A note on exponential dispersion models which are invariant under length-biased sampling

Abstract

A note on exponential dispersion models which are invariant under length-biased sampling Bar-Lev, S.K.; van der Duyn Schouten, F.A. Publication date: 2003 Link to publication Citation for published version (APA):Bar-Lev, S. K., & van der Duyn Schouten, F. A. (2003). A note on exponential dispersion models which are invariant under length-biased sampling. (CentER Discussion Paper; Vol. 2003-84). Tilburg: Operations research. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract Length-biased sampling situations may occur in clinical trials, reliability, queueing models, survival analysis and population studies where a proper sampling frame is absent. In such situations items are sampled at rate proportional to their "length" so that larger values of the quantity being measured are sampled with higher probabilities. More specifically, if f (x) is a p.d.f. presenting a parent population composed of nonnegative valued items then the sample is practically drawn from a distribution with p.d.f. g(x) = xf (x)/E(X) describing the lengthbiased population. In this case the distribution associated with g is termed a length-biased distribution. In this note we present a unified approach for characterizing exponential dispersion models which are invariant, up to translations, under various types of length-biased sampling. The approach is rather simple as it reduces such invariance problems into differential equations in terms of the derivatives of the associated variance functions.

Associated Natural Exponential Families and Elliptic Functions

“…Recall a definition appearing in Hassairi (1992) and Barlev, Bshouty and Letac (1994). We say that two NEF F 1 and F 2 on the real line belong to the same orbit if there exists a Moebius transformation y = (ax + b)/(cx + d) such that ad − bc = 1 and such that on a suitable interval for m we have V F 1 (m) = (cm + d) 3…”