Improved Likelihood Inference for Discrete Data

Abstract

Summary. Discrete data, particularly count and contingency table data, are typically analyzed using methods that are accurate to first order, such as normal approximations for maximum likelihood estimators. By contrast continuous data can quite generally be analyzed using third order procedures, with major improvements in accuracy and with intrinsic separation of information concerning parameter components. This paper extends these higher order results to discrete data, yielding a methodology that is widely applicable and accurate to second order. The extension can be described in terms of an approximating exponential model expressed in terms of a score variable. The development is outlined and the flexibility of the approach illustrated by examples.

Accurate directional inference for vector parameters

“…We propose a directional test for the vector parameter of interest, that is computed using one-dimensional integration. For discrete responses this extends the development of Davison et al (2006), and several examples below concern testing hypotheses in contingency tables. For continuous responses the work extends the directional test of Cheah et al (1994).…”

“…Since these approximations have bounded relative error both in the centre of the distribution and in large deviation regions, they provide highly accurate inferences well into the distribution tails. A review of this literature and several examples are given in Brazzale et al (2007) and Brazzale and Davison (2008); the discrete case is considered in more generality in Davison et al (2006).…”

Accurate Directional Inference for Vector Parameters in Linear Exponential Families

“…Since these approximations have bounded relative error both in the centre of the distribution and in large deviation regions, they provide highly accurate inferences well into the distribution tails. A review of this literature and several examples are given in Brazzale et al (2007) and Brazzale and Davison (2008); the discrete case is considered in more generality in Davison et al (2006). A development for vector parameters of interest, parallel to that of r * , was given in Skovgaard (2001).…”

“…Based on higher order asymptotic theory for likelihood, we propose a directional test whose p-value is computed using one-dimensional integration. For discrete responses this extends the development of Davison et al (2006), and some of our examples concern testing in contingency tables. For continuous responses the work extends the directional test of Cheah et al (1994).…”

Three Examples of Accurate Likelihood Inference

“…Equivalently, for location-scale and scale models with continuous responses one may take z k = (y k − μ k )/σ k and z k = y k /σ k , respectively, where the location and scale parameters μ k and σ k may vary with k; this encompasses regression formulations in which μ k = μ(x T β) and σ k = σ (x T k γ ) are functions of covariate vectors x k . For a discrete exponential family model, we may take (Davison, Fraser, and Reid 2006)…”