| Title:
|
Some properties and applications of probability distributions based on MacDonald function (English) |
| Author:
|
Kropáč, Oldřich |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
27 |
| Issue:
|
4 |
| Year:
|
1982 |
| Pages:
|
285-302 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| Summary lang:
|
Russian |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the paper the basic analytical properties of the MacDonald function (the modified Bessel function of the second kind) are summarized and the properties of some subclasses of distribution functions based on MacDonald function, especially of the types $x^nK_n(x), x\geq 0, \left|x\right|^n K_n(x\left|x\right|), x\in \bold R$ and $x^{n+1}K_n(x), x\geq 0$ are discussed. The distribution functions mentioned are useful for analytical modelling of composed (mixed) distributions, especially for products of random variables having distributions of the exponential type. Extensive and useful applications may be found in the field of non-Gaussian random processes, the marginal and joint probability densities of which and of their envelopes may be described by means of the types discussed. (English) |
| Keyword:
|
MacDonald function |
| Keyword:
|
Bessel function of the second kind |
| Keyword:
|
composed distributions |
| MSC:
|
33A40 |
| MSC:
|
33C10 |
| MSC:
|
60E99 |
| MSC:
|
62E15 |
| idZBL:
|
Zbl 0491.60021 |
| idMR:
|
MR0666907 |
| DOI:
|
10.21136/AM.1982.103973 |
| . |
| Date available:
|
2008-05-20T18:19:44Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103973 |
| . |
| Reference:
|
[1] H. Bateman A. Erdélyi: Higher transcendental functions.Vol. 2. New York, McGraw-Hill, 1953. MR 0058756 |
| Reference:
|
[2] H. Bateman A. Erdélyi: Tables of integral transforms, Vol. I, II.New York, McGraw-Hill, 1954. MR 0061695 |
| Reference:
|
[3] K. C. Chu: Estimation and decision for linear systems with elliptical random processes.IEEE Trans. Autom. Control, AC-18, 1973, pp. 499-505. Zbl 0263.93049, MR 0441500, 10.1109/TAC.1973.1100374 |
| Reference:
|
[4] C. C. Craig: On the frequency function xy.Ann. Math. Statist., 7, 1936, pp. 1 - 15. 10.1214/aoms/1177732541 |
| Reference:
|
[5] E. M. Elderton: Table of the product moment $T_m$ function.Biometrika, 21, 1929, pp. 194-201. 10.1093/biomet/21.1-4.194 |
| Reference:
|
[6] Jahnke-Emde-Lösch: Tafeln höherer Funktionen.Stuttgart, Teubner, 1960. |
| Reference:
|
[7] N. L. Johnson S. Kotz: Continuous multivariate distributions.New York, J. Wiley, 1972. |
| Reference:
|
[8] O. Kropáč: Relations between distributions of random vibratory processes and distributions of their envelopes.Aplik. Matem., 17, 1972, pp. 75-112. MR 0299032 |
| Reference:
|
[9] O. Kropáč: Rozdělení s náhodnými parametry a jejich inženýrské aplikace.Strojn. Čas. 33, 1980, pp. 597-622. |
| Reference:
|
[10] O. Kropáč: A unified model for non-stationary and/or non-Gaussian random processes.J. Sound Vibr., 79, 1981, pp. 11 - 21. MR 0634635, 10.1016/0022-460X(81)90326-6 |
| Reference:
|
[11] O. Kropáč: Some general properties of elliptically symmetric random processes.Kybernetika, 17, 1981, pp. 401-412. MR 0648212 |
| Reference:
|
[12] S. Kullback: The distribution laws of the difference and quotient of variables independently distributed in Pearson III laws.Ann. Math. Statist., 7, 1936, pp. 51 - 53. 10.1214/aoms/1177732546 |
| Reference:
|
[13] D. K. McGraw J. F. Wagner: Elliptically symmetric distributions.IEEE Trans. Inform. Theory, IT-14, 1968, pp. 110-120. 10.1109/TIT.1968.1054081 |
| Reference:
|
[14] K. Pearson G. B. Jeffery E. M. Elderton: On the distribution of the first product moment-coefficient, in samples drawn from an indefinitely large normal population.Biometrika, 21, 1929, pp. 164-193. 10.1093/biomet/21.1-4.164 |
| Reference:
|
[15] K. Pearson S. A. Stouffer F. N. David: Further applications in statistics of the $T_m (x)$ Bessel function.Biometrika, 24, 1932, pp. 293 - 350. |
| Reference:
|
[16] D. Teichrow: The mixture of normal distributions with different variances.Ann. Math. Statist., 28, 1957, pp. 510-512. MR 0089551, 10.1214/aoms/1177706981 |
| Reference:
|
[17] G. N. Watson: A treatise on the theory of Bessel functions.Cambridge, Univ. Press, 1944 (2nd edit.). Zbl 0063.08184, MR 0010746 |
| . |
