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Goodness-of-fit tests in parametric regression based on the estimation of the error distribution. (English) Zbl 1196.62049
Summary: Consider a heteroscedastic regression model \(Y=m(X)+\sigma (X)\varepsilon \), where \(m(X)=E(Y|X)\) and \(\sigma ^{2}(X)= \text{Var} (Y|X)\) are unknown, and the error \(\varepsilon \) is independent of the covariate \(X\). We propose a new type of test statistic for testing whether the regression curve \(m(\cdot )\) belongs to some parametric family of regression functions. The proposed test statistic measures the distance between the empirical distribution function of the parametric and of the nonparametric residuals. The asymptotic theory of the proposed test is developed, and the proposed testing procedure is illustrated by means of a small simulation study and the analysis of a data set.

MSC:
62G10 Nonparametric hypothesis testing
62G30 Order statistics; empirical distribution functions
62G20 Asymptotic properties of nonparametric inference
62F05 Asymptotic properties of parametric tests
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[1] Akritas MG, Van Keilegom I (2001) Nonparametric estimation of the residual distribution. Scand J Stat 28:549–568 · Zbl 0980.62027 · doi:10.1111/1467-9469.00254
[2] Alcalá JT, Cristóbal JA, González Manteiga W (1999) Goodness-of-fit test for linear models based on local polynomials. Stat Probab Lett 42:39–46 · Zbl 0946.62016 · doi:10.1016/S0167-7152(98)00184-9
[3] Bellver C (1987) Influence of particulate pollution on the positions of neutral points in the sky in Seville (Spain). Atmos Environ 21:699–702 · doi:10.1016/0004-6981(87)90051-5
[4] Chen SX, Härdle W, Li M (2003) An empirical likelihood goodness-of-fit test for time series. J Roy Stat Soc Ser B 65:663–678 · Zbl 1063.62064 · doi:10.1111/1467-9868.00408
[5] Choi E, Hall P, Rousson V (2000) Data sharpening methods for bias reduction in nonparametric regression. Ann Stat 28:1339–1355 · Zbl 1105.62336 · doi:10.1214/aos/1015957396
[6] Cleveland WS (1993) Visualizing data. Hobart Press, Summit
[7] Dette H (1999) A consistent test for the functional form of a regression based on a difference of variance estimators. Ann Stat 27:1012–1040 · Zbl 0957.62036 · doi:10.1214/aos/1018031266
[8] Dette H, Munk A (1998) Validation of linear regression models. Ann Stat 26:778–800 · Zbl 0930.62041 · doi:10.1214/aos/1028144860
[9] Dette H, Munk A, Wagner T (2000) Testing model assumptions in multivariate linear regression models. J Nonparametric Stat 12:309–342 · Zbl 1033.62037 · doi:10.1080/10485250008832811
[10] Fan J, Zhang C, Zhang J (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Ann Stat 29:153–193 · Zbl 1029.62042 · doi:10.1214/aos/996986505
[11] González Manteiga W, Prada Sánchez JM, Romo J (1994) The bootstrap–a review. Comput Stat 9:165–205 · Zbl 0938.62047
[12] Hall P, DiCiccio TJ, Romano JP (1989) On smoothing and the bootstrap. Ann Stat 17:692–704 · Zbl 0672.62051 · doi:10.1214/aos/1176347135
[13] Härdle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21:1926–1947 · Zbl 0795.62036 · doi:10.1214/aos/1176349403
[14] Hart JD (1997) Nonparametric smoothing and lack-of-fit tests. Springer, New York · Zbl 0886.62043
[15] Jennrich RI (1969) Asymptotic properties of non-linear least-squares estimators. Ann Math Stat 40:633–643 · Zbl 0193.47201 · doi:10.1214/aoms/1177697731
[16] Koul HL, Lahiri SN (1994) On bootstrapping M-estimated residual processes in multiple linear regression models. J Multivar Anal 49:255–265 · Zbl 0795.62063 · doi:10.1006/jmva.1994.1025
[17] Mammen E (2000) Resampling methods for nonparametric regression. In: Schimek MG (ed) Smoothing and regression. Wiley, New York · Zbl 0980.62031
[18] Sánchez Sellero C (2001) Inferencia estadística en datos con censura y/o truncamiento. PhD thesis, University of Santiago de Compostela
[19] Seber GAF, Wild CJ (1989) Nonlinear regression. Wiley, New York
[20] Stute W (1997) Nonparametric model checks for regression. Ann Stat 25:613–641 · Zbl 0926.62035 · doi:10.1214/aos/1031833666
[21] Stute W, González Manteiga W, Presedo Quindimil M (1998) Bootstrap approximations in model checks for regression. J Am Stat Assoc 93:141–149 · Zbl 0902.62027 · doi:10.2307/2669611
[22] Van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, Cambridge · Zbl 0910.62001
[23] Van Keilegom I, Veraverbeke N (2002) Density and hazard estimation in censored regression models. Bernoulli 8:607–625 · Zbl 1007.62029
[24] White H (1981) Consequences and detection of misspecified nonlinear regression models. J Am Stat Assoc 76:419–433 · Zbl 0467.62058 · doi:10.2307/2287845
[25] White H (1982) Maximum likelihood estimation of misspecified models. Econometrica 50:1–25 · Zbl 0478.62088 · doi:10.2307/1912526
[26] Wu CF (1981) Asymptotic theory of nonlinear least-squares estimation. Ann Stat 9:501–513 · Zbl 0475.62050 · doi:10.1214/aos/1176345455
[27] Zhang CM (2003) Adaptive tests of regression functions via multiscale generalized likelihood ratios. Can J Stat 31:151–171 · Zbl 1040.62035 · doi:10.2307/3316065
[28] Zhang CM, Dette H (2004) A power comparison between nonparametric regression tests. Stat Probab Lett 66:289–301 · Zbl 1102.62049 · doi:10.1016/j.spl.2003.11.005
[29] Zhu L-X (2003) Model checking of dimension-reduction type for regression. Stat Sin 13:283–296 · Zbl 1015.62042
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