An exponential subfamily which admits UMPU tests based on a single test statistic

Let f(x:θ)=a(x)exp{θ1u1(x)+θ2u2(x)+c(θ)},θ=(θ1,θ2)∈⊖⊂R2, be a density with respect to the Lebesgue measure on the real line which characterizes a two-parameter exponential family of distributions. Let (θ1,η2) be the mixed parameters, where η2=E{u2(X)}. Assume that θ2 can be represented as θ2=−θ1φ′(η2) where φ′(η2)=dφ(η2)/dη2 for some function φ(η2). Let (X1,⋯,Xn) be independent random variables having a common density f(x:θ) and set Ti=∑nj=1ui(Xj),i=1,2. It is shown that if u2(x) is a 1-1 function then the random variables T2 and Zn=T1−nφ(T2/n) are independent and that the statistic Zn is ancillary for θ2 in the presence of θ1 (i.e. the density of Zn depends on θ1 only). Furthermore, the density of Zn belongs to the one-parameter exponential family with natural parameter θ1. These results enable us to construct uniformly most powerful unbiased (UMPU) tests for various hypotheses concerning the parameter θ1 which are based on the statistic Zn.