### 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) |
---|---|

Pages (from-to) | 603-620 |

Number of pages | 18 |

Journal | Biometrika |

Volume | 97 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2010 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Biometrika*,

*97*(3), 603-620. https://doi.org/10.1093/biomet/asq031

**On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems.** / Chen, Yong; Liang, Kung Yee.

Research output: Contribution to journal › Article

*Biometrika*, vol. 97, no. 3, pp. 603-620. https://doi.org/10.1093/biomet/asq031

}

TY - JOUR

T1 - On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems

AU - Chen, Yong

AU - Liang, Kung Yee

PY - 2010/9

Y1 - 2010/9

N2 - This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η) H0 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 H0 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 θ0, or the true value of the nuisance parameter φ0, 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.

AB - This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η) H0 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 H0 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 θ0, or the true value of the nuisance parameter φ0, 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.

KW - Asymptotic distribution

KW - Boundary problem

KW - Frailty survival model

KW - Likelihood ratio test

KW - Nuisance parameter

KW - Pseudolikelihood

KW - Teratological experiment

KW - Variance component model

UR - http://www.scopus.com/inward/record.url?scp=77955880322&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955880322&partnerID=8YFLogxK

U2 - 10.1093/biomet/asq031

DO - 10.1093/biomet/asq031

M3 - Article

C2 - 22822249

AN - SCOPUS:77955880322

VL - 97

SP - 603

EP - 620

JO - Biometrika

JF - Biometrika

SN - 0006-3444

IS - 3

ER -