Hotelling's test for testing one Euclidean population mean vector.
hotel1T2(x, M, a = 0.05, R = 999, graph = FALSE)A list including:
The sample mean vector.
The test statistic, the p-value, the critical value and the degrees of freedom of the F distribution (numerator and denominator). This is given if no bootstrap calibration is employed.
The bootstrap p-value is bootstrap is employed.
The runtime of the bootstrap calibration.
A matrix containing Euclidean data.
The significance level, set to 0.05 by default.
The hypothesized mean vector.
If R is 1 no bootstrap calibration is performed and the classical p-value via the F distribution is returned. If R is greater than 1, the bootstrap p-value is returned.
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 hypothesis test is that a mean vector is equal to some specified vector \(H_0:\pmb{\mu}=\pmb{\mu}_0\). We assume that \(\pmb{\Sigma}\) is unknown. The first approach to this hypothesis test is parametrically, using the Hotelling's \(T^2\) test Mardia, Bibby and Kent (1979, pg. 125-126). The test statistic is given by $$ T^2=\frac{\left(n-p\right)n}{\left(n-1\right)p}\left(\bar{{\bf X}}-\pmb{\mu}\right)^T{\bf S}^{-1}\left(\bar{{\bf X}}-\pmb{\mu} \right). $$ Under the null hypothesis, the above test statistic follows the \(F_{p,n-p}\) distribution. The bootstrap version of the one-sample multivariate generalization of the simple t-test is also included in the function. An extra argument (R) indicates whether bootstrap calibration should be used or not. If R=1, then the asymptotic theory applies, if R>1, then the bootstrap p-value will be applied and the number of re-samples is equal to R.
Mardia K.V., Kent J.T. and Bibby J.M. (1979). Multivariate analysis. London: Academic Press.
eel.test1, el.test1, james, hotel2T2, maov, el.test2
x <- matrix( rnorm( 100 * 4), ncol = 4)
hotel1T2(x, numeric(4), R = 1)
hotel1T2(x, numeric(4), R = 999, graph = TRUE)
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