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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Hoeffding inequality for a product of two random variables
Let $X_1,X_2, \dots, X_{m_1}$, $Y_1,Y_2, \dots, Y_{m_2}$ be $m_1 + m_2$ independent random variables from a probabilistic space $\mathcal{X}$, let $h: \mathcal{X} \to \{-1,1\}$
I'm interested in point ...
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U -statistics for bi variate sample problem
Let $(X_1, Y_1), (X_2, Y_2),....,(X_n, Y_n)$ be iid random variables with joint distribution function $F(x, y)$ and $F(x), G(x)$ be the marginal distribution functions of $X_1$ and $Y_1$ respectively. ...
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Asymptotic distribution of $U$-statistics
Let $(X_1, Y_1), ...., (X_n, Y_n)$ be iid random vectors with marginal distributions functions $F(x)$ and $G(x)$ (both are continuous distributions) respectively such that $F(0)=G(0)=\frac{1}{2}$. ...
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$U$-statistics and their limiting distributions
Let $X_1,X_2, . . . ,X_n$ be i.i.d. observations from a continuous distribution
$F$. Consider the parametric function $\mathbb{P}([\text{min}(X_1,X_2) > X3])$. Find the U-Statistics and its ...
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Find the U-statistics for a parameter
$X_1,\ldots,X_n, i.i.d.$
$p=P\left(X>0\right),\theta=p\left(1-p\right) $
How can I find the U-statistics of $\theta$?
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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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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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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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(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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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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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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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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