# Test the case weighted regression estimator by Empirical Likelihood

### Description

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

* ∑_{d=1} \log Δ F(y_i) + ∑_{d=0} \log [1-F(y_i)].*

### Usage

1 |

### Arguments

`x` |
a matrix of size N by q. Random design matrix. |

`y` |
a vector of length N, containing the censored responses. |

`delta` |
a vector (length N) of either 1's or 0's. delta=1 means y is uncensored; delta=0 means y is right censored. |

`beta0` |
a vector of length q. The value of the regression coefficient to be tested in the linear model |

.

`psifun` |
the estimating function. The definition of it determines the type of estimator under testing. |

### Details

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 = ∑ δ_i Δ F(Y_i) X_i ( Y_i - X_i β0 ) *

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.

### Value

A list with the following components:

`"-2LLR"` |
the -2 log likelihood ratio; have approximate chisq
distribution under |

`P-val` |
the p-value using the chi-square approximation. |

### Author(s)

Mai Zhou.

### References

Zhou, M.; Bathke, A. and Kim, M. (2012).
Empirical likelihood analysis of the case weighted estimator in
heteroscastic AFT model.
*Statistica Sinica*, **22**, 295-316.

### Examples

1 | ```
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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