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Frequentist-Bayes Lack-of-Fit Tests Based on Laplace Approximations

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Abstract

The null hypothesis that all of a function’s Fourier coefficients are 0 is tested in frequentist fashion using as test statistic a Laplace approximation to the posterior probability of the null hypothesis. Testing whether or not a regression function has a prescribed linear form is one application of such a test. In contrast to BIC, the Laplace approximation depends on prior probabilities, and hence allows the investigator to tailor the test to particular kinds of alternative regression functions. On the other hand, using diffuse priors produces new omnibus lack-of-fit statistics.

The new omnibus test statistics are weighted sums of exponentiated squared (and normalized) Fourier coefficients, where the weights depend on prior probabilities. Exponentiation of the Fourier components leads to tests that can be exceptionally powerful against high frequency alternatives. Evidence to this effect is provided by a comprehensive simulation study, in which one new test that had good power at high frequencies also performed comparably to some other well-known omnibus tests at low frequency alternatives.

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References

  • Aerts, M., Claeskens, G., Hart, J.D., 1999. Testing the fit of a parametric function. J. Amer. Statist. Assoc., 94, 869–879.

    MathSciNet  MATH  Google Scholar 

  • Aerts, M., Claeskens, G., Hart, J.D., 2000. Testing lack of fit in multiple regression. Biometrika, 87, 405–424.

    MathSciNet  MATH  Google Scholar 

  • Aerts, M., Claeskens, G., Hart, J.D., 2004. Bayesian-motivated tests of function fit and their asymptotic frequentist properties. Ann. Statist., 32, 2580–2615.

    MathSciNet  MATH  Google Scholar 

  • Baraud Y., Huet, S., Laurent, B., 2003. Adaptive tests of linear hypotheses by model selection. Ann. Statist., 31, 225–251.

    MathSciNet  MATH  Google Scholar 

  • Bayarri, M. J., Berger, J.O., 2004. The interplay of Bayesian and frequentist analysis. Statist. Sci., 19, 58–80.

    MathSciNet  MATH  Google Scholar 

  • Bickel, P.J., Ritov, Y., Stoker, T.M., 2006. Tailor-made tests for goodness-of-fit to semiparametric hypotheses. Ann. Statist., 34, 721–741.

    MathSciNet  MATH  Google Scholar 

  • Bogdan, M., 2001. Data driven versions of Neyman’s test for uniformity based on a Bayesian rule. J. Statist. Comput. Simul., 68, 203–222.

    MathSciNet  MATH  Google Scholar 

  • Buckley, M.J., 1991. Detecting a smooth signal: optimality of cusum based procedures. Biometrika, 78, 253–262.

    MathSciNet  Google Scholar 

  • Chang, M., Chow, S.C., 2005. A hybrid Bayesian adaptive design for dose response trials. J. Biopharma-ceutical Statist., 15, 677–691.

    MathSciNet  Google Scholar 

  • Claeskens, G., Hjort, N.L., 2004. Goodness of fit via nonparametric likelihood ratios. Scandinavian Journal of Statistics, 31, 487–513.

    MathSciNet  MATH  Google Scholar 

  • Cline, D., 1983. Infinite series of random variables with regularly varying tails. Technical Report #83-24, University of British Columbia, Institute of Applied Mathematics and Statistics, http://www.stat.tamu.edu/~dcline/papers/infiniteseries.pdf.

    Google Scholar 

  • Cole, G.M., 1997. Water Boundaries. Wiley & Sons, Inc., New York.

    Google Scholar 

  • Conrad, J., Botner, O., Hallgren, A., Pérez de los Heros, C., 2003. Including systematic uncertainties in confidence interval construction for poisson statistics. Phys. Rev. D, 67, 012002.

    Google Scholar 

  • De Bruijn, N.G., 1970. Asymptotic Methods in Analysis. North-Holland, Amsterdam.

    Google Scholar 

  • Dette, H., Munk, A., 1998. Validation of linear regression models. Ann. Statist., 26, 778–800.

    MathSciNet  MATH  Google Scholar 

  • Dette, H., 1999. A consistent test for the functional form of a regression based on a difference of variance estimators. Ann. Statist., 27, 1012–1040.

    MathSciNet  MATH  Google Scholar 

  • Dümbgen, L., Spokoiny, V.G., 2001. Multiscale testing in the nonlinear structural errors-in-variables model. Ann. Statist., 29, 124–152.

    MathSciNet  MATH  Google Scholar 

  • Eubank, R.L., 2000. Testing for no effect by cosine series methods. Scandinavian Journal of Statistic, 27, 747–763.

    MathSciNet  MATH  Google Scholar 

  • Eubank, R.L., Hart, J.D., 1993. Commonality of cusum, von Neumann and smoothing-based goodness-of-fit tests. Biometrika, 80, 89–98.

    MathSciNet  MATH  Google Scholar 

  • Fan, J., Huang, L.S., 2001. Goodness-of-fit tests for parametric regression models. J. Amer. Statist. Assoc., 96, 640–652.

    MathSciNet  MATH  Google Scholar 

  • Fan, J., Zhang, C., Zhang, J., 2001. Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist., 29, 153–193.

    MathSciNet  MATH  Google Scholar 

  • Good, I.J., 1957. Saddle-point methods for the multinomial distribution. Ann. Math. Statist., 28, 861–881.

    MathSciNet  MATH  Google Scholar 

  • Guerre, E., Lavergne, P., 2005. Data-driven rate-optimal specification testing in regression models. Ann. Statist., 33, 840–870.

    MathSciNet  MATH  Google Scholar 

  • Hall, P., Kay, J.W., Titterington, D.M., 1990. Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika, 77, 521–528.

    MathSciNet  MATH  Google Scholar 

  • Hart, J.D., 1997. Nonparametric Smoothing and Lack-of-Fit Tests. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Horowitz, J.L., Spokoiny, V.G., 2001. An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica, 69, 599–631.

    MathSciNet  MATH  Google Scholar 

  • Inglot, T., Ledwina, T., 1996. Asymptotic optimality of data driven Neyman’s tests. Ann. Statist., 24, 1982–2019.

    MathSciNet  MATH  Google Scholar 

  • Inglot, T., Ledwina, T., 2006. Data driven score tests for a homoscedastic linear regression model: asymptotic results. Probability and Mathematical Statistics, 26, 41–61.

    MathSciNet  MATH  Google Scholar 

  • Janssen, A., 2000. Global power functions of goodness of fit tests. Ann. Statist., 28, 239–253.

    MathSciNet  MATH  Google Scholar 

  • Kallenberg, W.C.M., 1983. Intermediate efficiency, theory and examples. Ann. Statist., 11, 170–182.

    MathSciNet  MATH  Google Scholar 

  • Kallenberg, W.C.M., 2002. The penalty in data driven Neyman’s tests. Mathematical Methods of Statistics, 11, 323–340.

    MathSciNet  Google Scholar 

  • Kass, R.E., Raftery, A.E., 1995. Bayes factors. J. Amer. Statist. Assoc., 90, 773–795.

    MathSciNet  MATH  Google Scholar 

  • Kass, R.E., Wasserman, L., 1995. A reference Bayesian test for nested hypotheses and its relationship to the schwarz criterion. J. Amer. Statist. Assoc., 90, 928–934.

    MathSciNet  MATH  Google Scholar 

  • Kuchibhatla, M., Hart, J.D., 1996. Smoothing-based lack-of-fit tests: variations on a theme. J. Nonparametr. Statist., 7, 1–22.

    MathSciNet  MATH  Google Scholar 

  • Ledwina, T., 1994. Data-driven version of Neyman’s smooth test of fit. J. Amer. Statist. Assoc., 89, 1000–1005.

    MathSciNet  MATH  Google Scholar 

  • Lee, G., Hart, J.D., 2000. Model selection criteria with data dependent penalty, with application to data-driven Neyman smooth tests. Nonparametric Statistics, 12, 683–707.

    MathSciNet  MATH  Google Scholar 

  • Lehmann, E., 1959. Testing Statistical Hypotheses. John Wiley & Sons, New York.

    MATH  Google Scholar 

  • Lucito, R., West, J., Reiner, A., Alexander, D., Esposito, D., Mishra B., Powers, S., Norton, L., Wigler, M., 2000. Detecting gene copy number fluctuations in tumor cells by microarray analysis of genomic representations. Genome Research, 10, 1726–1736.

    Google Scholar 

  • Mallows, C.L., 1973. Some comments on c p. Technometrics, 15, 661–675.

    MATH  Google Scholar 

  • Neyman, J., 1937. ‘Smooth’ test for goodness of fit. Skandinavisk Aktuarietidskrift, 20, 149–199.

    MATH  Google Scholar 

  • Ogden, R.T., 1997. Essential Wavelets for Statistical Applications and Data Analysis. Birkhäuser, Boston.

    MATH  Google Scholar 

  • Rayner, J.C.W., Best, D.J., 1989. Smooth Tests of Goodness of Fit. Oxford University Press, New York.

    MATH  Google Scholar 

  • Snijders, A., Nowak, N., Segraves, R., Blackwood, S., Brown, N., Conroy, J., Hamilton, G., Hindle, A., Huey, B., Kimura, K., Law, S., Myambo, K., Palmer, J., Ylstra, B., Yue, J., Gray, J., Jain, A., Pinkel, D., Albertson, D., 2001. Assembly of microarrays for genome-wide measurement of DNA copy number. Nature Genetics, 29, 263–264.

    Google Scholar 

  • Spokoiny, V.G., 1996. Adaptive hypothesis testing using wavelets. Ann. Statist., 24, 2477–2498.

    MathSciNet  MATH  Google Scholar 

  • Stute, W., 1997. Nonparametric model checks for regression. Ann. Statist., 25, 613–641.

    MathSciNet  MATH  Google Scholar 

  • Tierney, L., Kadane, J.B., 1986. Accurate approximations for posterior moments and marginal densities. J. Amer. Statist. Assoc., 81, 82–86.

    MathSciNet  MATH  Google Scholar 

  • Verdinelli, I., Wasserman, L., 1998. Bayesian goodness-of-fit testing using infinite-dimensional exponential families. Ann. Statist., 26, 1215–1241.

    MathSciNet  MATH  Google Scholar 

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  1. Department of Statistics, Texas A&M University, College Station, TX, 77843-3143, USA

    Jeffrey D. Hart

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  1. Jeffrey D. Hart
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Correspondence to Jeffrey D. Hart.

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Hart, J.D. Frequentist-Bayes Lack-of-Fit Tests Based on Laplace Approximations. J Stat Theory Pract 3, 681–704 (2009). https://doi.org/10.1080/15598608.2009.10411954

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  • DOI: https://doi.org/10.1080/15598608.2009.10411954

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