The Annals of Statistics

An Exponential Subfamily which Admits UMPU Tests Based on a Single Test Statistic

Shaul K. Bar-Lev and Benjamin Reiser

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Let $f(x: \theta) = a(x) \exp\{\theta_1u_1(x) + \theta_2u_2(x) + c(\theta)\}, \theta = (\theta_1, \theta_2) \in \ominus \subset R^2$, be a density with respect to the Lebesgue measure on the real line which characterizes a two-parameter exponential family of distributions. Let $(\theta_1, \eta_2)$ be the mixed parameters, where $\eta_2 = E\{u_2(X)\}$. Assume that $\theta_2$ can be represented as $\theta_2 = -\theta_1 \varphi'(\eta_2)$ where $\varphi'(\eta_2) = d\varphi(\eta_2)/d\eta_2$ for some function $\varphi(\eta_2)$. Let $(X_1, \cdots, X_n)$ be independent random variables having a common density $f(x: \theta)$ and set $T_i = \sum^n_{j=1} u_i(X_j), i = 1, 2$. It is shown that if $u_2(x)$ is a 1-1 function then the random variables $T_2$ and $Z_n = T_1 - n\varphi(T_2/n)$ are independent and that the statistic $Z_n$ is ancillary for $\theta_2$ in the presence of $\theta_1$ (i.e. the density of $Z_n$ depends on $\theta_1$ only). Furthermore, the density of $Z_n$ belongs to the one-parameter exponential family with natural parameter $\theta_1$. These results enable us to construct uniformly most powerful unbiased (UMPU) tests for various hypotheses concerning the parameter $\theta_1$ which are based on the statistic $Z_n$.

Article information

Ann. Statist., Volume 10, Number 3 (1982), 979-989.

First available in Project Euclid: 12 April 2007

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Primary: 62F05: Asymptotic properties of tests
Secondary: 62E10: Characterization and structure theory 62E15: Exact distribution theory

Exponential family ancillary uniformly most powerful unbiased tests


Bar-Lev, Shaul K.; Reiser, Benjamin. An Exponential Subfamily which Admits UMPU Tests Based on a Single Test Statistic. Ann. Statist. 10 (1982), no. 3, 979--989. doi:10.1214/aos/1176345888.

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