Abstract
Testing procedures for assessing specific parametric model forms, or for checking the plausibility of simplifying assumptions, play a central role in the mathematical treatment of the uncertain. No certain answers are obtained by testing methods, but at least the uncertainty of these answers is properly quantified. This is the case for tests designed on the two most general data generating mechanisms in practice: distribution/density and regression models. Testing proposals are usually formulated on the Euclidean space, but important challenges arise in non-Euclidean settings, such as when directional variables (i.e., random vectors on the hypersphere) are involved. This work reviews some of the smoothing-based testing procedures for density and regression models that comprise directional variables. The asymptotic distributions of the revised proposals are presented, jointly with some numerical illustrations justifying the need of employing resampling mechanisms for effective test calibration.
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References
Bai ZD, Rao CR, Zhao LC (1988) Kernel estimators of density function of directional data. J Multivar Anal 27(1):24–39
Bickel PJ, Rosenblatt M (1973) On some global measures of the deviations of density function estimates. Ann Stat 1(6):1071–1095
Boente G, Rodríguez D, González-Manteiga W (2014) Goodness-of-fit test for directional data. Scand J Stat 41(1):259–275
Durbin J (1973) Weak convergence of the sample distribution function when parameters are estimated. Ann Stat 1:279–290
Elderton WP (1902) Tables for testing the goodness of fit of theory to observation. Biometrika 1(2):155–163
Fan J, Gijbels I (1996) Local polynomial modelling and its applications, vol 66. Monographs on statistics and applied probability. Chapman and Hall, London
Fan Y (1994) Testing the goodness of fit of a parametric density function by kernel method. Econom Theory 10(2):316–356
Fisher NI, Lee AJ (1981) Nonparametric measures of angular-linear association. Biometrika 68(3):629–636
García-Portugués E, Crujeiras RM, González-Manteiga W (2013) Kernel density estimation for directional-linear data. J Multivar Anal 121:152–175
García-Portugués E, Barros AMG, Crujeiras RM, González-Manteiga W, Pereira J (2014) A test for directional-linear independence, with applications to wildfire orientation and size. Stoch Environ Res Risk Assess 28(5):1261–1275
García-Portugués E, Crujeiras RM, González-Manteiga W (2015) Central limit theorems for directional and linear data with applications. Stat Sin 25:1207–1229
García-Portugués E, Van Keilegom I, Crujeiras R, González-Manteiga W (2016) Testing parametric models in linear-directional regression. Scand J Stat 43(4):1178–1191
González-Manteiga W, Crujeiras RM (2013) An updated review of goodness-of-fit tests for regression models. Test 22(3):361–411
Hall P, Watson GS, Cabrera J (1987) Kernel density estimation with spherical data. Biometrika 74(4):751–762
Härdle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21(4):1926–1947
Liddell IG, Ord JK (1978) Linear-circular correlation coefficients: some further results. Biometrika 65(2):448–450
Mardia KV (1976) Linear-circular correlation coefficients and rhythmometry. Biometrika 63(2):403–405
Mardia KV, Jupp PE (2000) Directional statistics, 2nd edn. Wiley series in probability and statistics. Wiley, Chichester
Pearson K (1900) On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. The London, Edinburgh, and Dublin Philos Mag and J Sci Ser-5 50(302):157–175
Pearson K (1916) On the application of “goodness of fit” tables to test regression curves and theoretical curves used to describe observational or experimental data. Biometrika 11(3):239–261
Rosenblatt M (1975) A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Ann Stat 3(1):1–14
Zhao L, Wu C (2001) Central limit theorem for integrated square error of kernel estimators of spherical density. Sci China Ser A 44(4):474–483
Acknowledgements
The authors acknowledge the support of project MTM2016-76969-P from the Spanish State Research Agency (AEI), Spanish Ministry of Economy, Industry and Competitiveness, and European Regional Development Fund (ERDF). We also thank Eduardo Gil, Eva Gil, Juan J. Gil, and María Angeles Gil for inviting us to contribute to this volume, in memory of Pedro.
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García-Portugués, E., Crujeiras, R.M., González-Manteiga, W. (2018). Smoothing-Based Tests with Directional Random Variables. In: Gil, E., Gil, E., Gil, J., Gil, M. (eds) The Mathematics of the Uncertain. Studies in Systems, Decision and Control, vol 142. Springer, Cham. https://doi.org/10.1007/978-3-319-73848-2_17
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DOI: https://doi.org/10.1007/978-3-319-73848-2_17
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