TY - JOUR

T1 - Distributional characterizations through scaling relations

AU - Pakes, Anthony

AU - Navarro, J.

PY - 2007

Y1 - 2007

N2 - Investigated here are aspects of the relation between the laws of X and Y where X is represented as a randomly scaled version of Y. In the case that the scaling has a beta law, the law of Y is expressed in terms of the law of X. Common continuous distributions are characterized using this beta scaling law, and choosing the distribution function of Y as a weighted version of the distribution function of X, where the weight is a power function. It is shown, without any restriction on the law of the scaling, but using a one-parameter family of weights which includes the power weights, that characterizations can be expressed in terms of known results for the power weights. Characterizations in the case where the distribution function of Y is a positive power of the distribution function of X are examined in two special cases. Finally, conditions are given for existence of inverses of the length-bias and stationary-excess operators.

AB - Investigated here are aspects of the relation between the laws of X and Y where X is represented as a randomly scaled version of Y. In the case that the scaling has a beta law, the law of Y is expressed in terms of the law of X. Common continuous distributions are characterized using this beta scaling law, and choosing the distribution function of Y as a weighted version of the distribution function of X, where the weight is a power function. It is shown, without any restriction on the law of the scaling, but using a one-parameter family of weights which includes the power weights, that characterizations can be expressed in terms of known results for the power weights. Characterizations in the case where the distribution function of Y is a positive power of the distribution function of X are examined in two special cases. Finally, conditions are given for existence of inverses of the length-bias and stationary-excess operators.

U2 - 10.1111/j.1467-842X.2007.00468.x

DO - 10.1111/j.1467-842X.2007.00468.x

M3 - Article

VL - 49

SP - 115

EP - 135

JO - Australian & New Zealand Journal of Statistics

JF - Australian & New Zealand Journal of Statistics

SN - 1369-1473

IS - 2

ER -