A proof of the conjecture on positive skewness of generalised inverse Gaussian distributions

We exploit a reparameterisation of the marginal covariance matrix arising in longitudinal studies (Pourahmadi, 1999, 2000) to model, jointly, the mean and covariance structures in terms of three polynomial functions of time. By reanalysing Kenwardʹs (1987) cattle data, we compare model selection procedures based on regressogram estimation with these based on a global search of the model space. Using a BIC-based model selection criterion to identify the optimum degree triple of the three polynomials, we show that the use of a saturated mean model is not optimal and explain why regressogram-based model estimation may be misleading. We also suggest a new computational method for finding the global optimum based on a criterion involving three pairwise saturated profile likelihoods.