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Bootstrap based goodness-of-fit tests for binary multivariate regression models. (English) Zbl 1485.62102

Summary: We consider a binary multivariate regression model where the conditional expectation of a binary variable given a higher-dimensional input variable belongs to a parametric family. Based on this, we introduce a model-based bootstrap (MBB) for higher-dimensional input variables. This test can be used to check whether a sequence of independent and identically distributed observations belongs to such a parametric family. The approach is based on the empirical residual process introduced by W. Stute [Ann. Stat. 25, No. 2, 613–641 (1997; Zbl 0926.62035)]. In contrast to W. Stute and L.-X. Zhu’s approach [Scand. J. Stat. 29, No. 3, 535–545 (2002; Zbl 1035.62073)], a transformation is not required. Thus, any problems associated with non-parametric regression estimation are avoided. As a result, the MBB method is much easier for users to implement. To illustrate the power of the MBB based tests, a small simulation study is performed. Compared to the approach of Stute and Zhu [loc. cit.], the simulations indicate a slightly improved power of the MBB based method. Finally, both methods are applied to a real data set.

MSC:

62J12 Generalized linear models (logistic models)
62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
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[1] Agresti, A., Categorical data analysis, second edn. Wiley Series in Probability and Statistics (2002), New York: Wiley-Interscience [John Wiley & Sons], New York
[2] Bass, RF, Stochastic Processes. Cambridge Series in Statistical and Probabilistic Mathematics (2011), New York: Cambridge University Press, New York
[3] Billingsley, P., Convergence of probability measures. second edn. Wiley series in probability and statistics: probability and statistics (1999), New York: John Wiley & Sons Inc., New York
[4] Dikta, G.; Kvesic, M.; Schmidt, C., Bootstrap approximations in model checks for binary data, Journal of the American Statistical Association, 101, 474, 521-530 (2006) · Zbl 1119.62332
[5] Härdle, W.; Stoker, TM, Investigating smooth multiple regression by the method of average derivatives, Journal of the American Statistical Association, 84, 408, 986-995 (1989) · Zbl 0703.62052
[6] Kosorok, M., Introduction to empirical processes and semiparametric inference (2008), New York: Springer, New York · Zbl 1180.62137
[7] Robins, JM; Rotnitzky, A.; Zhao, LP, Estimation of regression coefficients when some regressors are not always observed, Journal of the American Statistical Association, 89, 427, 846-866 (1994) · Zbl 0815.62043
[8] Serfling, R., Approximation theorems of mathematical statistics. [nachdr.] edn.Wiley series in probability and mathematical statistics : probability and mathematical statistics (1980), NY: Wiley, NY · Zbl 0538.62002
[9] Singh, K. (1981). On the Asymptotic Accuracy of Efron’s Bootstrap. The Annals of Statistics,9(6), 1187-1195. · Zbl 0494.62048
[10] Stute, W., Nonparametric model checks for regression, The Annals of Statistics, 25, 2, 613-641 (1997) · Zbl 0926.62035
[11] Stute, W.; Zhu, LX, Model checks for generalized linear models, Scandinavian Journal of Statistics, 29, 3, 535-545 (2002) · Zbl 1035.62073
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.