Comparing higher order likelihood inference for location-scale models

For parametric models, the third order asymptotic theories for approximating tail probabilities are extremely accurate even for small sample size. These methods only require the likelihood function and the observed sample. Two third order asymptotic methods developed by Skovgaard in 1996 and Fraser and Reid in 1999 are compared and applied to location-scale family model in this dissertation. The Fraser and Reid method and the Skovgaard method have similar ideas except the canonical parameterization is different.

Based on the special structure of location-scale model, a simple and accurate method is developed by transforming all the scale parameters into location type parameters. However, the general formulas to calculate the confidence intervals for location or scale parameter in the Fraser and Reid method and the Skovgaard method are also derived. The Behrens-Fisher problem with an assumption that the ratio of the two variances is known which is first considered by Schechtman and Sherman (2007) is revisited. Our proposed third order methods exhibit significant advantage over some existing first order methods especially for small sample size. All of these results will be illustrated through numerical studies.