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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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 ...
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Is K-W test statistics a U statistics?

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:...
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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 ...
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13 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 ...
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Why $E[E[p_{N}(Z_n,Z_m)|Z_m]]\neq E[E[p_{N}(Z_n,Z_m)|Z_n]]$ here? What went wrong?

We have a U-statistic defined as $U=\frac{1}{N(N-1)}\sum_{n\neq m}p_{N}(Z_n,Z_m)$, where observations $\{Z_i\}_{i=1}^{N}$ are i.i.d. following distribution $F(z)$ with density $f(z)$. $p_{N}(Z_n,Z_m)=\...
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1answer
80 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 ...
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What is the two sample U-statistic of the given statistic?

Let $X_1,X_2$ be two independent random variables having distribution function F and $Y_1,Y_2$ be another two independent variables with d.f. G. Also $F$ and $G$ are independent. A statistic is ...
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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 $\...
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27 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 ...
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73 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-...
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U statistics for product kernels

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 ...
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72 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 ...
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1answer
93 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$...