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A one-sample U-statistic of fixed degree $m$ has a symmetric U-statistic kernel $h$ and averages it over all distinct subsets of an iid sample:It is unbiased for $\theta =\mathbb{E}h(X_1,\dots ,X_m)$. With finite second moment, define the first Hoeffding projection $h_1(x)=\mathbb{E}[h(x,X_2,\dots ,X_m)]-\theta$. Orthogonality of the higher projections givesIf ${\zeta }_1=\mathrm{Var}(h_1(X))>0$, the central limit theorem therefore proves the U-statistic central limit theoremEquivalently the influence function of a U-statistic is $m{h}_1(x)$, obtained by differentiating its population functional $\int hd{F}^m$ in each distribution argument. If ${\zeta }_1=0$, the root-$n$ limit degenerates and the second or higher projection determines a different, often nonnormal, limit.

For example, $h(x,y)=(x-y{)}^2/2$ gives the unbiased sample variance with denominator $n-1$. Its first projection is $[(x-\mu {)}^2-{\sigma }^2]/2$, so the influence function is $(x-\mu {)}^2-{\sigma }^2$. When the fourth moment is finite, the leading variance of the sample variance is $(\mathbb{E}(X-\mu {)}^4-{\sigma }^4)/n$. This example connects unbiased symmetric estimation to the functional influence calculation.

Let $X_i=Y_i-{\theta }_0$. Under the symmetric continuous null hypothesis, positive and negative signs have probability $1/2$ and zeros have probability zero. For the sign test, take $N_{+}={\sum }_i{1}_{\{X_i>0\}}$ and reject for small values: $N_{+}\sim \mathrm{Bin}(n,1/2)$ under the null. Choose $k$ with ${\mathbb{P}}_0(N_{+}\le k)\le \alpha$; randomization at the next value attains exact size if desired. The lower tail is appropriate for $\theta <{\theta }_0$.

For the Wilcoxon signed-rank test, rank $|X_i|$ increasingly and put $W_{+}={\sum }_i{R}_i{1}_{\{X_i>0\}}$. Continuous observations have no absolute-value ties almost surely. Conditional on the absolute values, the signs are independent fair signs because symmetry makes the sign independent of magnitude. Hence $W_{+}$ has the distribution of ${\sum }_{r=1}^{n}r{B}_r$ for independent Bernoulli random variables $B_r$ with parameter $1/2$. Again reject in the lower tail, using this exact subset-sum distribution or a normal approximation. Independence gives the requested null moments:For a symmetric kernel $h$ of fixed order $m$, a one-sample U-statistic is $U_n={\left(\genfrac{}{}{0ex}{}{n}{m}\right)}^{-1}{\sum }_{i_1<\cdots <i_m}h(X_{i_1},\dots ,X_{i_m})$. State the U-statistic central limit theorem: if $\mathbb{E}{h}^2<\infty$, $\eta =\mathbb{E}h$, and ${\zeta }_1=\mathrm{Var}(\mathbb{E}[h\mid X_1])>0$, then $\sqrt{n}(U_n-\eta )\Rightarrow N(0,m^2{\zeta }_1)$. The nondegeneracy condition is important.

For $h(x,y)=1_{\{x+y>0\}}$, its order-two U-statistic satisfies the exact Wilcoxon signed-rank statistic identityTo prove it, in an unordered pair the sign of the sum is the sign of whichever observation has larger absolute value. Counting the pairs for which a positive observation is larger contributes its rank minus one; its own positive-sign indicator supplies the remaining one.

Write $F_0$ for the distribution function of $X$ under the null. Symmetry gives $\eta =1/2$ andThe probability integral transform makes $F_0(X_1)$ uniform on $(0,1)$, so ${\zeta }_1=1/12$. Therefore $\sqrt{n}(U_n-1/2)\Rightarrow N(0,1/3)$. The centred $N_{+}$ is $O_p(\sqrt{n})$ and is negligible on the scale $n^{3/2}$ of $W_{+}$. Combining the exact decomposition and its mean givesThe lower-tail approximate level-$\alpha$ rejection rule compares this standardized statistic with ${\Phi }^{-1}(\alpha )$; a half-unit continuity correction can refine discrete-tail calculations.

For a continuous sample, the statistic also equals ${\sum }_i{1}_{\{X_i>0\}}+{\sum }_{i<j}{1}_{\{X_i+X_j>0\}}$. In each pair the sign of the sum is the sign of the observation with larger absolute value, proving the identity by counting its smaller-rank partners. Under a continuous symmetric null, the pair kernel has projection $F(x)-1/2$ and projection variance $1/12$. The U-statistic central limit theorem yields asymptotic normality of $W_{+}$, with variance asymptotic to $n^3/12$.