Empirical likelihood hypothesis testing for two mean vectors {mvhtests}

Empirical likelihood hypothesis testing for two mean vectors

Description

Empirical likelihood hypothesis testing for two mean vectors.

Usage

el.test2(y1, y2, R = 0, ncores = 1, graph = FALSE)

Arguments

y1

A matrix containing the Euclidean data of the first group.

y2

A matrix containing the Euclidean data of the second group.

R

If R is 0, the classical chi-square distribution is used, if R = 1, the corrected chi-square distribution (James, 1954) is used and if R = 2, the modified F distribution (Krishnamoorthy and Yanping, 2006) is used. If R is greater than 3 bootstrap calibration is performed.

ncores

How many to cores to use.

graph

A boolean variable which is taken into consideration only when bootstrap calibration is performed. IF TRUE the histogram of the bootstrap test statistic values is plotted.

Details

The H_0 is that \pmb{\mu}_1 = \pmb{\mu}_2 and the two constraints imposed by EL are

\frac{1}{n_j}\sum_{i=1}^{n_j}\left\lbrace\left[1+\pmb{\lambda}_j^T\left({\bf x}_{ji}-\pmb{\mu} \right)\right]^{-1}\left({\bf x}_{ij}-\pmb{\mu}\right)\right\rbrace={\bf 0},

where j=1,2 and the \pmb{\lambda}_js are Lagrangian parameters introduced to maximize the above expression. Note that the maximization of is with respect to the \pmb{\lambda}_js. The probabilities of the j-th sample have the following form

p_{ji}=\frac{1}{n_j} \left[1+\pmb{\lambda}_j^T \left({\bf x}_{ji}-\pmb{\mu} \right)\right]^{-1}

. The log-likelihood ratio test statistic can be written as

\Lambda=\sum_{j=1}^2\sum_{i=1}^{n_j}\log{n_jp_{ij}}.

The test is implemented by searching for the mean vector that minimizes the sum of the two one sample EL test statistics. See el.test1 for the test statistic in the one-sample case.

Value

A list including:

test

The empirical likelihood test statistic value.

modif.test

The modified test statistic, either via the chi-square or the F distribution.

dof

Thre degrees of freedom of the chi-square or the F distribution.

pvalue

The asymptotic or the bootstrap p-value.

mu

The estimated common mean vector.

runtime

The runtime of the bootstrap calibration.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris [e-mail].

References

Amaral G.J.A., Dryden I.L. and Wood A.T.A. (2007). Pivotal bootstrap methods for k-sample problems in directional statistics and shape analysis. Journal of the American Statistical Association, 102(478): 695–707.

Owen A. B. (2001). Empirical likelihood. Chapman and Hall/CRC Press.

Owen A.B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75(2): 237–249.

Preston S.P. and Wood A.T.A. (2010). Two-Sample Bootstrap Hypothesis Tests for Three-Dimensional Labelled Landmark Data. Scandinavian Journal of Statistics, 37(4): 568–587.

See Also

eel.test2, maovjames, maov, hotel2T2, james

Examples

el.test2( y1 = as.matrix(iris[1:25, 1:4]), y2 = as.matrix(iris[26:50, 1:4]), R = 0 )
el.test2( y1 = as.matrix(iris[1:25, 1:4]), y2 = as.matrix(iris[26:50, 1:4]), R = 1 )
el.test2( y1 =as.matrix(iris[1:25, 1:4]), y2 = as.matrix(iris[26:50, 1:4]), R = 2 )