1\name{mvI.test} 2\alias{mvI.test} 3\alias{mvI} 4\title{ Energy Statistic Test of Independence} 5\description{ 6 Computes the multivariate nonparametric E-statistic and test of independence 7 based on independence coefficient \eqn{\mathcal I_n}{I_n}.} 8\usage{ 9 mvI.test(x, y, R) 10 mvI(x, y) 11} 12\arguments{ 13 \item{x}{ matrix: first sample, observations in rows} 14 \item{y}{ matrix: second sample, observations in rows} 15 \item{R}{ number of replicates} 16} 17\details{ 18 Computes the coefficient \eqn{\mathcal I}{I_n} and performs a nonparametric 19 \eqn{\mathcal E}{E}-test of independence. The test decision is obtained via 20 bootstrap, with \code{R} replicates. 21 The sample sizes (number of rows) of the two samples must agree, and 22 samples must not contain missing values. The statistic 23 \eqn{\mathcal E = n \mathcal I^2}{E = I^2} is a ratio of V-statistics based 24 on interpoint distances \eqn{\|x_{i}-y_{j}\|}{||x_{i}-y_{j}||}. 25 See the reference below for details. 26} 27\value{ 28\code{mvI} returns the statistic. \code{mvI.test} returns 29 a list with class 30 \code{htest} containing 31 \item{ method}{ description of test} 32 \item{ statistic}{ observed value of the test statistic \eqn{n\mathcal I_n^2}{n I_n^2}} 33 \item{ estimate}{ \eqn{\mathcal I_n}{I_n}} 34 \item{ replicates}{ replicates of the test statistic} 35 \item{ p.value}{ approximate p-value of the test} 36 \item{ data.name}{ description of data} 37} 38\references{ 39 Bakirov, N.K., Rizzo, M.L., and Szekely, G.J. (2006), A Multivariate 40 Nonparametric Test of Independence, \emph{Journal of Multivariate Analysis} 41 93/1, 58-80, \cr 42 \doi{10.1016/j.jmva.2005.10.005} 43 } 44 \note{ 45 Historically this is the first energy test of independence. The 46 distance covariance test \code{\link{dcov.test}}, distance correlation 47 \code{\link{dcor}}, and related methods are more recent (2007,2009). 48 The distance covariance test is faster and has different properties than 49 \code{mvI.test}. Both methods are based on a population independence coefficient 50 that characterizes independence and both tests are statistically consistent. 51 } 52\author{ Maria L. Rizzo \email{mrizzo @ bgsu.edu} and 53Gabor J. Szekely 54} 55 \seealso{ 56 \code{ \link{indep.test} } 57 \code{ \link{mvI.test} } 58 \code{ \link{dcov.test} } 59 \code{ \link{dcov} } 60 } 61 \keyword{ htest } 62 \keyword{ multivariate } 63 \keyword{ nonparametric } 64 \concept{ independence } 65 \concept{ energy statistics } 66 67