## Abstract

Consider inference about a scalar parameter of interest 0 in the presence of a vector nuisance parameter. Inference about 0 is often based on a pseudolikelihood function. In this paper, the general problem of constructing a pseudo-loglikelihood function H(0) is considered. Conditions are given under which H has the same properties as a genuine loglikelihood function for a model without a nuisance parameter. When these conditions are satisfied to a given order of approximation, H is said to be a y th-order local loglikelihood function. The theory of local loglikelihood functions is developed and it is shown that second-order versions of these have a number of desirable properties. Several commonly used pseudolikelihood functions are studied from this point of view. One commonly used pseudolikelihood function is profile likelihood in which parameters other than 0 are replaced by their maximum likelihood estimates. A second aspect of the paper is to consider the use of other estimates in this context. Examples are given which suggest that inference about 0 may be improved if a method other than maximum likelihood is used, particularly when the number of other parameters is large relative to the sample size.

Original language | English (US) |
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Pages (from-to) | 507-522 |

Number of pages | 16 |

Journal | Biometrika |

Volume | 85 |

Issue number | 3 |

DOIs | |

State | Published - 1998 |

## Keywords

- Adjusted profile likelihood
- Asymptotic theory
- Local inference
- Profile likelihood

## ASJC Scopus subject areas

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