# Questions tagged [u-statistics]

an estimator arising in the theory of unbiased estimation, arising as the mean of a statistic computed over all ordered subsamples of a given size.

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### Actual difference between the statistic results from scipy.stats.ranksums and scipy.stats.mannwhitneyu

So, I have been trying to test if two independent samples come from the distribution, i.e. if they are greater or less than one another. Eventually I found out the Mann Whitney U Test is the ...
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### Variance of a U-statistic with random kernel

The variance of a U-statistic $\widehat{\Theta}$ (with fixed kernel) amounts to $Var(\widehat{\Theta}) = \sum_{c=1}^m \alpha_c \kappa_c - (1 - \alpha_0)\Theta^2$, where all parameters are defined as ...
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### How to find the symmetric kernel for the given U-statistic?

The U-statistic is given by \begin{equation} \widehat{\Delta}=\frac{1}{\binom{n_1}{2}\binom{n_2}{2}}\sum_{1\leq i_1<i_2\leq n_1}\sum_{1\leq j_1<j_2\leq n_2}f(X_{i_1},X_{i_2},Y_{j_1},Y_{j_2}), \...
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### Equivalence of the completeness of the order statistics and the uniqueness of symmetric unbiased estimators

I am reading A.J. Lee's 1990 book "U-statistics: Theory and Practice". There is an equation on page 6 that I cannot explain why it holds, and I hope somebody could help me. Here is the ...
The K-W H test statistic is given by: $$H=(N-1) \frac{\sum_{i=1}^{g} n_{i}\left(\bar{r}_{i \cdot}-\bar{r}\right)^{2}}{\sum_{i=1}^{g} \sum_{j=1}^{n_{i}}\left(r_{i j}-\bar{r}\right)^{2}}, \text { where:... 1 vote 0 answers 65 views ### estimation of covariance of function of two i.i.d. data points Given i.i.d. data: X_1,\dots,X_n living in some space \mathcal{X} and drawn according to distribution P, and symmetric functions f,g: \mathcal{X} \times \mathcal{X} \to \mathbb{R}, I want to ... 1 vote 0 answers 19 views ### The asymptotic properties of V-statistic for mixing multivariate process Suppose \{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d is a \alpha- or \rho-mixing process. Let h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R} be the symmetric kernel ... 5 votes 1 answer 197 views ### (From van der Vaart's Asymptotic Statistics, page 161, U-statistic) Why we can always replace the function h with a symmetric one? I'm reading the following Chapter from van der Vaart's Asymptotic Statistics, Section 12.1 page 161 (see the screenshot below). For the h function that it mentioned, I have two questions regarding ... 2 votes 1 answer 43 views ### U-stat with random kernel U-statistics assume that the kernel remain fixed. I wonder if theorems in u-stat still hold true when the kernel is random. For instance, I estimate the kernel h using data. The estimated kernel is ... 2 votes 1 answer 112 views ### Can the variance of a U-statistic be of the order O(\frac{1}{n^2})? It is not that easy to find estimators T_n such that \mbox{Var}[T_n] \sim O(n^{-B}) with B = 2. In most cases, B=1.Here n is the sample size. It seems, according to this paper on U-... 0 votes 0 answers 75 views ### Critical value for Mann's test against trend To test that a sample X_1,\ldots,X_n are i.i.d against that the distributions of X_i are stochastically increasing in i, how to find the distribution of the test statistic and the critical value ... 3 votes 1 answer 127 views ### Showing a Corollary of Hoeffding's Theorem I am currently reading Jun Shao's Mathematical Statistics, and in his discussion of U statistics, he proves that Var(U_n) =  n\choose m$$^{-1} \sum_{k=1}^m $$m \choose k$$n - m \choose m-k\zeta_k$... 1 vote 0 answers 46 views ### The degree of nonparametric estimation kernels and induced$U$-statistics The definition of kernels in nonparametric can be formulated as follows. [Randles&Wolfe] pp.61-62. A parameter$\gamma$is said to be estimable of degree$r$for the family of distributions$\...
Background Let $X$ be a real, univariate random variable with probability distribution function $p(x)$. Let $x_1, \ldots, x_n$ be a sample of size $n$ drawn from $X$. U statistics provide an unbiased ...