# Parametric Inference for Models with Dependent Truncation Data

### Description

Maximum likelihood estimation (MLE) for dependent truncation data under the bivariate normal distribution. A bivariate normal distribution is assumed for bivariate random variables (L, X). The truncated data (L_j, X_j), subject to L_j<=X_j for all j=1, ..., n, are used to obtain the MLE for the population parameters of (L, X).

### Usage

1 | ```
PMLE.Normal(l.trunc, x.trunc, testimator = FALSE,GOF=TRUE)
``` |

### Arguments

`l.trunc` |
vector of truncation variables satisfying l.trunc<=x.trunc |

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

`testimator` |
if TRUE, testimator is computed instead of MLE |

`GOF` |
if TRUE, goodness-of-fit test is performed |

### Details

PMLE.Normal performs the maximum likelihood estimation for dependently left-truncated data under the bivariate normal distribution. "PMLE.Normal" implements the methodologies developed in Emura T. & Konno Y. (2012, Statistical Papers 53, 133-149)and can produce the maximum likelihood estimates and their standard errors. Furthermore, "PMLE.Normal" tests the independence assumption between truncation variable and variable of interest via likelihood ratio test. The MLE is obtained by minimizing -logL using "nlm", where L is the log-likelihood.

### Value

`mu_L ` |
mean of L and its standard error |

`mu_X ` |
mean of X and its standard error |

`var_L ` |
variance of L and its standard error |

`var_X ` |
variance of X and its standard error |

`cov_LX ` |
covariance between L and X and its standard error |

`c ` |
inclusion probability, defined by c=Pr(L<=X), and its standard error |

`test ` |
Likelihood ratio statistic and p-value |

`C ` |
Cramer-von Mises goodness-of-fit test statistics |

`K ` |
Kolmogorov-Smirnov goodness-of-fit test statistics |

### Author(s)

Takeshi EMURA

### References

Emura T, Konno Y (2012), Multivariate Normal Distribution Approaches for Dependently Truncated Data. Statistical Papers 53 (No.1), 133-149.

Emura T, Konno Y (2014), Erratum to: Multivariate Normal Distribution Approaches for Dependently Truncated Data, Statistical Papers 55 (No.4): 1233-36

### Examples

1 2 3 | ```
l.trunc=c(1,2,3,4,5,6,7,8,8)
x.trunc=c(2,4,4,5,5,7,7,9,10)
PMLE.Normal(l.trunc,x.trunc,testimator=FALSE)
``` |