| Exponential empirical likelihood for a one sample mean vector hypothesis testing {mvhtests} | R Documentation |
Exponential empirical likelihood for a one sample mean vector hypothesis testing
Description
Exponential empirical likelihood for a one sample mean vector hypothesis testing.
Usage
eel.test1(x, mu, tol = 1e-06, R = 1)
Arguments
x |
A matrix containing Euclidean data. |
mu |
The hypothesized mean vector. |
tol |
The tolerance value used to stop the Newton-Raphson algorithm. |
R |
The number of bootstrap samples used to calculate the p-value. If R = 1 (default value), no bootstrap calibration is performed |
Details
Exponential empirical likelihood or exponential tilting was first introduced by Efron (1981) as a way to perform a "tilted" version of the bootstrap for the one sample mean hypothesis testing. Similarly to the empirical likelihood, positive weights p_i, which sum to one, are allocated to the observations, such that the weighted sample mean {\bf \bar{x}} is equal to some population mean \pmb{\mu}_0, under the H_0. Under H_1 the weights are equal to \frac{1}{n}, where n is the sample size. Following Efron (1981), the choice of p_is will minimize the Kullback-Leibler distance from H_0 to H_1
D\left(L_0,L_1\right)=\sum_{i=1}^np_i\log\left(np_i\right),
subject to the constraint \sum_{i=1}^np_i{\bf x}_i=\pmb{\mu}_0. The probabilities take the form
p_i=\frac{e^{\pmb{\lambda}^T{\bf x}_i}}{\sum_{j=1}^ne^{\pmb{\lambda}^T{\bf x}_j}}
and the constraint becomes
\frac{\sum_{i=1}^ne^{\pmb{\lambda}^T{\bf x}_i}\left({\bf x}_i-\pmb{\mu}_0\right)}{\sum_{j=1}^ne^{\pmb{\lambda}^T{\bf x}_j}}=0 \Rightarrow \frac{\sum_{i=1}^n{\bf x}_ie^{\pmb{\lambda}^T{\bf x}_i}}{\sum_{j=1}^ne^{\pmb{\lambda}^T{\bf x}_j}}-\pmb{\mu}_0={\bf 0}.
A numerical search over \pmb{\lambda} is required. Under H_0 \Lambda \sim \chi^2_d, where d denotes the number of variables. Alternatively the bootstrap p-value may be computed.
Value
A list including:
p |
The estimated probabilities. |
lambda |
The value of the Lagrangian parameter |
iter |
The number of iterations required by the newton-Raphson algorithm. |
info |
The value of the log-likelihood ratio test statistic along with its corresponding p-value. |
runtime |
The runtime of the process. |
Author(s)
Michail Tsagris.
R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.
References
Efron B. (1981) Nonparametric standard errors and confidence intervals. Canadian Journal of Statistics, 9(2): 139–158.
Jing B.Y. and Wood A.T.A. (1996). Exponential empirical likelihood is not Bartlett correctable. Annals of Statistics, 24(1): 365–369.
Owen A. B. (2001). Empirical likelihood. Chapman and Hall/CRC Press.
See Also
el.test1, hotel1T2, james, hotel2T2, maov, el.test2
Examples
x <- as.matrix( iris[, 1:4] )
eel.test1(x, numeric(4) )
el.test1(x, numeric(4) )