| Title:
|
Estimating an even spherical measure from its sine transform (English) |
| Author:
|
Hoffmann, Lars Michael |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2009 |
| Pages:
|
67-78 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Boolean model |
| Keyword:
|
convex cylinder |
| Keyword:
|
direction distribution |
| Keyword:
|
least square estimator |
| Keyword:
|
parameter estimation |
| Keyword:
|
Poisson process |
| Keyword:
|
spherical measure |
| Keyword:
|
sine transform |
| MSC:
|
52A22 |
| MSC:
|
60D05 |
| MSC:
|
60G10 |
| MSC:
|
62H11 |
| MSC:
|
62M30 |
| MSC:
|
65D15 |
| idZBL:
|
Zbl 1211.62092 |
| idMR:
|
MR2476022 |
| DOI:
|
10.1007/s10492-009-0005-9 |
| . |
| Date available:
|
2010-07-20T12:47:24Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140350 |
| . |
| Reference:
|
[1] Billingsley, P.: Convergence of Probability Measures.Wiley New York (1968). Zbl 0172.21201, MR 0233396 |
| Reference:
|
[2] Gardner, R. J., Kiderlen, M., Milanfar, P.: Convergence of algorithms for reconstructing convex bodies and directional measures.Ann. Stat. 34 (2006), 1331-1374. Zbl 1097.52503, MR 2278360, 10.1214/009053606000000335 |
| Reference:
|
[3] Hoffmann, L. M.: Mixed measures of convex cylinders and quermass densities of Boolean models.Submitted. Zbl 1180.52011 |
| Reference:
|
[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. Zbl 1113.52013, MR 1962583 |
| Reference:
|
[5] Kallenberg, O.: Foundations of Modern Probability, 2nd. ed.Springer New York (2002). Zbl 0996.60001, MR 1876169 |
| Reference:
|
[6] Kiderlen, M.: Non-parametric estimation of the directional distribution of stationary line and fibre processes.Adv. Appl. Probab. 33 (2001), 6-24. Zbl 0998.62080, MR 1825313, 10.1239/aap/999187894 |
| Reference:
|
[7] Kiderlen, M., Pfrang, A.: Algorithms to estimate the rose of directions of a spatial fibre system.J. Microsc. 219 (2005), 50-60. MR 2196184, 10.1111/j.1365-2818.2005.01493.x |
| Reference:
|
[8] Schladitz, K., Peters, S., Reinel-Bitzer, D., Wiegmann, A., Ohser, J.: Design of accoustic trim based on geometric modeling and flow simulation for non-woven.Comp. Mat. Sci. 38 (2006), 56-66. 10.1016/j.commatsci.2006.01.018 |
| Reference:
|
[9] Schneider, R.: Convex Bodies: the Brunn-Minkowski Theory.Cambridge University Press Cambridge (1993). Zbl 0798.52001, MR 1216521 |
| Reference:
|
[10] Schneider, R., Weil, W.: Integralgeometrie.Teubner Stuttgart (1992), German. Zbl 0762.52001, MR 1203777 |
| Reference:
|
[11] Schneider, R., Weil, W.: Stochastische Geometrie.Teubner Stuttgart (2000), German. Zbl 0964.52009, MR 1794753 |
| Reference:
|
[12] Spiess, M., Spodarev, E.: Anisotropic dilated Poisson $k$-flat processes.Submitted. |
| Reference:
|
[13] Stoyan, D., Kendall, W. S., Mecke, J.: Stochastic Geometry and Its Applications, 2nd ed.John Wiley & Sons Chichester (1995). Zbl 0838.60002, MR 0895588 |
| . |
