Use the empirical likelihood ratio and Wilks theorem to test if the regression coefficient is equal to beta.
The log empirical likelihood been maximized is $$ \sum_{d=1} \log \Delta F(e_i) + \sum_{d=0} \log [1-F(e_i)];$$ where \(e_i\) are the residuals.
bjtest(y, d, x, beta)A list with the following components:
the -2 loglikelihood ratio; have approximate chisq distribution under \(H_o\).
the log empirical likelihood, under estimating equation.
the log empirical likelihood of the Kaplan-Meier of e's.
the probabilities that max the empirical likelihood under estimating equation.
a vector of length N, containing the censored responses.
a vector (length N) of either 1's or 0's. d=1 means y is uncensored; d=0 means y is right censored.
a matrix of size N by q.
a vector of length q. The value of the regression coefficient to be tested in the model \(y_i = \beta x_i + \epsilon_i\)
Mai Zhou.
The above likelihood should be understood as the likelihood of the error term, so in the regression model the error epsilon should be iid.
This version can handle the model where beta is a vector (of length q).
The estimation equations used when maximize the empirical likelihood is $$ 0 = \sum d_i \Delta F(e_i) (x \cdot m[,i])/(n w_i) $$ which was described in detail in the reference below.
Buckley, J. and James, I. (1979). Linear regression with censored data. Biometrika, 66 429-36.
Zhou, M. and Li, G. (2008). Empirical likelihood analysis of the Buckley-James estimator. Journal of Multivariate Analysis 99, 649-664.
Zhou, M. (2016) Empirical Likelihood Method in Survival Analysis. CRC Press.
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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