Computation of confidence intervals in regression utilizing uncertain prior information

We consider a linear regression model with regression parameter beta =(beta_1, …, beta_p) and independent and identically N(0, sigma^2)distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau = c^T beta - t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. Kabaila and Giri (2009c) present a new frequentist 1-alpha confidence interval for theta that utilizes this prior information. This interval has expected length that (a) is relatively small when the prior information about tau is correct and (b) has a maximum value that is not too large. It coincides with the standard 1-alpha confidence interval (obtained by fitting the full model to the data) when the data strongly contradicts the prior information. At first sight, the computation of this new confidence interval seems to be infeasible. However, by the use of the various computational devices that are presented in detail in the present paper, this computation becomes feasible and practicable.

💡 Research Summary

The paper addresses the construction of a frequentist 1‑α confidence interval for a linear combination θ = aᵀβ of regression coefficients in the classical linear model Y = Xβ + ε, where ε ∼ N(0, σ²I). In addition to the data, the analyst possesses uncertain prior information that another linear combination τ = cᵀβ − t equals zero. The vectors a and c are linearly independent, so the two quantities convey distinct information about β. Traditional practice ignores τ and uses the full‑model least‑squares estimator β̂ to form the standard interval θ̂ ± σ̂ z_{1‑α/2}, which can be unnecessarily wide when the prior information is correct.

Kabaila and Giri (2009c) propose a new interval that adapts its width according to the observed value of τ̂, the least‑squares estimator of τ. The interval is defined as

 C = { θ : |θ̂ − θ| ≤ σ̂ q(τ̂/σ̂) },

where q(·) is a non‑negative function to be chosen. The design goals for q are: (i) when τ̂/σ̂ is close to zero (i.e., the data support the prior), q should be small, yielding a shorter expected length; (ii) when |τ̂/σ̂| is large (the data contradict the prior), q should revert to the standard normal critical value z_{1‑α/2}, so that the interval coincides with the usual one; and (iii) the interval must maintain coverage at least 1 − α for all possible β and σ².

These objectives lead to a constrained optimisation problem: minimise the expected length E