# zbMATH — the first resource for mathematics

Distribution free testing for conditional distributions given covariates. (English) Zbl 1379.62036
Summary: Given a sample $$(\xi_i,X_i)_{i=1}^n$$ of i.i.d. pairs, consider $$F_x$$ and $$Q_x$$ as two different models for conditional distribution of $$\xi_i$$ given $$X_i=x$$. The paper shows how the empirical process for testing $$F_x$$ can be transformed into a process with the same asymptotic behavior as the empirical process for testing $$Q_x$$ thereby rendering the two testing problems equivalent.

##### MSC:
 62G10 Nonparametric hypothesis testing 62F03 Parametric hypothesis testing 62F05 Asymptotic properties of parametric tests 62G20 Asymptotic properties of nonparametric inference
Full Text:
##### References:
 [1] Delgado, M. A.; Stute, W., Distribution-free specification tests of conditional models, J. Econometrics, 143, 37-55, (2008) · Zbl 1418.62193 [2] Khmaladze, Estate, Unitary transformations, empirical processes and distribution free testing, Bernoulli, 22, 563-588, (2016) · Zbl 1345.60094 [3] Khmaladze, Estate V., The use of omega-square tests for testing parametric hypotheses, Theory Probab. Appl., XXIV, 283-302, (1979) [4] Khmaladze, Estate V., Martingale approach to nonparametric goodness of fit tests, Theory Probab. Appl., XXVI, 240-258, (1981) [5] Khmaladze, Estate V., Goodness of fit problem and scanning innovation martingales, Ann. Statist., 21, 798-830, (1993) · Zbl 0801.62043 [6] Khmaladze, Estate V., Note on distribution free testing for discrete distributions, Ann. Statist., 41, 2979-2993, (2013) · Zbl 1294.62095 [7] Khmaladze, Estate V.; Koul, Hira L., Martingale transforms goodness of fit tests in regression models, Ann. Statist., 32, 955-1034, (2004) · Zbl 1092.62052 [8] Koenker, Roger, Quantile Regression, (2005), Cambridge University Press [9] Koenker, Roger; Xiao, Z., Inference on the quantile regression process, Econometrica, 70, 1583-1612, (2002) · Zbl 1152.62339 [10] Koul, Hira L., (Weighted Empirical Processes in Dynamic Nonlinear Models, Lecture Notes in Statistics, Vol. 166, (2002), Springer) · Zbl 1007.62047 [11] Kuo, H.-H., (Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, (1975), Springer) [12] Nguyen, Thuong T. M., New approach to distribution free tests in contingency tables, Metrika, 80, 153-170, (2017) · Zbl 1394.62074 [13] Pyke, Ronald, A uniform central limit theorem for partial-sum processes indexed by sets, (Probability, Statistics and Analysis, London Math. Soc. Lecture Note Ser. Vol. 79, (1983), Cambridge University Press), 219-240 · Zbl 0497.60030 [14] Roberts, Leigh, On Distribution Free Testing in Bernoulli Trials with Distributions Depending on Covariates, (2016), VUW, SEF Reports [15] Stute, W., Nonparametric model checks for regression, Ann. Statist., 25, 613-641, (1997) · Zbl 0926.62035 [16] van der Vaart, A. W., Asymptotic Statistics, (1998), Cambridge University Press · Zbl 0910.62001 [17] van der Vaart, Aad W.; Wellner, Jon A., Weak Convergence and Empirical Processes, (1996), Springer · Zbl 0862.60002
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.