We consider a general regression model without a scale parameter. We construct a confidence interval for a scalar parameter of interest that utilises the uncertain prior information that a distinct scalar parameter takes a specified value. This confidence interval has good coverage properties. It also has scaled expected length, where the scaling is with respect to the usual confidence interval, that is, (a) it is substantially less than 1 when the prior information is correct, (b) it has a maximum value that is not too large and (c) it is close to 1 when the data and prior information are highly discordant. We use the R package ciuupi, which computes a confidence interval that utilises uncertain prior information in a linear regression model with known error variance, to build this confidence interval. This work appears in [Reference Kabaila and Ranathunga3] and has been submitted for publication.
We also solve the problem of numerically evaluating the expected value of a smooth bounded function of a chi-distributed random variable, divided by the square root of the number of degrees of freedom, using Mori’s transformation followed by the trapezoidal rule, which is exponentially convergent for suitable integrands. This problem arises in simultaneous inference, selection and ranking of populations, the evaluation of multivariate t probabilities and the assessment of the coverage and expected volume properties of nonstandard confidence regions. This work appears in [Reference Kabaila and Ranathunga4].
We apply this solution in the new R package ciuupi2 [Reference Ranathunga and Kabaila5] that computes the Kabaila and Giri [Reference Kabaila and Giri1] confidence interval which utilises uncertain prior information in a linear regression model with unknown error variance. Previous computations of this interval used MATLAB programs that were time-consuming to run. By writing these programs in R, the computation time is greatly reduced and they become freely available. We also assess a new definition of scaled expected length.
Finally, we compare the computations of the log-likelihood function for generalised linear mixed models using (a) adaptive Gauss–Hermite quadrature and (b) importance sampling, where both methods share the same initial step. This work appears in [Reference Kabaila, Ranathunga and Nguyen2].
