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 pip_i, which sum to one, are allocated to the observations, such that the weighted sample mean xˉ{\bf \bar{x}} is equal to some population mean μ0\pmb{\mu}_0, under the H0H_0. Under H1H_1 the weights are equal to 1n\frac{1}{n}, where nn is the sample size. Following Efron (1981), the choice of pisp_is will minimize the Kullback-Leibler distance from H0H_0 to H1H_1

D(L0,L1)=i=1npilog(npi), D\left(L_0,L_1\right)=\sum_{i=1}^np_i\log\left(np_i\right),

subject to the constraint i=1npixi=μ0\sum_{i=1}^np_i{\bf x}_i=\pmb{\mu}_0. The probabilities take the form

pi=eλTxij=1neλTxj p_i=\frac{e^{\pmb{\lambda}^T{\bf x}_i}}{\sum_{j=1}^ne^{\pmb{\lambda}^T{\bf x}_j}}

and the constraint becomes

i=1neλTxi(xiμ0)j=1neλTxj=0i=1nxieλTxij=1neλTxjμ0=0. \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 λ\pmb{\lambda} is required. Under H0H_0 Λχd2\Lambda \sim \chi^2_d, where dd 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 λ\lambda.

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) )

[Package mvhtests version 1.0 Index]