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References

Published online by Cambridge University Press:  05 May 2012

Abhishek Bhattacharya
Affiliation:
Indian Statistical Institute, Kolkata
Rabi Bhattacharya
Affiliation:
University of Arizona
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Nonparametric Inference on Manifolds
With Applications to Shape Spaces
, pp. 229 - 234
Publisher: Cambridge University Press
Print publication year: 2012

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References

Amit, Y. 2002. 2D Object Detection and Recognition: Models, Algorithms and Networks. Lecture Notes in Biomathematic, MIT Press, Cambridge, MA.
Anderson, C.R. 1997. Object recognition using statistical shape analysis. Ph.D. Thesis,University of Leeds.
Babu, G. J. and Singh, K. 1984. On one term Edgeworth correction by Efron's bootstrap. Sankhya Ser. A, 46, 219–232.Google Scholar
Bandulasiri, A. and Patrangenaru, V. 2005. Algorithms for nonparametric inference on shape manifolds. Proc. JSM 2005, Minneapolis, pp. 1617–1622.Google Scholar
Bandulasiri, A., Bhattacharya, R. N., and Patrangenaru, V. 2009. Nonparametric inference on shape manifolds with applications in medical imaging. J. Multivariate Analysis, 100, 1867–1882.Google Scholar
Barron, A. R., Schervish, M., and Wasserman, L. 1999. The consistency of posterior distribution in nonparametric problems. Ann. Statist., 27, 536–561.Google Scholar
Barron, A. R. 1989. Uniformly powerful goodness of fit tests. Ann. Statist., 17, 107–124.Google Scholar
Beran, R. J. 1968. Testing for uniformity on a compact homogeneous space. J. Appl.Probability, 5, 177–195.Google Scholar
Beran, R. J. 1987. Prepivoting to reduce level error of confidence sets. Biometrica, 74, 457–468.Google Scholar
Beran, R. J. and Fisher, N. I. 1998. Nonparametric comparison of mean axes. Ann.Statist., 26, 472–493.Google Scholar
Berthilsson, R. and Astrom, K. 1999. Extension of affine shape. J. Math. Imaging Vision, 11, 119–136.Google Scholar
Berthilsson, R. and Heyden, A. 1999. Recognition of planar objects using the density of affine shape. Computer Vision and Image Understanding, 76, 135–145.Google Scholar
Bhattacharya, A. 2008a. Nonparametric statistics on manifolds with applications toshape spaces. Ph.D. Thesis, University of Arizona.
Bhattacharya, A. 2008b. Statistical analysis on manifolds: a nonparametric approach for inference on shape spaces. Sankhya, 70-A, part 2, 223–266.Google Scholar
Bhattacharya, A. and Bhattacharya, R. N. 2008a. Nonparametric statistics on manifolds with application to shape spaces. In Pushing the Limits of Contemporary Statistics: Contributions in Honor of J.K. Ghosh. IMS Collections, 3, 282–301.Google Scholar
Bhattacharya, A. and Bhattacharya, R. N. 2008b. Statistical on Riemannian manifolds: asymptotic distribution and curvature. Proceedings of the American Mathematical Society, 136, 2957–2967.Google Scholar
Bhattacharya, A. and Bhattacharya, R. N. 2009. Statistical on manifolds with application to shape spaces. In Perspectives in Mathematical Sciences I: Probability and Mathematics, edited by N. S., Narasimha Sastry, T. S. S. R. K., Rao, M., Delampady and B., Rajeev. Indian Statistical Institute, Bangalore, 41–70.
Bhattacharya, A. and Dunson, D. 2010a. Nonparametric Bayesian density estimation on manifolds with applications to planar shapes. Biometrika, 97, 851–865.Google Scholar
Bhattacharya, A. and Dunson, D. 2010b. Nonparametric Bayes classification and testing on manifolds with applications on hypersphere. Discussion paper, Department of Statistical Science, Duke University.
Bhattacharya, A. and Dunson, D. 2011. Strong consistency of nonparametric Bayes density estimation on compact metric spaces. Ann. Institute of Statistical Mathematics, 63.Google Scholar
Bhattacharya, R. N. and Ghosh, J.K. 1978. On the validity of the formal Edgeworth expansion. Ann. Statist., 6, 434–451.Google Scholar
Bhattacharya, R. N. and Patrangenaru, V. 2003. Large sample theory of intrinsic and extrinsic sample means on manifolds. Ann. Statist., 31, 1–29.Google Scholar
Bhattacharya, R. N. and Patrangenaru, V. 2005. Large sample theory of intrinsic and extrinsic sample means on manifolds—II. Ann. Statist., 33, 1225–1259.Google Scholar
Bhattacharya, R. N. and Patrangenaru, V. 2012. A Course in Mathematical Statisticsand Large Sample Theory. Springer Series in Statistics. To appear.
Bhattacharya, R. N. and Qumsiyeh, M. 1989. Second order and Lp-comparisons between the bootstrap and empirical Edgeworth expansion methodologies. Ann.Statist., 17, 160–169.Google Scholar
Bhattacharya, R. N. and Waymire, E. 2007. A Basic Course in Probability Theory. Universitext, Springer, New York.
Bhattacharya, R. N. 1977. Refinements of the multidimensional central limit theorem and applications. Ann. Probability, 5, 1–27.Google Scholar
Bhattacharya, R. N. 1987. Some aspects of Edgeworth expansions in statistics and probability. In New Perspectives in Theoretical and Applied Statistics, edited by M., Puri, J., Villaplana, and W., Wertz. Wiley, New York.
Bhattacharya, R. N. 2007. On the uniqueness of intrinsic mean. Unpublished manuscript.
Bhattacharya, R. N. and Denker, M. 1990. Asymptotic Statistics. Vol. 14. DMV Seminar, Birkhauser, Berlin.
Bhattacharya, R. N. and Ranga Rao, R. 2010. Normal Approximation and Asymptotic Expansions. SIAM, Philadelphia.
Bickel, P. J. and Doksum, K. A. 2001. Mathematical Statistics, 2nd ed. Prentice Hall, Upper Saddle River, NJ.
Bingham, C. 1974. An antipodally symmetric distribution on the sphere. Ann. Statist., 2, 1201–1225.Google Scholar
Bookstein, F. 1978. The Measurement of Biological Shape and Shape Change. Lecture Notes in Biomathematics, Springer, Berlin.
Bookstein, F. L. 1986. Size and shape spaces of landmark data (with discussion). Statistical Science, 1, 181–242.Google Scholar
Bookstein, F. L. 1989. Principal warps: thin plate splines and the decomposition of deformations. Pattern Analysis and Machine Intelligence, 11, 567–585.Google Scholar
Bookstein, F. L. 1991. Morphometric Tools for Landmark data: Geometry and Biology. Cambridge University Press, Cambridge.
Boothby, W. M. 1986. An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. Academic Press, New York.
Burgoyne, C. F., Thompson, H.W., Mercante, D. E., and Amin, R. E. 2000. Basic issues in the sensitive and specific detection in optic nerve head surface change within longitudinal LDT TopSS image: Introduction to the LSU experimental glaucoma (LEG) study. In The Shape of Glaucoma, Quantitative Neural Imaging Techniques, edited by H. G., Lemij and J. S., Shuman, 1–37. Kugler Publications, The Hague, The Netherlands.
Casella, G. and Berger, R. L. 2001. Statistical Inference. Duxbury Press, Pacific Grove, CA.
Chandra, T. and Ghosh, J. K. 1979. Valid asymptotic expansion for the likelihood ratio statistics and other pertubed chi-square variables. Sankhya, Ser. A., 41, 22–47.Google Scholar
Chikuse, Y. 2003. Statistics on Special Manifolds. Springer, New York.
Diaconis, P., Holmes, S. and Shakshahani, M. 2012. Sampling from a manifold. To appear.
Dieudonné, J. 1970. Treatise on Analysis. Vol. 2. Academic Press, New York.
Dimitric, I. 1996. A note on equivariant embeddings of Grassmannians. Publ. Inst. Math (Beograd) (N.S.), 59, 131–137.Google Scholar
Dimroth, E. 1963. Fortschritte der Gefugestatistik. Neues Jahrbuch der Mineralogie, Montashefte 13, 186–192.Google Scholar
Do Carmo, M. 1992. Riemannian Geometry. Birkhäuser, Boston.
Dryden, I. L. and Mardia, K. V. 1992. Size and shape analysis of landmark data. Biometrica, 79, 57–68.Google Scholar
Dryden, I. L. and Mardia, K. V. 1998. Statistical Shape Analysis. Wiley, New York.
Dryden, I. L., Faghihi, M. R., and Taylor, C. C. 1997. Procrustes shape analysis of spatial point patterns. J. Roy. Statist. Soc. Ser. B, 59, 353–374.Google Scholar
Dryden, I. L., Kume, A., Le, H., and Wood, A. T. A. 2008. A multi-dimensional scaling approach to shape analysis. Biometrika, 95(4), 779–798.Google Scholar
Dunford, N. and Schwartz, J. 1958. Linear Operators—I. Wiley, New York.
Dunson, D. B. and Bhattacharya, A. 2010. Nonparametric Bayes regression and classification through mixtures of product kernels. Bayesian Statistics, 9, 145–164.Google Scholar
Efron, B. 1979. Bootstrap methods: another look at jackknife. Ann. Statist., 1, 1–26.Google Scholar
Ellingson, L., Ruymgaart, F. H., and Patrangenaru, V. 2011. Nonparametric estimation for extrinsic mean shapes of planar contours. To appear.
Embleton, B. J. J. and McDonnell, K. L. 1980. Magnetostratigraphy in the Sydney Basin, SouthEastern Australia. J. Geomag. Geoelectr., 32, 304.Google Scholar
Escobar, M. D. and West, M. 1995. Bayesian density-estimation and inference using mixtures. J. Am. Statist. Assoc., 90, 577–588.Google Scholar
Ferguson, T. S. 1973. A Bayesian analysis of some nonparametric problems. Ann. Statist., 1, 209–230.Google Scholar
Ferguson, T. S. 1974. Prior distributions on spaces of probability measures. Ann. Statist., 2, 615–629.Google Scholar
Ferguson, T. S. 1996. A Course in Large Sample Theory. Texts in Statistical ScienceSeries. Chapman & Hall, London.
Fisher, N. I., Hall, P., Jing, B., and Wood, A. T. A. 1996. Improved pivotal methods for constructing confidence regions with directional data. J. Amer. Statist. Assoc, 91, 1062–1070.Google Scholar
Fisher, N. I. 1993. Statistical Analysis of Circular Data. Cambridge University Press, Cambridge.
Fisher, N. I., Lewis, T., and Embleton, B. J. J. 1987. Statistical Analysis of Spherical Data. Cambridge University Press, Cambridge.
Fisher, R. A. 1953. Dispersion on a sphere. Proc. Roy. Soc. London Ser. A, 217, 295–305.Google Scholar
Fréchet, M. 1948. Lés élements aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré, 10, 215–310.Google Scholar
Gallot, S., Hulin, D., and Lafontaine, J. 1990. Riemannian Geometry. Universitext, Springer, Berlin.
Ghosh, J. K. and Ramamoorthi, R. 2003. Bayesian Nonparametrics. Springer, New York.
Goodall, C. R. 1991. Procrustes methods in the statistical analysis of shape (with discussion). J. Roy. Statist, Ser. B, 53, 285–339.Google Scholar
Hall, P. 1992. The Bootstrap and Edgeworth Expansion. Springer, New York.
Hendriks, H. and Landsman, Z. 1996. Asymptotic tests for mean location on manifolds. C.R. Acad. Sci. Paris Sr. I Math., 322, 773–778.Google Scholar
Hendriks, H. and Landsman, Z. 1998. Mean location and sample mean location on manifolds: asymptotics, tests, confidence regions. J. Multivariate Anal., 67, 227–243.Google Scholar
Hjort, N., Holmes, C., Muller, P., and Walker, S. G. 2010. Bayesian Nonparametrics. Cambridge University Press. Cambridge.
Hopf, H., and Rinow, W. 1931. Über den Begriff der vollständigen differentialgeometrischen Flache. Comment. Math. Helv., 3, 209–225.Google Scholar
Hopkins, J. W. 1966. Some considerations in multivariate allometry. Biometrics, 22, 747–760.Google Scholar
Huckemann, S., Hotz, T., and Munk, A. 2010. Intrinsic shape analysis: geodesic PCA for Riemannian manifolds modulo isometric Lie group actions (with discussions). Statist. Sinica, 20, 1–100.Google Scholar
Irving, E. 1963. Paleomagnetism of the Narrabeen Chocolate Shale and the Tasmanian Dolerite. J. Geophys. Res., 68, 2282–2287.Google Scholar
Irving, E. 1964. Paleomagnetism and Its Application to Geological and Geographical Problems. Wiley, New York.
Ishwaran, H. and Zarepour, M. 2002. Dirichlet prior sieves in finite normal mixtures. Statistica Sinica, 12, 941–963.Google Scholar
Johnson, R. A. and Wehrly, T. 1977. Measures and models for angular correlation and angular-linear correlation. J. Royal Stat. Soc. B, 39, 222–229.Google Scholar
Karcher, H. 1977. Riemannian center of mas and mollifier smoothing. Comm. Pure Appl. Math., 30, 509–554.Google Scholar
Kendall, D. G. 1977. The diffusion of shape. Adv. Appl. Probab., 9, 428–430.Google Scholar
Kendall, D. G. 1984. Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc., 16, 81–121.Google Scholar
Kendall, D. G. 1989. A survey of the statistical theory of shape. Statist. Sci., 4, 87–120.Google Scholar
Kendall, D. G., Barden, D., Carne, T. K., and Le, H. 1999. Shape and Shape Theory. Wiley, New York.
Kendall, W. S. 1990. Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. London Math. Soc, 61, 371–406.Google Scholar
Kent, J. T. 1992. New directions in shape analysis. In The Art of Statistical Science, edited by K. V., Mardia. Wiley, Chichester.
Kent, J. T. 1994. The complex Bingham distribution and shape analysis. J. Roy. Statist. Soc. Ser. B, 56, 285–299.Google Scholar
Kent, J. T. and Mardia, K. V. 1997. Consistency of Procustes estimators. J. Roy. Statist. Soc. Ser. B, 59, 281–290.Google Scholar
Krim, H. and Yezzi, A., eds. 2006. Statistics and Analysis of Shapes. Birkhauser, Boston.
Lahiri, S. N. 1994. Two term Edgeworth expansion and bootstrap approximation for multivariate studentized M-estimators. Sankhya Ser. A., 56, 201–226.Google Scholar
Le, H. 2001. Locating Fréchet means with application to shape spaces. Adv. Appl.Prob., 33, 324–338.Google Scholar
LeCam, L. 1973. Convergence of estimates under dimensionality restrictions. Ann. Statist., 1, 38–53.Google Scholar
Lee, J. and Ruymgaart, F. H. 1996. Nonparametric curve estimation on Stiefel manifolds. J. Nonparametri. Statist., 6, 57–68.Google Scholar
Lee, J. M. 1997. Riemannian Manifolds: An Introduction to Curvature. Springer, New York.
Lele, S. 1991. Some comments on coordinate free and scale invariant methods in morphometrics. Amer. J. Physi. Anthropology, 85, 407–418.Google Scholar
Lele, S. 1993. Euclidean distance matrix analysis (EDMA): estimation of mean form and mean form difference. Math. Geology, 25, 573–602.Google Scholar
Lele, S. and Cole, T. M. 1995. Euclidean distance matrix analysis: a statistical review. In Current Issues in Statistical Shape Analysis, edited by K. V., Mardia and C. A., Gill. University of Leeds Press, Leeds, 49–53.
Lewis, J. L., Lew, W. D., and Zimmerman, J. R. 1980. A non-homogeneous anthropometric scaling method based on finite element principles. J. Biomech., 13, 815–824.Google Scholar
Lo, A. Y. 1984. On a class of Bayesian nonparametric estimates. 1. Density estimates.Ann. Statist., 12, 351–357.Google Scholar
Mardia, K. V. and Jupp, P. E. 2000. Directional Statistics. Wiley, New York.
Mardia, K. V. and Patrangenaru, V. 2005. Directions and projective shapes. Ann. Statist., 33, 1666–1699.Google Scholar
Micheas, A. C., Dey, D. K., and Mardia, K. V. 2006. Complex elliptic distributions with applications to shape analysis. J. Statist. Plan. and Inf., 136, 2961–2982.Google Scholar
Millman, R. and Parker, G. 1977. Elements of Differential Geometry. Prentice-Hall, Upper Saddle River, NJ.
NOAA National Geophysical Data Center Volcano Location Database, 1994. http:// www.ngdc.noaa.gov/nndc/struts/results?
Oller, J. M. and Corcuear, J. M. 1995. Intrinsic analysis of statistical estimation. Ann. Statist., 23, 1562–1581.Google Scholar
Parthasarathy, K. R. 1967. Probability Measures on Metric Spaces. Academic Press. New York.
Patrangenaru, V. 1998. Asymptotic statistics on manifolds and their applications. Ph.D. Thesis, Indiana University, Bloomington.
Patrangenaru, V., Liu, X., and Sagathadasa, S. 2010. A nonparametric approach to 3D shape analysis from digital camera images-I. J. Multivariate Analysis, 101, 11–31.Google Scholar
Prentice, M. J. 1984. A distribution-free method of interval estimation for unsigned directional data. Biometrica, 71, 147–154.Google Scholar
Prentice, M. J. and Mardia, K. V. 1995. Shape changes in the plane for landmark data. Ann. Statist., 23, 1960–1974.Google Scholar
Schwartz, L. 1965. On Bayes procedures. Z. Wahrsch. Verw. Gebiete, 4, 10–26.Google Scholar
Sepiashvili, D., Moura, J. M. F., and Ha, V. H. S. 2003. Affine-permutation symmetry: invariance and shape space. Proceedings of the 2003 Workshop on Statistical Signal Processing, St. Louis, MO, 293–296.
Sethuraman, J. 1994. A constructive definition of Dirichlet priors. Statist. Sinica, 4, 639–50.Google Scholar
Singh, K. 1981. On the asymptotic accuracy of Efron's bootstrap. Ann. Statist., 9, 1187–1195.Google Scholar
Small, C. G. 1996. The Statistical Theory of Shape. Springer, New York.
Sparr, G. 1992. Depth-computations from polihedral images. In Proc. 2nd European Conf. on Computer Vision, edited by G., Sandimi. Springer, New York, 378–386. Also in Image and Vision Computing, 10, 683–688.
Sprent, P. 1972. The mathematics of size and shape. Biometrics., 28, 23–37.Google Scholar
Stoyan, D. 1990. Estimation of distances and variances in Bookstein's landmark model. Biometrical J., 32, 843–849.Google Scholar
Sugathadasa, S. 2006. Affine and projective shape analysis with applications. Ph.D. dissertation, Texas Tech University.
von Mises, R. 1918. Uber die “Ganzzaligkeit” der atomgewichte und verwandte fragen. Phys. Z., 19, 490–500.Google Scholar
Watson, G. S. 1965. Equatorial distributions on a sphere. Biometrica, 52, 193–201.Google Scholar
Watson, G. S. 1983. Statistics on Spheres. Vol. 6. University Arkansas Lecture Notes in the Mathematical Sciences, Wiley, New York.
Wu, Y. and Ghosal, S. 2008. Kullback–Liebler property of kernel mixture priors in Bayesian density estimation. Electronic J. of Statist., 2, 298–331.Google Scholar
Wu, Y. and Ghosal, S. 2010. The L1-consistency of Dirichlet mixtures in multivariate Bayesian density estimation on Bayes procedures. J. Mutivar. Analysis, 101, 2411–2419.Google Scholar
Yau, C., Papaspiliopoulos, O., Roberts, G. O., and Holmes, C. 2011. Nonparametric Hidden Markov Models with application to the analysis of copy-number-variation in mammalian genomes. J. Roy. Statist. Soc. B, 73, 37–57.Google Scholar
Ziezold, H. 1977. On expected figures and a strong law of large numbers for random elements in quasi-metric spaces. Transactions of the Seventh Prague Conference on Information Theory, Statistical Functions, Random Processes and of the Eighth European Meeting of Statisticians, A, 591–602. Tech. Univ. Prague, Prague.

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