| Title:
|
Approximated maximum likelihood estimation of parameters of discrete stable family (English) |
| Author:
|
Slámová, Lenka |
| Author:
|
Klebanov, Lev B. |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
50 |
| Issue:
|
6 |
| Year:
|
2014 |
| Pages:
|
1065-1076 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this article we propose a method of parameters estimation for the class of discrete stable laws. Discrete stable distributions form a discrete analogy to classical stable distributions and share many interesting properties with them such as heavy tails and skewness. Similarly as stable laws discrete stable distributions are defined through characteristic function and do not posses a probability mass function in closed form. This inhibits the use of classical estimation methods such as maximum likelihood and other approach has to be applied. We depart from the $\mathcal{H}$-method of maximum likelihood suggested by Kagan (1976) where the likelihood function is replaced by a function called informant which is an approximation of the likelihood function in some Hilbert space. For this method only some functionals of the distribution are required, such as probability generating function or characteristic function. We adopt this method for the case of discrete stable distributions and in a simulation study show the performance of this method. (English) |
| Keyword:
|
discrete stable distribution |
| Keyword:
|
parameter estimation |
| Keyword:
|
maximum likelihood |
| MSC:
|
60E07 |
| MSC:
|
60E10 |
| MSC:
|
62F12 |
| MSC:
|
62G05 |
| MSC:
|
65C60 |
| idZBL:
|
Zbl 1308.60022 |
| idMR:
|
MR3301786 |
| DOI:
|
10.14736/kyb-2014-6-1065 |
| . |
| Date available:
|
2015-01-13T10:10:26Z |
| Last updated:
|
2016-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144123 |
| . |
| Reference:
|
[1] Devroye, L.: A triptych of discrete distributions related to the stable law..Stat. Probab. Lett. 18 (1993), 349-351. Zbl 0794.60007, MR 1247445, 10.1016/0167-7152(93)90027-G |
| Reference:
|
[2] Feuerverger, A., McDunnough, P.: On the efficiency of empirical characteristic function procedure..J. Roy. Stat. Soc. Ser. B 43 (1981), 20-27. MR 0610372 |
| Reference:
|
[3] Gerlein, O. V., Kagan, A. M.: Hilbert space methods in classical problems of mathematical statistics..J. Soviet Math. 12 (1979), 184-213. Zbl 0354.62007, 10.1007/BF01262718 |
| Reference:
|
[4] Kagan, A. M.: Fisher information contained in a finite-dimensional linear space, and a correctly posed version of the method of moments (in Russian)..Problemy Peredachi Informatsii 12 (1976), 20-42. MR 0413340 |
| Reference:
|
[5] Klebanov, L. B., Melamed, I. A.: Several notes on Fisher information in presence of nuisance parameters..Statistics: J. Theoret. Appl. Stat. 9 (1978), 85-90. Zbl 0381.62007, MR 0506482 |
| Reference:
|
[6] Klebanov, L. B., Slámová, L.: Integer valued stable random variables..Stat. Probab. Lett. 83 (2013), 1513-1519. Zbl 1283.60022, MR 3048317, 10.1016/j.spl.2013.02.016 |
| Reference:
|
[7] Slámová, L., Klebanov, L. B.: Modelling financial returns with discrete stable distributions..In: Proc. 30th International Conference Mathematical Methods in Economics (J. Ramík and D. Stavárek, eds.), Silesian University in Opava, School of Business Administration in Karviná, 2012, pp. 805-810. |
| Reference:
|
[8] Steutel, F. W., Harn, K. van: Discrete analogues of self-decomposability and stability..Ann. Probab. 7 (1979), 893-899. MR 0542141, 10.1214/aop/1176994950 |
| . |
