Empirical likelihood hypothesis testing for two mean vectors.
el.test2(y1, y2, R = 0, ncores = 1, graph = FALSE)A list including:
The empirical likelihood test statistic value.
The modified test statistic, either via the chi-square or the F distribution.
Thre degrees of freedom of the chi-square or the F distribution.
The asymptotic or the bootstrap p-value.
The estimated common mean vector.
The runtime of the bootstrap calibration.
A matrix containing the Euclidean data of the first group.
A matrix containing the Euclidean data of the second group.
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.
How many to cores to use.
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.
Michail Tsagris.
R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.
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.
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.
eel.test2, maovjames, maov, hotel2T2, james
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 )
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