## Abstract

This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η) H_{0} γ = γo, based on the pseudolikelihood L(θ,φ) where φ is a consistent estimator of φ, the nuisance parameter. We show that the asymptotic distribution of T under H_{0} is a weighted sum of independent chi-squared variables. Some sufficient conditions are provided for the limiting distribution to be a chi-squared variable. When the true value of the parameter of interest θ_{0}, or the true value of the nuisance parameter φ_{0}, lies on the boundary of parameter space, the problem is shown to be asymptotically equivalent to the problem of testing the restricted mean of a multivariate normal distribution based on one observation from a multivariate normal distribution with misspecified covariance matrix, or from a mixture of multivariate normal distributions. A variety of examples are provided for which the limiting distributions of T may be mixtures of chi-squared variables. We conducted simulation studies to examine the performance of the likelihood ratio test statistics in variance component models and teratological experiments.

Original language | English (US) |
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Pages (from-to) | 603-620 |

Number of pages | 18 |

Journal | Biometrika |

Volume | 97 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2010 |

## Keywords

- Asymptotic distribution
- Boundary problem
- Frailty survival model
- Likelihood ratio test
- Nuisance parameter
- Pseudolikelihood
- Teratological experiment
- Variance component model

## ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Agricultural and Biological Sciences (miscellaneous)
- Agricultural and Biological Sciences(all)
- Statistics, Probability and Uncertainty
- Applied Mathematics