springer, New York

    MATH  Google Scholar 

  • Efron B, Tibshirani R (1995) An introduction to the bootstrap. Chapman & Hall, New York

    Google Scholar 

  • Eubank RL (1999) Spline smoothing and nonparametric regression, 2nd edn. Dekker, New York

    MATH  Google Scholar 

  • Eubank RL, Hart JD (1992) Testing goodness-of-fit in regression via order selection criteria. Ann Stat 20:1412–1425

    MATH  MathSciNet  Google Scholar 

  • Eubank RL, LaRiccia VN (1992) Asymptotic comparison of Cramér–von Mises and nonparametric function estimation techniques for testing goodness-of-fit. Ann Stat 20:2071–2086

    MATH  MathSciNet  Google Scholar 

  • Fan J (1991) On the estimation of quadratic functionals. Ann Stat 19:1273–1294

    MATH  Google Scholar 

  • Fan J (1996) Test of significance based on wavelet thresholding and Neyman’s truncation. J Am Stat Assoc 91:674–688

    MATH  Google Scholar 

  • Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman & Hall, London

    MATH  Google Scholar 

  • Fan J, Huang L (2001) Goodness-of-fit test for parametric regression models. J Am Stat Assoc 96:640–652

    MATH  MathSciNet  Google Scholar 

  • Fan J, Huang T (2005) Profile Likelihood Inferences on semiparametric varying-coefficient partially linear models. Bernoulli 11:1031–1057

    Article  MATH  MathSciNet  Google Scholar 

  • Fan J, Jiang J (2005) Nonparametric inference for additive models. J Am Stat Assoc 100:890–907

    MATH  MathSciNet  Google Scholar 

  • Fan J, Kreutzberger E (1998) Automatic local smoothing for spectral density estimation. Scand J Stat 25:359–369

    MATH  MathSciNet  Google Scholar 

  • Fan J, Li R (2004) New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. J Am Stat Assoc 99:710–723

    MATH  MathSciNet  Google Scholar 

  • Fan J, Yao Q (2003) Nonlinear time series: nonparametric and parametric methods. Springer, New York

    MATH  Google Scholar 

  • Fan J, Zhang C (2003) A re-examination of diffusion estimations with applications to financial model validation. J Am Stat Assoc 98:118–134

    MATH  MathSciNet  Google Scholar 

  • Fan J, Zhang J (2004) Sieve empirical likelihood ratio tests for nonparametric functions. Ann Stat 32:1858–1907

    MATH  MathSciNet  Google Scholar 

  • Fan J, Zhang W (1999) Statistical estimation in varying coefficient models. Ann Stat 27:1491–1518

    MATH  MathSciNet  Google Scholar 

  • Fan J, Zhang W (2004) Generalized likelihood ratio tests for spectral density. Biometrika 91:195–209

    MathSciNet  MATH  Google Scholar 

  • Fan J, Yao Q, Tong H (1996) Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83:189–206

    MATH  MathSciNet  Google Scholar 

  • Fan J, Härdle W, Mammen E (1998) Direct estimation of additive and linear components for high dimensional data. Ann Stat 26:943–971

    MATH  Google Scholar 

  • Fan J, Zhang C, Zhang J (2001) Generalized likelihood ratio statistics and Wilks’ phenomenon. Ann Stat 29:153–193

    MATH  MathSciNet  Google Scholar 

  • Fan J, Yao Q, Cai Z (2003) Adaptive varying-coefficient linear models. J R Stat Soc Ser B 65:57–80

    MATH  MathSciNet  Google Scholar 

  • Fisher RA (1922) On the mathematical foundations of theoretical statistics. Philos Trans R Soc Ser A 222–326

  • Friedman JH, Stuetzle W (1981) Projection pursuit regression. J Am Stat Assoc 76:817–823

    MathSciNet  Google Scholar 

  • Glad IK (1998) Parametrically guided non-parametric regression. Scand J Stat 25:649–668

    MathSciNet  Google Scholar 

  • Grama I, Nussbaum M (2002) Asymptotic equivalence for nonparametric regression. Math Methods Stat 11:1–36

    MATH  MathSciNet  Google Scholar 

  • Gu C (2002) Smoothing spline ANOVA models. Springer, New York

    MATH  Google Scholar 

  • Haggan V, Ozaki T (1981) Modeling nonlinear vibrations using an amplitude-dependent autoregression time series model. Biometrika 68:189–196

    MATH  MathSciNet  Google Scholar 

  • Hall P (1993) The bootstrap and edgeworth expansion. Springer, New York

    Google Scholar 

  • Hall P, Marron JS (1988) Variable window width kernel estimates of probability densities. Probab Theory Relat Fields 80:37–49

    MATH  MathSciNet  Google Scholar 

  • Hansen LP (1982) Large sample properties of generalized method of moments estimators. Econometrica 50:1029–1054

    MATH  MathSciNet  Google Scholar 

  • Härdle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21:1926–1947

    MATH  Google Scholar 

  • Härdle W, Liang H, Gao J (2000) Partially linear models. Springer, Heidelberg

    MATH  Google Scholar 

  • Härdle W, Herwartz H, Spokoiny VG (2003) Time inhomogeneous multiple volatility modelling. J Financ Econom 1:55–95

    Google Scholar 

  • Hart JD (1997) Nonparametric smoothing and lack-of-fit tests. Springer, New York

    MATH  Google Scholar 

  • Hastie TJ, Tibshirani RJ (1990) Generalized additive models. Chapman & Hall, New York

    MATH  Google Scholar 

  • Hastie TJ, Tibshirani RJ (1993) Varying-coefficient models. J R Stat Soc Ser B 55:757–796

    MATH  MathSciNet  Google Scholar 

  • Hjort N, Glad IK (1995) Nonparametric density estimation with a parametric start. Ann Stat 23:882–904

    MATH  MathSciNet  Google Scholar 

  • Hong Y, Li H (2005) Nonparametric specification testing for continuous-time models with applications to term structure of interest. Rev Financ Stud 18:37–84

    MathSciNet  Google Scholar 

  • Hoover DR, Rice JA, Wu CO, Yang L-P (1998) Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85:809–822

    MATH  MathSciNet  Google Scholar 

  • Horowitz JL, Mammen E (2004) Nonparametric estimation of an additive model with a link function. Ann Stat 32:2412–2443

    MATH  MathSciNet  Google Scholar 

  • Horowitz JL, Spokoiny VG (2001) An adaptive, rate-optimal test of a parametric model against a nonparametric alternative. Econometrica 69:599–631

    MATH  MathSciNet  Google Scholar 

  • Horowitz JL, Spokoiny VG (2002) An adaptive, rate-optimal test of linearity for median regression models. J Am Stat Assoc 97:822–835

    MATH  MathSciNet  Google Scholar 

  • Huang JZ, Wu CO, Zhou L (2002) Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika 89:111–128

    MATH  MathSciNet  Google Scholar 

  • Inglot T, Ledwina T (1996) Asymptotic optimality of data-driven Neyman’s tests for uniformity. Ann Stat 24:1982–2019

    MATH  MathSciNet  Google Scholar 

  • Ingster YI (1993a) Asymptotic minimax hypothesis testing for nonparametric alternatives. Math Methods Stat 2:85–114

    MATH  MathSciNet  Google Scholar 

  • Ingster YI (1993b) Asymptotic minimax hypothesis testing for nonparametric alternatives. Math Methods Stat 3:171–189

    MathSciNet  Google Scholar 

  • Ingster YI (1993c) Asymptotic minimax hypothesis testing for nonparametric alternatives. Math Methods Stat 4:249–268;

    MathSciNet  Google Scholar 

  • Jiang J, Hui YV (2004) Spectral density estimation with amplitude modulation and outlier detection. Ann Inst Stat Math 56:611–630

    MATH  MathSciNet  Google Scholar 

  • Jiang J, Li J (2007) Two-stage local M-estimation of additive models. Sci China Ser A (to appear)

  • Jiang J, Zhou H, Jiang X, Peng J (2007) Generalized likelihood ratio tests for the structures of semiparametric additive models. Can J Stat 35:381–398

    Article  MATH  MathSciNet  Google Scholar 

  • Kallenberg WCM, Ledwina T (1997) Data-driven smooth tests when the hypothesis is composite. J Am Stat Assoc 92:1094–1104

    MATH  MathSciNet  Google Scholar 

  • Kauermann G, Tutz G (1999) On model diagnostics using varying coefficient models. Biometrika 86:119–128

    MATH  MathSciNet  Google Scholar 

  • Kooperberg C, Stone CJ, Truong YK (1995a) Rate of convergence for logspline spectral density estimation. J Time Ser Anal 16:389–401

    MATH  MathSciNet  Google Scholar 

  • Kooperberg C, Stone CJ, Truong YK (1995b) Logspline estimation of a possibly mixed spectral distribution. J Time Ser Anal 16:359–389

    MATH  MathSciNet  Google Scholar 

  • Lepski OV, Spokoiny VG (1999) Minimax nonparametric hypothesis testing: the case of an inhomogeneous alternative. Bernoulli 5:333–358

    MATH  MathSciNet  Google Scholar 

  • Li Q, Huang CJ, Li D, Fu T-T (2002) Semiparametric smooth coefficient models. J Bus Econom Stat 20:412–422

    MathSciNet  Google Scholar 

  • Liang K-Y, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73:13–22

    MATH  MathSciNet  Google Scholar 

  • Lin DY, Ying Z (2001) Semiparametric and nonparametric regression analysis of longitudinal data (with discussions). J Am Stat Assoc 96:103–126

    MATH  MathSciNet  Google Scholar 

  • Lin X, Carroll RJ (2001a) Semiparametric regression for clustered data using generalized estimating equations. J Am Stat Assoc 96:1045–1056

    MATH  MathSciNet  Google Scholar 

  • Lin X, Carroll RJ (2001b) Semiparametric regression for clustered data. Biometrika 88:1179–1865

    MATH  MathSciNet  Google Scholar 

  • McCullough P, Nelder JA (1989) Generalized linear models, 2nd edn. Chapman & Hall, New York

    Google Scholar 

  • Mercurio D, Spokoiny VG (2004) Statistical inference for time-inhomogeneous volatility models. Ann Stat 32:577–602

    MATH  MathSciNet  Google Scholar 

  • Murphy SA (1993) Testing for a time dependent coefficient in Cox’s regression model. Scand J Stat 20:35–50

    MATH  MathSciNet  Google Scholar 

  • Neyman J (1937) Smooth test for goodness of fit. Skand Aktuarietidskr 20:149–199

    MATH  Google Scholar 

  • Opsomer J-D (2000) Asymptotic properties of backfitting estimators. J Multivar Anal 73:166–179

    MATH  MathSciNet  Google Scholar 

  • Opsomer J-D, Ruppert D (1998) Fully automated bandwidth selection method for fitting additive models. J Am Stat Assoc 93:605–619

    MATH  MathSciNet  Google Scholar 

  • Paparoditis E (2000) Spectral density based goodness-of-fit tests in time series models. Scand J Stat 27:143–176

    MATH  MathSciNet  Google Scholar 

  • Pawitan Y, O’Sullivan F (1994) Nonparametric spectral density estimation using penalized Whittle likelihood. J Am Stat Assoc 89:600–610

    MATH  MathSciNet  Google Scholar 

  • Portnoy S (1988) Asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity. Ann Stat 16:356–366

    MATH  MathSciNet  Google Scholar 

  • Press H, Tukey JW (1956) Power spectral methods of analysis and their application to problems in airplane dynamics. Bell telephone system monograph 2606

  • Royall RM (1997) Statistical evidence: a likelihood paradigm. Chapman & Hall, London

    MATH  Google Scholar 

  • Shao J, Tu D (1996) The jackknife and bootstrap. Springer, New York

    Google Scholar 

  • Spokoiny VG (1996) Adaptive hypothesis testing using wavelets. Ann Stat 24:2477–2498

    MATH  MathSciNet  Google Scholar 

  • Stone CJ (1985) Additive regression and other nonparametric models. Ann Stat 13:689–705

    MATH  Google Scholar 

  • Vidakovic B (1999) Statistical modeling by wavelets. Wiley, New York

    MATH  Google Scholar 

  • Wahba G (1980) Automatic smoothing of the log periodogram. J Am Stat Assoc 75:122–132

    MATH  Google Scholar 

  • Wahba G (1990) Spline models for observational data. SIAM, Philadelphia

    MATH  Google Scholar 

  • Wand MP, Jones MC (1995) Kernel smoothing. Chapman & Hall, London

    MATH  Google Scholar 

  • Zhang CM (2003a) Adaptive tests of regression functions via multi-scale generalized likelihood ratios. Can J Stat 31:151–171

    MATH  Google Scholar 

  • Zhang CM (2003b) Calibrating the degrees of freedom for automatic data smoothing and effective curve checking. J Am Stat Assoc 98:609–628

    MATH  Google Scholar 

  • Zhang W, Lee SY, Song X (2002) Local polynomial fitting in semivarying coefficient models. J Multivar Anal 82:166–188

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of ORFE, Princeton University, Princeton, NJ, 08544, USA

    Jianqing Fan

  2. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, USA

    Jiancheng Jiang

Authors
  1. Jianqing Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiancheng Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianqing Fan.

Additional information

This invited paper is discussed in the comments available at: http://dx.doi.org/10.1007/s11749-007-0081-7, http://dx.doi.org/10.1007/s11749-007-0082-6, http://dx.doi.org/10.1007/s11749-007-0083-5, http://dx.doi.org/10.1007/s11749-007-0084-4, http://dx.doi.org/10.1007/s11749-007-0085-3, http://dx.doi.org/10.1007/s11749-007-0086-2, http://dx.doi.org/10.1007/s11749-007-0087-1, http://dx.doi.org/10.1007/s11749-007-0088-0, http://dx.doi.org/10.1007/s11749-007-0089-z.

The work was supported by the NSF grants DMS-0354223, DMS-0532370 and DMS-0704337.

The paper was initiated when Jiancheng Jiang was a research fellow at Princeton University.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fan, J., Jiang, J. Nonparametric inference with generalized likelihood ratio tests. TEST 16, 409–444 (2007). https://doi.org/10.1007/s11749-007-0080-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-007-0080-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation