Testing the goodness of fit of parametric regression models with random Toeplitz forms

Scandinavian Journal of Statistics

Volumn 29, Issue 3, 2002, Pages 501-533

  Munk, Axel ^a  

^a UNIVERSITY OF PADERBORN   (Germany)

Author keywords

Difference estimator; Goodness of fit; Jacknife; Model testing; Non linear least squares estimation; Precise hypotheses; Random design; Toeplitz matrice; Triangular array

Indexed keywords

EID: 0035995089     PISSN: 03036898     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9469.00303     Document Type: Article

References (78)

1. Achieser, N.J. (1956). Theory of approximation. Dover, New York.
2. Azzalini, A. & Bowman, A.W. (1993). On the use of nonparametric regression for checking linear relationships. J. Roy. Statist. Soc. Ser. B 55, 549-557.
3. Azzalini, A., Bowman, A.W. & Härdle, W. (1989). On the use of nonparametric regression for model checking. Biometrika 76, 1-11.
4. Barry, D. & Hartigan, J.A. (1990). An omnibus test for departure from constant mean. Ann. Statist. 18, 1340-57.
5. Bednarski, T. (1981). On optimal properties of the chi-square test for testing ε - validity of parametric models. Math. Operationsforsch. Statist. Ser. Statist. 12, 31-41.
6. Benzekri, S. & Brodeau, F. (1991). Asymptotic results for parametric estimation in inadequate two phases regression models. Statistics 22, 331-348.
7. Berk, K.N. (1973). A central limit theorem for m-dependent random variables with unbounded m. Ann. Probab. 1, 352-354.
8. Berkson, J. (1942). Tests of significance considered as evidence. J. Amer. Statist. Soc. 37, 325-335.
9. Brodeau, F. (1993). Tests for the choice of approximative models in nonlinear regression when the variance is unknown. Statistics 24, 95-106.
10. Bunke, H. (1981) A note on parameter estimation in inadequate nonlinear regression models. Math. Operationsforsch. Statist. Ser. Statist. 12, 7-11.
11. Bunke, H. & Schmidt, W.H. (1980). Asymptotic results on nonlinear approximation of regression functions and weighted least squares. Math. Operationsforsch. Statist. Ser. Statist. 11, 3-22.
12. Chen, J.C. (1994). Testing for no effect in nonparametric regression via spline smoothing techniques. Ann. Inst. Statist. Math. 46, 251-65.
13. Cox, D., Koh, E., Wahba, G. & Yandell, B.S. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Statist. 18, 113-119.
14. 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.
15. Dette, H. & Munk, A. (1998a). Validation of linear regression models. Ann. Statist. 26, 778-800.
16. Dette, H. & Munk, A. (1998b). A simple goodness of fit test for linear models under a random design assumption. Ann. Inst. Statist. Math. 50, 253-275.
17. Dette, H., Munk, A. & Wagner, T. (1998). Estimating the variance in nonparametric regression - what is a reasonable choice? J. Roy. Statist. Soc. Ser. B. 60, 751-764.
18. Dieboldt, J. (1995). A nonparametric test for the regression function: asymptotic theory. J. Statist. Plann. Inference 44, 1-17.
19. Dieboldt, J. & Zuber, J. (1999). Goodness of fit tests for nonlinear heteroscedastic regression models. Statist. Probab. Lett. 42, 53-60.
20. Djojosngito, R.A. (1994). On the use of cubic spline smoothing for testing parametric linear regression models. Comput. Statist. 9, 213-231.
21. Dorogovtsev, A.Ya., Zerek, N. & Kukush, A.G. (1987). Asymptotic properties of nonlinear regression estimators in Hilbert space. Theory Probab. Math. Statist. 35, 37-44.
22. Dorogovtsev, A. Ya., Zerek, N. & Kukush, A.G. (1988). Weak convergence of an estimator of an infinite-dimensional parameter to a normal distribution. Theory Probab. Math. Statist. 37, 45-51.
23. Draper, N.R. & Smith, H. (1981). Applied regression analysis. Wiley, New York.
24. Efron, B. & Tibshirani, R.J. (1993). An introduction to the bootstrap. Chapman & Hall, London.
25. Eubank, R.L. & Spiegelmann, C.H. (1990). Testing the goodness of a linear model via regression techniques. J. Amer. Statist. Assoc. 85, 387-392.
26. Eubank, R.L. & Hart, J.D. (1992). Testing goodness of fit in regression via order selection criteria. Ann. Statist. 20, 1412-1425.
27. Eubank, R.L., Hart, J.D. & LaRiccia, V.N. (1993). Testing goodness of fit via nonparametric function estimation techniques. Comm. Statist. Theory Methods 22, 3327-3354.
28. Fan, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19, 1273-1294.
29. Fan, Y. & Li, Q. (1996). Consistent model specification tests: omitted variables and semiparametric functional forms. Econometrics 64, 86-890.
30. Firth, D., Glosup, J. & Hinkley, D.V. (1991). Model checking with nonparametric curves. Biometrika 78, 245-252.
31. Gallant, A.R. (1987). Nonlinear statistical models. Wiley, New York.
32. Gasser, T., Sroka, L. & Jennen-Steinmetz, C. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73, 625-633.
33. Goutis, C. & Robert, Ch.P. (1998). Model choice in generalised linear models: a Bayesian approach via Kullback-Leibler projections. Biometrika 85, 29-37.
34. Hall, P., Kay, J.W. & Titterington, D.M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77, 521-528.
35. Hall, P., Kay, J.W. & Titterington, D.M. (1991). On estimation of noise variance in two dimensional signal processing. Adv. Appl. Probab. 23, 476-495.
36. Hall, P. & Marron, J.S. (1990). On variance estimation in nonparametric regression. Biometrika 77, 415-419.
37. Härdle, W. & Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21, 1926-1947.
38. Hart, J.D. (1997). Nonparametric smoothing and lack-of-fit tests. Springer, New York.
39. Hart, J.D. & Wehrly, T.E. (1992). Kernel regression when the boundary region is large, with an application to testing the adequacy of polynomial models. J. Amer. Statist. Assoc. 87, 1018-1024.
40. Hausman, J.A. (1978). Specification tests in econometrics. Econometrica 46, 1251-1271.
41. Hong, Y. & White, H. (1995). Consistent specification testing via nonparametric series regression. Econometrica 63, 1133-1159.
42. Hu, T.-C. & Taylor, R.L. (1997). On the strong law for arrays and for the bootstrap mean and variance. Internat. J. Math. Math. Sci. 20, 375-382.
43. Jennrich, R.I. (1969). Asymptotic properties of non-linear least squares estimators. Ann. Math. Statist. 40, 633-643.
44. Läuter, H. (1989). Note on the strong consistency of the least squares estimator in nonlinear regression. Statistics 20, 199-210.
45. MacKinnon, J.G. (1992). Model specification tests and artificial regressions. J. Econom. Lit. 30, 102-146.
46. Müller, H.G. (1992). Goodness of fit diagnostics for regression models. Scand. J. Statist. 19, 157-172.
47. Müller, H.G. & Stadtmüller, U. (1987) Estimation of heteroscedasticity in regression analysis. Ann. Statist. 15, 610-635.
48. Müller, H.G. & Stadtmüller, U. (1993). On variance function estimation with quadratic forms. J. Statist. Plann. Inference 35, 213-231.
49. Müller, H.G. & Zhao, P.L. (1995). On a semi-parametric variance function model and a test for heteroscedasticity. Ann. Statist. 23, 946-967.
50. Munk, A. & Czado, C. (1998). Nonparametric validation of similar distributions and the assessment of goodness of fit. J. Roy. Statist. Soc. Ser. B. 60, 223-241.
51. Neil, J.W. & Johnson, D.E. (1985). Testing linear regression function adequacy without replication. Ann. Statist. 13, 1482-1489.
52. Neumann, M.H. (1994). Fully data-driven nonparametric variance estimators. Statistics 25, 189-212.
53. Neyman, J. (1937). Smooth tests for goodness of fit. Skand. Aktuar. 20, 149-199.
54. Rice, J. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12, 1215-1230.
55. Richardson, G.D. & Bhattacharya, B.B. (1986). Consistent estimators in nonlinear regression for a non compact parameter space. Ann. Statist. 14, 1591-1596.
56. Rodriguez-Campos, M.C., González-Manteiga, W. & Cao, R. (1998). Testing the hypothesis of a generalized linear regression model using nonparametric regression estimation. J. Statist. Plann. Inference 67, 99-122.
57. Ruppert, D., Wand, M.P., Holst, U. & Hössjer, O. (1997). Local polynomial variance function estimation. Technometrics 39, 262-273.
58. Sacks, J. & Ylvisaker, D. (1970). Designs for regression problems with correlated errors III. Ann. Math. Statist. 41, 2057-2074.
59. Sacks, J. & Ylvisaker, D. (1981). Variance estimation for approximately linear models. Math. Operationsforsch. Statist. Ser. Statist. 12, 140-162.
60. Schmidt, W.H. (1979). Asymptotic results for estimation and testing variances in regression models. Math. Operationsforsch. Statist. Ser. Statist. 10, 209-236.
61. Seber, G.A.F. & Wild, C.J. (1989). Nonlinear regression. Wiley Series in Probability and Statistics. Wiley, New York.
62. Seifert, B., Gasser, T. & Wolf, A. (1993). Nonparametric estimation of residual variance revisited. Biometrika 80, 373-383.
63. Shillington, E.R. (1979). Testing lack of fit in regression without replication. Canad. J. Statist. 7, 137-146.
64. Shorack, G.R. & Wellner, J. (1986). Empirical processes with applications to statistics. Wiley, New York.
65. Staniswalis, J.G. & Severini, T.A. (1991). Diagnostics for assessing regression models. J. Amer. Statist. Assoc. 86, 684-692.
66. Stute, W. (1997). Nonparametric model checks for regression. Ann. Statist. 25, 613-641.
67. Stute, W., Gonzáles-Manteiga, W. & Presedo Quindimil, M. (1998). Bootstrap approximation in model checks for regression. J. Amer. Statist. Assoc. 93, 141-149.
68. Su, J.Q. & Wei, L.J. (1991). A lack of fit test for the mean function in a generalized linear model. J. Amer. Statist. Assoc. 86, 120-126.
69. Ullah, A. & Zinde-Walsh, V. (1992). On the estimation of residual variance in nonparametric regression. Nonparam. Statist. 1, 263-265.
70. von Neumann, J. (1941). Distribution of the ratio of the mean squared successive difference to the variance. Ann. Math. Statist. 12, 367-395.
71. von Neumann, J. (1942). A further remark concerning the distribution of the ratio of the mean squared successive difference to the variance. Ann. Math. Statist. 13, 86-88.
72. White, H. (1980). Nonlinear regression on cross-section data. Econometrica 48, 721-746.
73. White, H. (1981). Consequences and detection of misspecified nonlinear regression models. J. Amer. Statist. Assoc. 76, 419-433.
74. Witting, H. & Müller-Funk, U. (1995). Mathematische Statistik II. Teubner Verlag, Stuttgart.
75. Wooldridge, J.M. (1992). A test for a functional form against nonparametric alternatives. Econom. Theory 8, 452-475.
76. Wu, C.F. (1981). Asymptotic theory of nonlinear least squares estimation. Ann. Statist. 9, 501-513.
77. Zheng, J.X. (1997). A consistent test of a functional form via nonparametric estimation techniques. J. Econometrics 75, 263-289.
78. Zwanzig, S. (1980). The choice of approximative models in nonlinear regression. Math. Operationsforsch. Statist. Ser. Statist. 11, 23-47.