# Semiparametric Inference for Copula Models with Dependent Truncation Data

### Description

Nonparametric maximum likelihood estimation for dependent truncation data under the Frank copula models (Emura & Wang, 2012). The forms of the marginal distributions for X and Y are completely unspecified, but the parametric form of copula is specified as the Frank copula.

### Usage

1 2 | ```
NPMLE.Frank(x.trunc, y.trunc,
x.fix = median(x.trunc), y.fix = median(y.trunc), plotX = TRUE)
``` |

### Arguments

`x.trunc` |
vector of variables satisfying x.trunc<=y.trunc |

`y.trunc` |
vector of variables satisfying x.trunc<=y.trunc |

`x.fix` |
vector of fixed points at which marginal distribution function of X is calculated |

`y.fix` |
vector of fixed points at which marginal survival function of Y are calculated |

`plotX` |
if TRUE, the culumative distribution function for X is plotted |

### Details

The function produces the nonparametric maximum likelihood estimate (NPMLE) for the marginal distributions and the estimate of the association parameter under the Frank copula model. The method follows Emura & Wang (2012). The maximization of the likelihood (L) is conducted by minimizing -logL by "nlm". Standard errors are calculated from the diagonal elements in the observed Fisher information matrix, which is obtained from the output of the "nlm". The method can handle ties but cannot handle right-censoring.

### Value

`alpha ` |
estimate of association parameter |

`alpha_se ` |
standard error of the estimate of association parameter |

`Hx ` |
estimate of the marginal cumulative reverse-hazard function for X |

`Hx_se ` |
standard error of the estimate of the marginal cumulative reverse-hazard function for X |

`Ay ` |
estimate of the marginal cumulative hazard function for Y |

`Ay_se ` |
standard error of the estimate of the marginal cumulative hazard function for Y |

`Fx ` |
estimate of the marginal distribution function for X |

`Fx_se ` |
standard error of the estimate of the marginal distribution function for X |

`Sy ` |
estimate of the marginal survival function for Y |

`Sy_se ` |
standard error of the estimate of the marginal survival function for Y |

`conv ` |
an integer indicating why the optimization process terminated in "nlm": If conv=1, the maximization of the likelihood function is properly done. Please refer R function "nlm" for more details. |

`iteration` |
the number of iterations until convergence |

`Grad` |
L_2 norm for the gradient vector at the solution (close to zero if the solution is proper) |

`MinEigen` |
Minimum eigenvalue of the Hessian matrix at the solution (positive if the solution is proper) |

### Author(s)

Takeshi EMURA

### References

Emura T, Wang W (2012) Nonparametric maximum likelihood estimation for dependent truncation data based on copulas, Journal of Multivariate Analysis 110, 171-88

### Examples

1 2 3 | ```
x.trunc=c(0.53, 0.43, 1.63, 0.98, 0.62)
y.trunc=c(1.20, 4.93, 2.09, 2.57, 1.52)
NPMLE.Frank(x.trunc, y.trunc, x.fix =1, y.fix = 2, plotX = TRUE)
``` |