Multinomial interpoint distances
type: Item
instance of: scholarly article
title: Multinomial interpoint distances
zbMATH Open document ID: 1416.62289
author: Reza Modarres
published in: Statistical Papers
publication date: 2018-03-22T00:00:00Z
review text: Consider the following model: $r$ and $n$ are integers greater or equal than $2$. Define $\overline{(0,n)}$ as \[ \overline{(0,n)}:=\left\{ 0,1,\dots ,n\right\}. \] Let $E_{r,n}$ be given by \[ E_{r,n}:=\left\{(n_1,\dots,n_{r})^{\prime}\in \left(\overline{(0,n)}\right)^{r}\mid \sum_{i=1}^{r}n_{i}=n\right\}. \] $\mathbf{Z}=(Z_{1},\dots ,Z_{r})$ is a random field assuming values in $E_{r,n}$. Let $\mathbf{P}$ be given by \[ \mathbf{P}:=\left\{ \mathbf{p=}(p_{1},\dots ,p_{r})\in ( \left[ 0,1\right] )^{r}\mid \sum_{i=1}^{r}p_{i}=1\right\}. \] For each $\mathbf{p=}(p_{1},\dots ,p_{r})\in \mathbf{P}$, define \[ MT(r,n,\mathbf{p}):=\left\{ P\text{ is a probability on }E_{r,n} \text{ / }P(\left\{ (n_{1},\dots ,n_{r})\right\} )=n!\prod_{t=1}^{r}\frac{(p_{t})^{n_{t}}}{n_{t}!}\right\}. \] $N_{1}$ and $N_{2}$ are positive integers. $\mathbf{p}_{1}\mathbf{=}(p_{11},\dots ,p_{1r})\in \mathbf{P}$ and $\mathbf{p}_{2}\mathbf{=}(p_{21},\dots ,p_{2r})\in \mathbf{ P}$. $\mathbf{X}_{1},\dots ,\mathbf{X}_{N_{1}}$ are independent and identically $ MT(r,n,\mathbf{p}_{1})$-distributed random vectors. $\mathbf{Y}_{1},\dots ,\mathbf{Y}_{N_{2}}$ are independent and identically $ MT(r,n,\mathbf{p}_{2})$-distributed random vectors. For $1\leq i<j\leq N_{1}$ the squared interpoint distance (ID) between $ \mathbf{X}_{i}$ and $\mathbf{X}_{j}$ is the random variable defined by \[ d_{(x)ij}^{2}:=(\mathbf{X}_{i}-\mathbf{X}_{j})^{\prime}(\mathbf{X}_{i}-\mathbf{X}_{j})=\sum_{t=1}^{r}(X_{it}-X_{jt})^{2}. \] For $1\leq k<h\leq N_{2}$ the squared interpoint distance (ID) between $ \mathbf{Y}_{k}$ and $\mathbf{Y}_{h}$ is the random variable defined by \[ d_{(y)kh}^{2}:=(\mathbf{Y}_{k}-\mathbf{Y}_{h})^{\prime}(\mathbf{Y}_{k}-\mathbf{Y}_{h})=\sum_{t=1}^{r}(Y_{kt}-Y_{ht})^{2}. \] For $1\leq i\leq N_{1}$ and $1\leq k\leq N_{2}$ the squared interpoint distance (ID) between $\mathbf{X}_{i}$ and $\mathbf{Y}_{k}$ is the random variable defined by \[ d_{(xy)ik}^{2}:=(\mathbf{X}_{i}-\mathbf{Y}_{k})^{\prime}(\mathbf{X…
Mathematics Subject Classification ID: 62H10
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 6852102
zbMATH Keywords: interpoint distance
zbMATH Keywords: multinomial
zbMATH Keywords: high dimension
zbMATH Keywords: testing
reviewed by: Oscar Bustos
describes a project that uses: mstat
describes a project that uses: Stata
MaRDI profile type: Publication
full work available at URL: s00362-016-0766-7
OpenAlex ID: W2307909705
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DOI: 10.1007/S00362-016-0766-7
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description: scientific article; zbMATH DE number 6852102