Article
| Title: | Generalized length biased distributions (English) |
| Author: | Lingappaiah, Giri S. |
| Language: | English |
| Journal: | Aplikace matematiky |
| ISSN: | 0373-6725 |
| Volume: | 33 |
| Issue: | 5 |
| Year: | 1988 |
| Pages: | 354-361 |
| Summary lang: | English |
| Summary lang: | Russian |
| Summary lang: | Czech |
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| Category: | math |
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| Summary: | Generalized length biased distribution is defined as $h(x)=\phi_r (x)f(x), x>0$, where $f(x)$ is a probability density function, $\phi_r (x)$ is a polynomial of degree $r$, that is, $\phi_r (x)=a_1(x/\mu'_1)+ \ldots + a_r(x^r/\mu'_r)$, with $a_i>0, i=1,\ldots ,r, a_1+\ldots + a_r=1, \mu'_i=E(x^i)$ for $f(x), i=1,2 \ldots, r$. If $r=1$, we have the simple length biased distribution of Gupta and Keating [1]. First, characterizations of exponential, uniform and beta distributions are given in terms of simple length biased distributions. Next, for the case of generalized distribution, the distribution of the sum of $n$ independent variables is put in the closed form when $f(x)$ is exponential. Finally, Bayesian estimates of $a_1, \ldots, a_r$ are obtained for the generalized distribution for general $f(x), x>1$. (English) |
| Keyword: | characterizations |
| Keyword: | exponential |
| Keyword: | uniform |
| Keyword: | beta distributions |
| Keyword: | length biased distributions |
| Keyword: | Bayesian estimates |
| MSC: | 62E10 |
| MSC: | 62E15 |
| MSC: | 62F15 |
| idZBL: | Zbl 0665.62016 |
| idMR: | MR0961313 |
| DOI: | 10.21136/AM.1988.104316 |
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| Date available: | 2008-05-20T18:35:12Z |
| Last updated: | 2020-07-28 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/104316 |
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| Reference: | [1] Ramesh Gupta, Jerome P. Keating: Relations for reliability measures under length biased sampling.Scand. J. Stat. 13 (1986), 49-56. MR 0844034 |
| Reference: | [2] G. S. Lingappaiah: On the Dirichlet Variables.J. Stat. Research, 11 (1977), 47-52. MR 0554878 |
| Reference: | [3] G. S. Lingappaiah: On the generalized inverted Dirichlet distribution.Demonstratio Math. 9 (1976), 423-433. MR 0428542 |
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