Data-driven Sobolev tests of uniformity on compact Riemannian manifolds

Annals of Statistics

Volumn 36, Issue 3, 2008, Pages 1246-1260

  Jupp, P E ^a  

^a UNIVERSITY OF ST ANDREWS   (United Kingdom)

Author keywords

Consistency; Directional statistics; Invariance; Penalized score test

Indexed keywords

EID: 51049101508     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053607000000541     Document Type: Article

References (24)

1. BERAN, R. (1979). Exponential models for directional data. Ann. Statist. 7 1162-1178. MR0550142
2. BOGDAN, M., BOGDAN, K. and FUTSCHIK, A. (2002). A data driven smooth test for circular uniformity. Ann. Inst. Statist. Math. 54 29-44. MR1893540
3. BOOTHBY, W. (2003). An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. Academic Press, Orlando. MR0861409
4. CHAVEL, I. (1984). Eigenvalues in Riemannian Geometry. Academic Press, Orlando. MR0768584
5. -1. Biometrika 78 573-582. MR1130925
6. DUISTERMAAT, J. J. and GUILLEMIN, V. W. (1975). The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29 39-79. MR0405514
7. GINE, M. E. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Statist. 3 1243-1266. MR0388663
8. HELGASON, S. (1978). Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York. MR0514561
9. HENDRIKS, H. (1990). Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions. Ann. Statist. 18 832-849. MR1056339
10. HÖRMANDER. L. (1968). The spectral function of an elliptic operator. Acta Math.121193-218. MR0609014
11. INGLOT, T. (1999). Generalized intermediate efficiency of goodness-of-fit tests. Math. Methods Statist. 8 487-509. MR1755897
12. INGLOT, T., KALLENBERG, W. C. M. and LEDWINA, T. (1997). Data driven smooth tests for composite hypotheses. Ann. Statist. 25 1222-1250. MR1447749
13. INGLOT, T. and LEDWINA, T. (1996). Asymptotic optimality of data-driven Neyman's test. Ann. Statist. 24 1982-2019. MR1421157
14. INGLOT, T. and LEDWINA, T. (2001). Intermediate approach to comparison of some goodness-of-fit tests. Ann. Inst. Statist. Math. 53 810-834. MR1880814
15. KALLENBERG, W. C. M. and LEDWINA, T. (1995). Consistency and Monte Carlo simulation of a data driven version of smooth goodness-of-fit tests. Ann. Statist. 23 1594-1608. MR 1370299
16. KALLENBERG, W. C. M. and LEDWINA, T. (1997). Data-driven smooth tests when the hypothesis is composite. J. Amer. Statist. Assoc. 92 1094-1104. MR1482L40
17. KALLENBERG, W. C. M. and LEDWINA, T. (1997). Data driven smooth tests for composite hypotheses: comparison of powers. J. Statist. Comput. Simulation 59 101-121. MR1488856
18. LEDWINA, T. (1994). Data-driven versions of Neyman's smooth test of fit. J. Amer. Statist. Assoc. 89 1000-1005. MR1294744
19. MARDIA, K. V. and JUPP, P. E. (2000). Directional Statistics. Wiley, Chichester. MR 1828667
20. MINAKSHISUNDARAM, S. and PLEUEL, Å. (1949). Some properties of the eigenfunctions of the Laplace-operator on Riemannnian manifolds. Canad. J. Math. 1 242-256. MR0031145
21. NEYMAN, J. (1937). 'Smooth test for goodness of fit. Skand. Aktuarietidsskrift 20 149-199.
22. PRENTICE, M. J. (1978). On invariant tests of uniformity for directions and orientations. Ann. Statist. 6 169-176. MR0458721
23. ROE, J. (1988). Elliptic Operators, Topology and Asymptotic Methods. Longman, Harlow. MR0960889
24. SCHWARZ, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461-464. MR0468014