# Generalized length biased distributions

Aplikace matematiky (1988)

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

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topLingappaiah, 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

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

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