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 |
Keywords:
binary regression model; bootstrap based test; goodness-of-fit test; marked empirical processReferences:
| [1] | Agresti, A., Categorical data analysis, second edn. Wiley Series in Probability and Statistics (2002), New York: Wiley-Interscience [John Wiley & Sons], New York · Zbl 1018.62002 |
| [2] | Bass, RF, Stochastic Processes. Cambridge Series in Statistical and Probabilistic Mathematics (2011), New York: Cambridge University Press, New York · Zbl 1247.60001 |
| [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 · Zbl 0944.60003 |
| [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 · doi:10.1198/016214505000001032 |
| [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 · doi:10.1007/978-0-387-74978-5 |
| [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 · doi:10.1080/01621459.1994.10476818 |
| [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 · doi:10.1214/aos/1031833666 |
| [11] | Stute, W.; Zhu, LX, Model checks for generalized linear models, Scandinavian Journal of Statistics, 29, 3, 535-545 (2002) · Zbl 1035.62073 · doi:10.1111/1467-9469.00304 |
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