An Exact Distribution-Free Test for Means of Nonnegative Random Variables

Let $X=(X_1,\dots ,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\dots ,D_n)\sim \mathrm{Dir}(1,\dots ,1)$ be independent of $X$, and define $K(x)=\mathbb{P}\{{\sum }_{i=1}^n{x}_i{D}_i\le 1\}$. We prove that, for every $n\ge 1$, whenever $\mathbb{E}{X}_i\le 1$ for every $i$, $\mathbb{P}\{K(X)\le \alpha \}\le \alpha$ for all $0\le \alpha \le 1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\mathbb{E}{X}_i\le 1$ for all $i$. This proves a conjecture of Gaffke (2005).