zbMATH — the first resource for mathematics

Goodness-of-fit tests in semiparametric transformation models. (English) Zbl 1342.62058
Summary: Consider a semiparametric transformation model of the form \(\Lambda_{\theta}(Y)= m(X) + \varepsilon\), where \(Y\) is a univariate dependent variable, \(X\) is a \(d\)-dimensional covariate, and \(\varepsilon\) is independent of \(X\) and has mean zero. We assume that \(\{\Lambda_{\theta} : \theta\in\Theta\}\) is a parametric family of strictly increasing functions, while \(m\) is an unknown regression function. The goal of the paper is to develop tests for the null hypothesis that \(m(\cdot)\) belongs to a certain parametric family of regression functions. We propose a Kolmogorov-Smirnov and a Cramér-von Mises type test statistic, which measure the distance between the distribution of \(\varepsilon\) estimated under the null hypothesis and the distribution of \(\varepsilon\) without making use of this null hypothesis. The estimated distributions are based on a profile likelihood estimator of \(\theta\) and a local polynomial estimator of \(m(\cdot)\). The limiting distributions of these two test statistics are established under the null hypothesis and under a local alternative. We use a bootstrap procedure to approximate the critical values of the test statistics under the null hypothesis. Finally, a simulation study is carried out to illustrate the performance of our testing procedures, and we apply our tests to data on the scattering of sunlight in the atmosphere.

62G08 Nonparametric regression and quantile regression
62G09 Nonparametric statistical resampling methods
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
PDF BibTeX Cite
Full Text: DOI
[1] Akritas, MG; Keilegom, I, Nonparametric estimation of the residual distribution, Scand J Stat, 28, 549-568, (2001) · Zbl 0980.62027
[2] Bellver, C, Influence of particulate pollution on the positions of neutral points in the sky in seville (Spain), Atmos Environ, 21, 699-702, (1987)
[3] Bickel, PJ; Doksum, K, An analysis of transformations revisited, J Am Stat Assoc, 76, 296-311, (1981) · Zbl 0464.62058
[4] Box, GEP; Cox, DR, An analysis of transformations, J R Stat Soc, 26, 211-252, (1964) · Zbl 0156.40104
[5] Carroll RJ, Ruppert D (1988) Transformation and weighting in regression. Chapman and Hall, New York · Zbl 0666.62062
[6] Cleveland WS (1993) Visualizing data. Hobart Press, Summit
[7] Colling B, Heuchenne C, Samb R, Van Keilegom I (2013) Estimation of the error density in a semiparametric transformation model. Ann Inst Stat Math (in press) · Zbl 1331.62214
[8] Dette, H; Neumeyer, N; Keilegom, I, A new test for the parametric form of the variance function in nonparametric regression, J R Stat Soc Ser B, 69, 903-917, (2007)
[9] González-Manteiga, W; Crujeiras, R, An updated review of goodness-of-fit tests for regression models (with discussion), TEST, 22, 361-447, (2013) · Zbl 1273.62086
[10] Härdle, W; Mammen, E, Comparing nonparametric versus parametric regression fits, Ann Stat, 21, 1926-1947, (1993) · Zbl 0795.62036
[11] Hart JD (1997) Nonparametric smoothing and lack-of-fit tests. Springer, New York · Zbl 0886.62043
[12] Heuchenne C, Samb R, Van Keilegom I (2014) Estimating the residual distribution in semiparametric transformation models (submitted, and available on http://www.uclouvain.be/en-369695.html, DP2014/11) · Zbl 1327.62257
[13] Horowitz, JL, Semiparametric estimation of a regression model with an unknown transformation of the dependent variable, Econometrica, 64, 103-137, (1996) · Zbl 0861.62029
[14] Horowitz, JL, Nonparametric estimation of a generalized additive model with an unknown link function, Econometrica, 69, 499-513, (2001) · Zbl 0999.62032
[15] Jacho-Chavez D, Lewbel A, Linton O (2008) Identification and nonparametric estimation of a transformation additively separable model. In: Technical report · Zbl 1431.62158
[16] Linton, O; Sperlich, S; Keilegom, I, Estimation of a semiparametric transformation model, Ann Stat, 36, 686-718, (2008) · Zbl 1133.62029
[17] Neumeyer, N, Smooth residual bootstrap for empirical processes of non-parametric regression residuals, Scand J Stat, 36, 204-228, (2009) · Zbl 1194.62051
[18] Neumeyer, N; Keilegom, I, Estimating the error distribution in nonparametric multiple regression with applications to model testing, J Multivar Anal, 101, 1067-1078, (2010) · Zbl 1185.62078
[19] Pardo-Fernández, JC; Keilegom, I; González-Manteiga, W, Testing for the equality of \(k\) regression curves, Stat Sin, 17, 1115-1137, (2007) · Zbl 1133.62031
[20] Sakia, RM, The box-Cox transformation technique: a review, Statistician, 41, 169-178, (1992)
[21] Silverman, BW; Young, GA, The bootstrap: to smooth or not to smooth, Biometrika, 74, 469-479, (1987) · Zbl 0654.62034
[22] Stute, W, Nonparametric model checks for regression, Ann Stat, 25, 613-641, (1997) · Zbl 0926.62035
[23] Keilegom, I; González-Manteiga, W; Sánchez Sellero, C, Goodness of fit tests in parametric regression based on the estimation of the error distribution, TEST, 17, 401-415, (2008) · Zbl 1196.62049
[24] Vanhems A, Van Keilegom I (2013) Semiparametric transformation model with endogeneity: a control function approach (submitted, and available on http://www.uclouvain.be/en-369695.html, DP2013/18)
[25] Zellner, A; Revankar, NS, Generalized production functions, Rev Econ Stud, 36, 241-250, (1969) · Zbl 0176.50103
[26] Zhang, CM, Adaptative tests of regression functions via multiscale generalized likelihood ratio, Can J Stat, 31, 151-171, (2003) · Zbl 1040.62035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.