Likelihood for component parameters

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Abstract
For a statistical model with data, likelihood for the scalar or vector full parameter θ, of dimension p say, is typically well defined and easily computed. In this paper, we investigate likelihood for a component parameter ψ(θ) of dimension d < p and make use of the recent likelihood theory that has been successful in producing highly accurate third‐order p‐values for scalar parameters of continuous models. The theory leads under moderate regularity to a definitive third‐order determination of likelihood for a component parameter ψ(θ) of dimension d, where 1 ≤ d ≤ p. We use the simple location model on the plane with standard normal errors to motivate the development. The example exhibits most of the key characteristics of the general case and the recent theory then extends the determination of likelihood to the general context. For the scalar interest parameter case with d = 1, the usual determinations are typically of second‐order accuracy; the example indicates how the new determination achieves third‐order accuracy. The implementation is straightforward and uses familiar ingredients to other determinations, such as the full maximum likelihood value θ̂, the constrained value θ̃ψ given ψ(θ) = ψ, and the observed information jλλ(θ̂ψ) for a complementing nuisance parameter λ(θ). It does however require a special version of the nuisance information jλλ(θ̂ψ), a version calibrated relative to a symmetric choice of the exponential‐type reparameterisation φ(θ) underlying the recent theory, but this is easily computed. Various examples are given and the motivating example is discussed in detail.

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