Estimating an even spherical measure from its sine transform

Lars Michael Hoffmann

Applications of Mathematics (2009)

  • Volume: 54, Issue: 1, page 67-78
  • ISSN: 0862-7940

Abstract

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To reconstruct an even Borel measure on the unit sphere from finitely many values of its sine transform a least square estimator is proposed. Applying results by Gardner, Kiderlen and Milanfar we estimate its rate of convergence and prove strong consistency. We close this paper by giving an estimator for the directional distribution of certain three-dimensional stationary Poisson processes of convex cylinders which have applications in material science.

How to cite

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Hoffmann, Lars Michael. "Estimating an even spherical measure from its sine transform." Applications of Mathematics 54.1 (2009): 67-78. <http://eudml.org/doc/37808>.

@article{Hoffmann2009,
abstract = {To reconstruct an even Borel measure on the unit sphere from finitely many values of its sine transform a least square estimator is proposed. Applying results by Gardner, Kiderlen and Milanfar we estimate its rate of convergence and prove strong consistency. We close this paper by giving an estimator for the directional distribution of certain three-dimensional stationary Poisson processes of convex cylinders which have applications in material science.},
author = {Hoffmann, Lars Michael},
journal = {Applications of Mathematics},
keywords = {Boolean model; convex cylinder; direction distribution; least square estimator; parameter estimation; Poisson process; spherical measure; sine transform; Boolean model; convex cylinder; directional distribution; least squares estimator; Poisson process},
language = {eng},
number = {1},
pages = {67-78},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Estimating an even spherical measure from its sine transform},
url = {http://eudml.org/doc/37808},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Hoffmann, Lars Michael
TI - Estimating an even spherical measure from its sine transform
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 67
EP - 78
AB - To reconstruct an even Borel measure on the unit sphere from finitely many values of its sine transform a least square estimator is proposed. Applying results by Gardner, Kiderlen and Milanfar we estimate its rate of convergence and prove strong consistency. We close this paper by giving an estimator for the directional distribution of certain three-dimensional stationary Poisson processes of convex cylinders which have applications in material science.
LA - eng
KW - Boolean model; convex cylinder; direction distribution; least square estimator; parameter estimation; Poisson process; spherical measure; sine transform; Boolean model; convex cylinder; directional distribution; least squares estimator; Poisson process
UR - http://eudml.org/doc/37808
ER -

References

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  1. Billingsley, P., Convergence of Probability Measures, Wiley New York (1968). (1968) Zbl0172.21201MR0233396
  2. Gardner, R. J., Kiderlen, M., Milanfar, P., 10.1214/009053606000000335, Ann. Stat. 34 (2006), 1331-1374. (2006) Zbl1097.52503MR2278360DOI10.1214/009053606000000335
  3. Hoffmann, L. M., Mixed measures of convex cylinders and quermass densities of Boolean models, Submitted. Zbl1180.52011
  4. Hug, D., Schneider, R., Stability results involving surface area measures of convex bodies, Rend. Circ. Mat. Palermo (2) Suppl. No. 70, part II (2002), 21-51. (2002) Zbl1113.52013MR1962583
  5. Kallenberg, O., Foundations of Modern Probability, 2nd. ed, Springer New York (2002). (2002) Zbl0996.60001MR1876169
  6. Kiderlen, M., 10.1239/aap/999187894, Adv. Appl. Probab. 33 (2001), 6-24. (2001) Zbl0998.62080MR1825313DOI10.1239/aap/999187894
  7. Kiderlen, M., Pfrang, A., 10.1111/j.1365-2818.2005.01493.x, J. Microsc. 219 (2005), 50-60. (2005) MR2196184DOI10.1111/j.1365-2818.2005.01493.x
  8. Schladitz, K., Peters, S., Reinel-Bitzer, D., Wiegmann, A., Ohser, J., 10.1016/j.commatsci.2006.01.018, Comp. Mat. Sci. 38 (2006), 56-66. (2006) DOI10.1016/j.commatsci.2006.01.018
  9. Schneider, R., Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press Cambridge (1993). (1993) Zbl0798.52001MR1216521
  10. Schneider, R., Weil, W., Integralgeometrie, Teubner Stuttgart (1992), German. (1992) Zbl0762.52001MR1203777
  11. Schneider, R., Weil, W., Stochastische Geometrie, Teubner Stuttgart (2000), German. (2000) Zbl0964.52009MR1794753
  12. Spiess, M., Spodarev, E., Anisotropic dilated Poisson k -flat processes, Submitted. 
  13. Stoyan, D., Kendall, W. S., Mecke, J., Stochastic Geometry and Its Applications, 2nd ed, John Wiley & Sons Chichester (1995). (1995) Zbl0838.60002MR0895588

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