| Title:
|
On the accuracy of approximation of the distribution of negative-binomial random sums by the gamma distribution (English) |
| Author:
|
Hung, Tran Loc |
| Author:
|
Hau, Tran Ngoc |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
54 |
| Issue:
|
5 |
| Year:
|
2018 |
| Pages:
|
921-936 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The main goal of this paper is to study the accuracy of approximation for the distributions of negative-binomial random sums of independent, identically distributed random variables by the gamma distribution. (English) |
| Keyword:
|
gamma distribution |
| Keyword:
|
negative-binomial random sums |
| Keyword:
|
Trotter's distance |
| MSC:
|
60F05 |
| MSC:
|
60G50 |
| idZBL:
|
Zbl 07031752 |
| idMR:
|
MR3893128 |
| DOI:
|
10.14736/kyb-2018-5-0921 |
| . |
| Date available:
|
2018-12-14T08:06:05Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147535 |
| . |
| Reference:
|
[1] Bevrani, H., Bening, V. E., Korolev, V. Yu.: On the accuracy of approximation of the negative-binomial distribution by the gamma distribution and convergence rate of the distributions of some statistics to the Student distribution..J. Math. Sci. 205 (2015), 1, 34-44. MR 3403276, 10.1007/s10958-015-2227-6 |
| Reference:
|
[2] Brown, M.: Error bound for exponential approximations of geometric convolutions..Ann. Probab. 18 (1990), 3, 1388-1402. MR 1062073, 10.1214/aop/1176990750 |
| Reference:
|
[3] Butzer, P. L., Hahn, L., U., Westphal: On the rate of approximation in the central limit theorem..J. Approx. Theory 13 (1975), 327-340. MR 0394809, 10.1016/0021-9045(75)90042-8 |
| Reference:
|
[4] Butzer, P. L., Hahn, L.: General theorems on rates of convergence in distribution of random variables I. General limit theorems..J. Multivariate Anal. 8 (1978), 181-201. MR 0497594, 10.1016/0047-259x(78)90071-4 |
| Reference:
|
[5] Butzer, P. L., Kirschfink, H., Schulz, D.: An extension of the Lindeberg-Trotter operator-theoretic approach to limit theorems for dependent random variables..Acta Sci. Math. 51 (1987), 423-433. MR 0940946 |
| Reference:
|
[6] Gavrilenko, S. V., Zubov, V. N., Korolev, V. Yu.: The rate of convergence of the distributions of regular statistics constructed from samples with negatively binomially distributed random sizes to the student distribution..J. Math. Sci. 220 (2017), 6, 701-713. MR 3595558, 10.1007/s10958-016-3213-3 |
| Reference:
|
[7] Grandell, J.: Risk Theory and geometric sums..Inform. Processes 2 (2002) 2, 180-181. |
| Reference:
|
[8] Gut, A.: Probability: A Graduate Course..Springer Texts in Statistics. Springer, New York 2005. Zbl 1267.60001, MR 2125120 |
| Reference:
|
[9] Hogg, R. V., McKean, J. W., Craig, A. T.: Introduction to Mathematical Statistics. Seventh edition..Pearson Education, Inc. 2013. MR 0137186 |
| Reference:
|
[10] Hung, T. L.: On a Probability metric based on Trotter operator..Vietnam J. Math. 35 (2007), 1, 22-33. MR 2317431 |
| Reference:
|
[11] Hung, T. L.: On the rate of convergence in limit theorems for geometric sums..Southeast Asian J. Sci. 2 (2013) 2, 117-130. |
| Reference:
|
[12] Kalashnikov, V.: Geometric Sums: Bounds for Rare Events with Applications..Kluwer Academic Publisher 1997. MR 1471479 |
| Reference:
|
[13] Kirschfink, H.: The generalized Trotter operator and weak convergence of dependent random variables in different probability metrics..Results Math. 15 (1989), 294-323. MR 0997067, 10.1007/BF03322619 |
| Reference:
|
[14] Korolev, V. Yu.: Convergence of random sequences with independent random indices. I (in Russian)..Teor. Veroyatnost. i Primenen. 39 (1994), 2, 313-333; translation in Theory Probab. Appl. 39 (1995), 2, 282-297. MR 1404685 |
| Reference:
|
[15] Kruglov, V. M., Korolev, V. Yu.: Limit theorems for Random Sums (in Russian)..Moskov. Gos. Univ., Moscow 1990. MR 1072999 |
| Reference:
|
[16] Malinowski, M. T.: Geometrically strictly semi-stable law as the limit laws..Discuss. Math. Probab. Statist. 27 (2007), 79-97. MR 2414775, 10.7151/dmps.1089 |
| Reference:
|
[17] Peköz, E., Röllin, A.: New rates for exponential approximation and the theorems of Rényi and Yaglom..Ann. Probab. 39 (2011), 2, 587-608. MR 2789507, 10.1214/10-aop559 |
| Reference:
|
[18] Peköz, E., Röllin, A., Ross, N.: Generalized Gamma approximation with rates for urns, walks and trees..Ann. Probab. 44 (2016) 3, 1776-1816. MR 3502594, 10.1214/15-aop1010 |
| Reference:
|
[19] Rényi, A.: Probability Theory..North-Holland Series in Applied Mathematics and Mechanics 10, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York. 1970. MR 0315747 |
| Reference:
|
[20] Rychlick, Z.: A central limit theorem for sums of a random number of independent random variables..Coll. Math. 35 (1976), 147-158. MR 0410866, 10.4064/cm-35-1-147-158 |
| Reference:
|
[21] Sunklodas, J. K.: On the normal approximation of a negative binomial random sum..Lith. Math. J. 55 (2015), 1, 150-158. MR 3323288, 10.1007/s10986-015-9271-2 |
| Reference:
|
[22] Trotter, H. F.: An elementary proof of the central limit theorem..Arch. Math. 10 (1959), 226-234. MR 0108847, 10.1007/BF01240790 |
| . |
