Use the empirical likelihood ratio and Wilks theorem to test if the
regression coefficient is equal to beta0,
by the case weighted estimation method.
The log empirical likelihood been maximized is $$ \sum_{d=1} \log \Delta F(y_i) + \sum_{d=0} \log [1-F(y_i)].$$
WRegTest(x, y, delta, beta0, psifun=function(t){t})A list with the following components:
the -2 log likelihood ratio; have approximate chisq distribution under \(H_0\).
the p-value using the chi-square approximation.
a matrix of size N by q. Random design matrix.
a vector of length N, containing the censored responses.
a vector (length N) of either 1's or 0's. delta=1 means y is uncensored; delta=0 means y is right censored.
a vector of length q. The value of the regression coefficient to be tested in the linear model
.
the estimating function. The definition of it determines the type of estimator under testing.
Mai Zhou.
The above likelihood should be understood as the likelihood of the
censored responses y and delta.
This version can handle the model where beta is a vector (of length q).
The estimation equations used when maximize the empirical likelihood is $$ 0 = \sum \delta_i \Delta F(Y_i) X_i \psi( Y_i - X_i \beta0 ) $$ which was described in detail in the reference below.
For median regression (Least Absolute Deviation) estimator, you should
define the
psifun as \(+1, -1\) or \(0\) when \(t\) is \(>0, <0 \)
or \( =0\).
For ordinary least squares estimator, psifun should be the identity function psifun <- function(t)t.
Zhou, M.; Kim, M. and Bathke, A. (2012). Empirical likelihood analysis of the case weighted estimator in heteroscastic AFT model. Statistica Sinica, 22, 295-316.
xx <- c(28,-44,29,30,26,27,22,23,33,16,24,29,24,40,21,31,34,-2,25,19)
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