Metrika (2011) 74:211–219 DOI 10.1007/s00184-010-0298-4 Bayes sequential estimation for a particular exponential family of distributions under LINEX loss Alicja Jokiel-Rokita Received: 19 July 2008 / Published online: 21 January 2010 © Springer-Verlag 2010 Abstract The problem of sequentially estimating an unknown distribution parame- ter of a particular exponential family of distributions is considered under LINEX loss function for estimation error and a cost c > 0 for each of an i.i.d. sequence of potential observations X 1 , X 2 ,... A Bayesian approach is adopted and conjugate prior distri- butions are assumed. Asymptotically pointwise optimal and asymptotically optimal procedures are derived. Keywords AO rule · APO rule · Bayes sequential estimation · LINEX loss function · Transformed chi-square family of distributions Mathematics Subject Classification (2000) 62L12 · 62F10 · 62F15 1 Introduction The aim of the Bayes sequential estimation is to derive an optimal sequential proce- dure consisting of an optimal stopping rule and a Bayes estimate. Usually, obtaining the Bayes estimate is possible in the problem. Then the Bayes sequential estimation problem reduces to finding an optimal (Bayes) stopping rule. Let {Y n , n ≥ 1} be a sequence of random variables defined on a probability space (, F , P ), where Y n is F n -measurable and F n ⊂ F n+1 ⊂···⊂ F is an increasing sequence of σ -fields. Define Z n (c) = Y n + cn, c > 0. Let N be the class of all stop- ping times N with respect to the filtration {F n , n ≥ 1}. A stopping rule N ∗ = N ∗ (c) A. Jokiel-Rokita (B ) Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wroclaw, Poland e-mail: alicja.jokiel-rokita@pwr.wroc.pl 123