Geodesic
Generalized length biased distributions
Applications of Mathematics, Tome 33 (1988) no. 5, pp. 354-361
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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 $f(x), x>1$.
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 $f(x), x>1$.
DOI : 10.21136/AM.1988.104316
Classification : 62E10, 62E15, 62F15
Keywords: characterizations; exponential; uniform; beta distributions; length biased distributions; Bayesian estimates 
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Lingappaiah, Giri S. Generalized length biased distributions. Applications of Mathematics, Tome 33 (1988) no. 5, pp. 354-361. doi: 10.21136/AM.1988.104316

[1] Ramesh Gupta, Jerome P. Keating: Relations for reliability measures under length biased sampling. Scand. J. Stat. 13 (1986), 49-56. | MR

[2] G. S. Lingappaiah: On the Dirichlet Variables. J. Stat. Research, 11 (1977), 47-52. | MR

[3] G. S. Lingappaiah: On the generalized inverted Dirichlet distribution. Demonstratio Math. 9 (1976), 423-433. | MR

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