Accurate Directional Inference for Vector Parameters in Linear Exponential Families

Abstract

We consider inference on a vector-valued parameter of interest in a linear exponential family, in the presence of a finite-dimensional nuisance parameter. 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). Examples and simulations illustrate the high accuracy of the method, which we compare with the usual likelihood ratio test and with an adjusted version due to Skovgaard (2001). In high-dimensional settings, such as covariance selection, the approach works essentially perfectly, whereas its competitors can fail catastrophically.

A directional look at F‐tests

“…The theory and methods for directional tests are given in Davison et al () and Fraser, Reid & Sartori (), and provided for completeness in the Supplementary Material. Here we introduce the necessary notation and key concepts.…”

“…The P ‐value for this directional approach is defined, as in Davison et al (, Section 3.2), by a ratio of two integrals, where h ( t ; ψ ) is defined below, d is the dimension of ψ , and t indexes points along the line, with t = 0 corresponding to the value u ψ , and t = 1 corresponding to the observed value u 0 . The upper bound, t max , of these integrals is the largest value of t where the corresponding sufficient statistic on the line between u ψ and u 0 still lies in the support of its distribution.…”

“…Skovgaard () derived a saddlepoint‐type expansion for directional tests in exponential family models. Davison et al () and Fraser, Reid & Sartori () showed how to calculate directional P ‐values via one‐dimensional integrals and illustrated this in a number of models. While the theory was developed in a standard asymptotic scenario, with n increasing and p fixed, empirical results have shown that the method is extremely accurate even in cases where the dimensions p and d are rather large relative to n , when standard first order methods, and some higher order improvements, generally fail.…”

p-Values: The Insight to Modern Statistical Inference

“…Now consider an exponential model with p-dimensional canonical parameter ϕ and a scalar parameter ψ = ψ(ϕ) of interest; the case with a vector interest parameter is discussed by Davison et al (2013) and Fraser et al (2016). As the null density for testing ψ = ψ0, we have the saddlepoint-based ancillary density (Equation 18) on the line L 0 = {u :λ(u) =λ 0 }, where the nuisance parameter constrained maximum likelihood estimateλ = λ ψ 0 is equal to its observed valueλ 0 under ψ = ψ0 or ϕ =φ.…”

Interview with Nancy Reid

“…And then Don and I have two papers on higher order asymptotics using directional tests that I really like: Fraser et al () and Davison et al (). Going back to an earlier idea of Don's doing multivariate testing through conditioning on a direction, we found a way to use higher order asymptotics to get really good results; that was a lot of fun.…”