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Keywords:
characterizations; exponential; uniform; beta distributions; length biased distributions; Bayesian estimates
characterizations; exponential; uniform; beta distributions; length biased distributions; Bayesian estimates
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 $f(x), x>1$.
References:
[1] Ramesh Gupta, Jerome P. Keating: Relations for reliability measures under length biased sampling. Scand. J. Stat. 13 (1986), 49-56. MR 0844034
[2] G. S. Lingappaiah: On the Dirichlet Variables. J. Stat. Research, 11 (1977), 47-52. MR 0554878
[3] G. S. Lingappaiah: On the generalized inverted Dirichlet distribution. Demonstratio Math. 9 (1976), 423-433. MR 0428542
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