# Generalized length biased distributions

Aplikace matematiky (1988)

• Volume: 33, Issue: 5, page 354-361
• ISSN: 0862-7940

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## Abstract

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Generalized length biased distribution is defined as $h\left(x\right)={\phi }_{r}\left(x\right)f\left(x\right),x>0$, where $f\left(x\right)$ is a probability density function, ${\phi }_{r}\left(x\right)$ is a polynomial of degree $r$, that is, ${\phi }_{r}\left(x\right)={a}_{1}\left(x/{\mu }_{1}^{\text{'}}\right)+...+{a}_{r}\left({x}^{r}/{\mu }_{r}^{\text{'}}\right)$, with ${a}_{i}>0,i=1,...,r,{a}_{1}+...+{a}_{r}=1,{\mu }_{i}^{\text{'}}=E\left({x}^{i}\right)$ for $f\left(x\right),i=1,2...,r$. 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 $f\left(x\right)$ is exponential. Finally, Bayesian estimates of ${a}_{1},...,{a}_{r}$ are obtained for the generalized distribution for general $f\left(x\right),x>1$.

## How to cite

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Lingappaiah, Giri S.. "Generalized length biased distributions." Aplikace matematiky 33.5 (1988): 354-361. <http://eudml.org/doc/15549>.

@article{Lingappaiah1988,
abstract = {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, $\phi _r (x)=a_1(x/\mu ^\{\prime \}_1)+ \ldots + a_r(x^r/\mu ^\{\prime \}_r)$, with $a_i>0, i=1,\ldots ,r, a_1+\ldots + a_r=1, \mu ^\{\prime \}_i=E(x^i)$ 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 $f(x), x>1$.},
author = {Lingappaiah, Giri S.},
journal = {Aplikace matematiky},
keywords = {characterizations; exponential; uniform; beta distributions; length biased distributions; Bayesian estimates; characterizations; exponential; uniform; beta distributions; length biased distributions; Bayesian estimates},
language = {eng},
number = {5},
pages = {354-361},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized length biased distributions},
url = {http://eudml.org/doc/15549},
volume = {33},
year = {1988},
}

TY - JOUR
AU - Lingappaiah, Giri S.
TI - Generalized length biased distributions
JO - Aplikace matematiky
PY - 1988
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 33
IS - 5
SP - 354
EP - 361
AB - 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, $\phi _r (x)=a_1(x/\mu ^{\prime }_1)+ \ldots + a_r(x^r/\mu ^{\prime }_r)$, with $a_i>0, i=1,\ldots ,r, a_1+\ldots + a_r=1, \mu ^{\prime }_i=E(x^i)$ 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 $f(x), x>1$.
LA - eng
KW - characterizations; exponential; uniform; beta distributions; length biased distributions; Bayesian estimates; characterizations; exponential; uniform; beta distributions; length biased distributions; Bayesian estimates
UR - http://eudml.org/doc/15549
ER -

## References

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1. Ramesh Gupta, Jerome P. Keating, Relations for reliability measures under length biased sampling, Scand. J. Stat. 13 (1986), 49-56. (1986) Zbl0627.62098MR0844034
2. G. S. Lingappaiah, On the Dirichlet Variables, J. Stat. Research, 11 (1977), 47-52. (1977) Zbl0418.62018MR0554878
3. G. S. Lingappaiah, On the generalized inverted Dirichlet distribution, Demonstratio Math. 9 (1976), 423-433. (1976) Zbl0343.62012MR0428542

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