Likelihood Asymptotics

Abstract

The paper gives an overview of modern likelihood asymptotics with emphasis on results and applicability. Only parametric inference in well-behaved models is considered and the theory discussed leads to highly accurate asymptotic tests for general smooth hypotheses. The tests are re®nements of the usual asymptotic likelihood ratio tests, and for one-dimensional hypotheses the test statistic is known as r à , introduced by BarndorffNielsen. Examples illustrate the applicability and accuracy as well as the complexity of the required computations. Modern likelihood asymptotics has developed by merging two lines of research: asymptotic ancillarity is the basis of the statistical development, and saddlepoint approximations or Laplace-type approximations have simultaneously developed as the technical foundation. The main results and techniques of these two lines will be reviewed, and a generalization to multi-dimensional tests is developed. In the ®nal part of the paper further problems and ideas are presented. Among these are linear models with non-normal error, non-parametric linear models obtained by estimation of the residual density in combination with the present results, and the generalization of the results to restricted maximum likelihood and similar structured models.

Accurate directional inference for vector parameters

“…Skovgaard (2001) notes that the accuracy of the χ 2 d approximation to (28) can be lost when the expected value is approximated using its asymptotic expansion, rather than computed analytically. Even the approximate version can be cumbersome to compute, as it involves arrays of third and fourth order cumulants (Lawley, 1956;McCullagh and Cox, 1986).…”

“…A development for vector parameters of interest, parallel to that of r * , was given in Skovgaard (2001). The resulting test statistic has a distribution close to χ 2 and was derived analogously to r * , so that the approximation is also accurate in large deviation regions.…”

“…Skovgaard (2001) shows that in addition to having good large-deviation properties, w * (ψ) is also easier to calculate than the Bartlett adjustment discussed in §6. Like the likelihood ratio w(ψ), (2) gives an omnibus measure of departure; all potential directions away from the hypothesis H ψ are averaged in the calculation of p-values.…”

“…Skovgaard (2001)'s w * gives 0.048, and a conditional simulation using the method of Kolassa and Tanner (1994) gives 0.051. The sample size in this example is too large for the methods to give very different p-values.…”

“…For continuous responses the work extends the directional test of Cheah et al (1994). Exponential family examples and simulations illustrate the high accuracy of the method, which we compare with an adjusted likelihood ratio test of Skovgaard (2001). In a high-dimensional covariance selection example the approach works essentially perfectly, whereas its competitors fail catastrophically.…”