# Calculate NPMLE with constriants for accelerated failure time model with given coefficients.

### Description

Use the empirical likelihood ratio and Wilks theorem to test if the regression coefficient is equal to beta.

*El(F)=∏_{i=1}^{n}(Δ F(T_i))^{δ_i}(1-F(T_i))^{1-δ_i}*

with constraints

*
∑_i g(T_i)Δ F(T_i)=0,\quad,i=1,2,…
*

Instead of EM algorithm, this function calculate the Kaplan-Meier estimator with mean constraints recursively to test *H_0:~β=β_0* in accelerated failure time model:

*
\log(T_i) = y_i = x_iβ^\top + ε_i,
*

where *ε* is distribution free.

### Usage

1 | ```
kmc.bjtest(y, d, x, beta,init.st="naive")
``` |

### Arguments

`y` |
Response variable vector (length n). |

`d` |
Status vector (length n), 0: right censored; 1 uncensored. |

`x` |
n by p explanatory variable matrix. |

`beta` |
The value of the regression coeffiecnt vector (length p) to be tested. |

`init.st` |
Type of methods to initialize the algorithm. Default uses 1/n |

### Details

The empirical likelihood is the likelihood of the error term when the coefficients are specified. Model assumptions are the same as requirements of a standard Buckley-James estimator.

### Value

a list with the following components:

`prob` |
the probabilities that max the empirical likelihood under estimating equation. |

`logel1` |
the log empirical likelihood without constraints, i.e. under Kaplan-Merier of residuals' |

`logel2` |
the log empirical likelihood with constraints, i.e. under null hypotheses or estimation equations. |

`"-2LLR"` |
the -2 loglikelihood ratio; have approximate chisq distribution under null hypotheses |

### Author(s)

Mai Zhou(mai@ms.uky.edu), Yifan Yang(yifan.yang@uky.edu)

### References

Buckley, J. and James, I. (1979). Linear regression with censored data. *Biometrika*, **66** 429-36

Zhou, M., & Li, G. (2008).
Empirical likelihood analysis of the Buckley-James estimator.
*Journal of multivariate analysis*, **99(4)**, 649-664.

Zhou, M. and Yang, Y. (2015).
A recursive formula for the Kaplan-Meier estimator with mean constraints and its application to empirical likelihood
*Computational Statistics*. **Online ISSN** 1613-9658.

### See Also

`plotkmc2D`

, `print.kmcS3`

`bjtest`

.

### Examples

1 2 |