Parker, Thomas (2010): A comparison of alternative approaches to sup-norm goodness of fit tests with estimated parameters.
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Goodness of fit tests based on sup-norm statistics of empirical processes have nonstandard limiting distributions when the null hypothesis is composite — that is, when parameters of the null model are estimated. Several solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985) can be applied to compute appropriate critical values for tests based on sup-norm statistics. The resulting tests have quite accurate size, a fact which has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin’s approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981) through the score function of the parametric model. Monte Carlo experiments suggest that these two testing strategies are roughly comparable to one another and more powerful than a simple bootstrap procedure.
|Item Type:||MPRA Paper|
|Original Title:||A comparison of alternative approaches to sup-norm goodness of fit tests with estimated parameters|
|Keywords:||Goodness of fit test; Estimated parameters; Gaussian process; Gauss-Markov process; Boundary crossing probability; Martingale transformation|
|Subjects:||C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General
C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General
C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C46 - Specific Distributions ; Specific Statistics
|Depositing User:||Thomas Parker|
|Date Deposited:||30. May 2010 06:37|
|Last Modified:||23. Feb 2013 07:00|
S. Aki. Some test statistics based on the martingale term of the empirical distribution function. Annals of the Institute of Statistical Mathematics, 38(1):1–21, 1986.
J. Bai. Testing parametric conditional distributions of dynamic models. Review of Economics and Statistics, 85(3):531–549, 2003.
P. Bickel, C. Klaassen, Y. Ritov, and J. Wellner. Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, 1993.
A. Buonocore, A. Nobile, and L. Ricciardi. A new integral equation for the evaluation of first-passage-time probability densities. Advances in Applied Probability, 19(4):784–800, 1987.
A. Cabaña and E. Cabaña. Transformed empirical processes and modified Kolmogorov-Smirnov tests for multivariate distributions. Annals of Statistics, 25(6):2388–2409, 1997.
E. del Barrio. Lectures on Empirical Processes: Theory and Statistical Applications, chapter Empirical and Quantile Processes in the Asymptotic Theory of Goodness-of-fit Tests, pages 1–92. EMS Series of Lectures in Mathematics. European Mathematical Society, 2007.
M. Delgado and W. Stute. Distribution-free specification tests of conditional models. Journal of Econometrics, 143(1):37–55, 2008.
E. Di Nardo, A. Nobile, E. Pirozzi, and L. Ricciardi. A computational approach to first-passage-time problems for Gauss-Markov processes. Advances in Applied Probability, 33(2):453–482, 2001.
J. Doob. Stochastic Processes. Wiley, 1953.
J. Durbin. Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. Journal of Applied Probability, 8 (3):431–453, 1971.
J. Durbin. Weak convergence of the sample distribution function when parameters are estimated. The Annals of Statistics, 1(2):279–290, 1973a.
J. Durbin. Distribution Theory for Tests Based on the Sample Distribution Function. Number 9 in Regional Conference Series in Applied Mathematics. SIAM, 1973b.
J. Durbin. Kolmogorov-Smirnov tests when parameters are estimated with applications to tests of exponentiality and tests on spacings. Biometrika, 62(1):5–22, 1975.
J. Durbin. The first-passage density of a continuous Gaussian process to a general boundary. Journal of Applied Probability, 22(1):99–122, 1985.
J. Durbin, M. Knott, and C. Taylor. Components of the Cramér-von Mises statistics. II. Journal of the Royal Statistical Society, Series B (Methodological), 37(2):216–237, 1975.
V. Fatalov. Asymptotics of large deviation probabilities for Gaussian fields. Journal of Contemporary Mathematical Analysis, 27(3):48–70, 1992.
V. Fatalov. Asymptotics of large deviation probabilities for Gaussian fields: Applications. Journal of Contemporary Mathematical Analysis, 28(5):21–44, 1993.
J. Haywood and E. Khmaladze. On distribution-free goodness-of-fit testing of exponentiality. Journal of Econometrics, 143(1):5–18, 2008.
Y. Hong and J. Liu. Generalized residual-based specification testing for duration models with censoring. Cornell University, 2007.
Y. Hong and J. Liu. Goodness-of-fit testing for duration models with censored grouped data. Cornell University, 2009.
E. Khmaladze. The use of ω2 tests for testing parametric hypotheses. Theory of Probability and its Applications, 24(2):283–301, 1979.
E. Khmaladze. Martingale approach in the theory of goodness-of-fit tests. Theory of Probability and its Applications, 26(2):240–257, 1981.
E. Khmaladze and H. Koul. Martingale transforms goodness-of-fit tests in regression models. The Annals of Statistics, 32(3):995–1034, 2004.
E. Khmaladze and H. Koul. Goodness-of-fit problem for errors in nonparametric regression: Distribution free approach. The Annals of Statistics, 37(6A):3165–3185, 2009.
R. Koenker and Z. Xiao. Inference on the quantile regression process. Econometrica, 70(4):1583–1612, 2002.
H. Koul. Weighted Empirical Processes in Dynamic Nonlinear Models, volume 166 of Lecture Notes in Statistics. Springer, 2nd edition, 2002.
H. Koul. Model diagnostics via martingale transforms: A brief review. In J. Fan and H. Koul, editors, Frontiers in Statistics, chapter 9, pages 183–206. Imperial College Press, 2006.
H. Koul and L. Sakhanenko. Goodness-of-fit testing in regression: A finite sample comparison of bootstrap methodology and Khmaladze transformation. Statistics & Probability Letters, 74(3):290–302, 2005.
E. Kulinskaya. Coefficients of the asymptotic distribution of the Kolmogorov-Smirnov statistic when parameters are estimated. Journal of Nonparametric Statistics, 5(1):43–60, 1995.
B. Li. Asymptotically distribution-free goodness-of-fit testing: A unifying view. Econometric Reviews, 28 (6):632–657, 2009.
R. Loynes. The empirical distribution function of residuals from generalised regression. The Annals of Statistics, 8(2):285–298, 1980.
G. Martynov. Goodness-of-fit tests for the Weibull and Pareto distributions. Paper presented at the Sixth International Conference on Mathematical Methods in Reliability, 2009.
M. Matsui and A. Takemura. Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE. Annals of the Institute of Statistical Mathematics, 57(1):183–199, 2005.
C. Mehr and J. McFadden. Certain properties of Gaussian processes and their first-passage times. Journal of the Royal Statistical Society, Series B (Methodological), 27(3):505–522, 1965.
G. Neuhaus. Weak Convergence Under Contiguous Alternatives when Parameters are Estimated: the Dk approach, volume 566 of Lecture Notes in Mathematics, pages 68–82. Springer, 1976.
G. Peskir. On integral equations arising in the first-passage problem for Brownian motion. Journal of Integral Equations and Applications, 14(4):397–423, 2002.
V. Piterbarg. Asymptotic Methods in the Theory of Gaussian Processes and Fields, volume 148 of Translations of Mathematical Monographs. American Mathematical Society, 1996.
W. Press, S. Teukolsky, W. Vetterling, and B. Flannery. Numerical Recipes in Fortran 77: The Art of Scientific Computing. Cambridge University Press, 2nd edition, 2001.
D. Rabinovitz. Estimating Durbin’s approximation. Biometrika, 80(3):671–680, 1993.
J. Romano. A bootstrap revival of some nonparametric distance tests. Journal of the American Statistical Association, 83(403):698–708, 1988.
G. Shorack and J. Wellner. Empirical Processes with Applications to Statistics. Wiley, 1986.
K. Song. Testing semiparametric conditional moment restrictions using conditional martingale transforms. Journal of Econometrics, 154(1):74–84, 2010.
Y. Tyurin. On the limit distribution of Kolmogorov-Smirnov statistics for a composite hypothesis. Mathematics of the USSR — Izvestiya, 25(3):619–646, 1985.
A. van der Vaart and J. Wellner. Weak Convergence and Empirical Processes. Springer, 1996.
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