##
**Consistency of semiparametric maximum likelihood estimators for two-phase sampling.**
*(English)*
Zbl 0976.62017

Summary: Semiparametric maximum likelihood estimators have recently been proposed for a class of two-phase, outcome-dependent sampling models. All of them were “restricted” maximum likelihood estimators, in the sense that the maximization is carried out only over distributions concentrated on the observed values of the covariate vectors.

In this paper, the authors give conditions for consistency of these restricted maximum likelihood estimators. They also consider the corresponding unrestricted maximization problems, in which the “absolute” maximum likelihood estimators may then have support on additional points in the covariate space. Their main consistency result also covers these unrestricted maximum likelihood estimators, when they exist for all sample sizes.

In this paper, the authors give conditions for consistency of these restricted maximum likelihood estimators. They also consider the corresponding unrestricted maximization problems, in which the “absolute” maximum likelihood estimators may then have support on additional points in the covariate space. Their main consistency result also covers these unrestricted maximum likelihood estimators, when they exist for all sample sizes.

### MSC:

62F10 | Point estimation |

62F12 | Asymptotic properties of parametric estimators |

62G05 | Nonparametric estimation |

### Keywords:

empirical processes; Gilvenko-Cantelli theorem; identifiability; missing data; mixture; outcome dependence; stratified sampling; two-phase sampling; maximum likelihood estimators; consistency
PDF
BibTeX
XML
Cite

\textit{A. van der Vaart} and \textit{J. A. Wellner}, Can. J. Stat. 29, No. 2, 269--288 (2001; Zbl 0976.62017)

### References:

[1] | Breslow, Design and analysis of two-phase studies with binary outcome applied to Wilms tumour prognosis, Applied Statistics 48 pp 457– (1999) · Zbl 0957.62091 |

[2] | Breslow, Maximum likelihood estimation of logistic regression parameters under two-phase, outcome-dependent sampling, Journal of the Royal Statistical Society Series B 59 pp 447– (1997) · Zbl 0886.62071 |

[3] | Breslow, Technical Report (2000) |

[4] | Dudley, Real Analysis and Probability. (1989) · Zbl 0686.60001 |

[5] | Ferguson, A Course in Large Sample Theory. (1996) |

[6] | Gilbert, Large sample theory of maximum likelihood estimates in semiparametric biased sampling models, The Annals of Statistics 28 pp 151– (2000) · Zbl 1106.60302 |

[7] | Gill, Large sample theory of empirical distributions in biased sampling models, The Annals of Statistics 16 pp 1069– (1988) · Zbl 0668.62024 |

[8] | Lawless, Semiparametric methods for response-selective and missing data problems in regression, Journal of the Royal Statistical Society Series B 61 pp 413– (1999) · Zbl 0915.62030 |

[9] | Le Cam, On some asymptotic properties of maximum likelihood estimates and related estimates, University of California Publications in Statistics 1 pp 277– (1953) · Zbl 0052.15404 |

[10] | McNeney, Asymptotic Efficiency in Semiparametric Models with non-i.i.d. Data (1998) |

[11] | Palmgren, Regression models for bivariate binary responses 101 (1989) |

[12] | Prentice, A case-cohort design for epidemiologic cohort studies and disease prevention trials, Biometrika 73 pp 1– (1986) · Zbl 0595.62111 |

[13] | Prentice, Logistic disease incidence models and case-control studies, Biometrika 66 pp 403– (1979) · Zbl 0428.62078 |

[14] | Scott, Fitting logistic models in stratified case-control studies, Biometrics 47 pp 497– (1991) · Zbl 0736.62093 |

[15] | Scott, Fitting regression models to case-control data by maximum likelihood, Biometrika 84 pp 57– (1997) · Zbl 1058.62505 |

[16] | Scott, Maximum likelihood for generalised case-control studies (1998) |

[17] | Self, Asymptotic distribution theory and efficiency results for case-cohort studies, The Annals of Statistics 16 pp 64– (1988) |

[18] | van der Vaart, Maximum likelihod estimation with partially censored observations, The Annals of Statistics 22 pp 1896– (1994) |

[19] | van der Vaart, Asymptotic Statistics. (1998) · Zbl 0910.62001 |

[20] | van der Vaart, Existence and consistency of maximum likelihood in upgraded mixture models, Journal of Multivariate Analysis 43 pp 133– (1992) · Zbl 0752.62026 |

[21] | van der Vaart, High Dimensional Probability II pp 113– (2000) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.