We consider a bivariate normal distribution with linear correlation ρ whose random components are discretized according to two assigned sets of thresholds. On the resulting bivariate ordinal random variable, one can compute Goodman and Kruskal’s gamma coefficient,γ which is a common measure of ordinal association. Given the known analytical monotonic relationship between Pearson’s ρ and Kendall’s rank correlation τ for the bivariate normal distribution, and since in the continuous case, Kendall’s τ coincides with Goodman and Kruskal’s γ, the change of this association measure before and after discretization is worth studying. We consider several experimental settings obtained by varying the two sets of thresholds, or, equivalently, the marginal distributions of the final ordinal variables. This study, confirming previous findings, shows how the gamma coefficient is always larger in absolute value than Kendall’s rank correlation; this discrepancy lessens when the number of categories increases or, given the same number of categories, when using equally probable categories. Based on these results, a proposal is suggested to build a bivariate ordinal variable with assigned margins and Goodman and Kruskal’s γ by ordinalizing a bivariate normal distribution. Illustrative examples employing artificial and real data are provided.

This study extends the joint estimation of revealed and stated preference data literature by accounting for truncation in the revealed preference data. The analytical model and estimation procedure are used to estimate the value of recreational red snapper fishing in the Gulf of Mexico. This recreational red snapper valuation is decomposed into its direct and indirect components. As expected, the value of recreational red snapper fishing using the joint revealed-stated preference model proposed in this analysis is bracketed on the upper limit by the value obtained using the contingent valuation method and on the lower limit by the travel cost method. The results also indicate that the joint model improves the precision of estimated recreational red snapper valuation.

A saddlepoint approximation is derived for the cumulative distribution function of the sample mean of n independent bivariate random vectors. The derivations use Lugannani and Rice's saddlepoint formula and the standard bivariated normal distribution function. The separate versions of the approximation for the discrete cases are also given. A Monte Carlo study shows that the new approximation is very accurate.

Suppose given an absolutely continuous distribution on the plane, and points P i, · · ·, Pj , Πt, · · ·, Πk chosen independently according to the given distribution. Denoting by Gj the convex hull of {P i, · · ·, Pj }, and by Γk the convex hull of (Πt, · · ·, Πk }, and writing pjk for the probability that Gj and Γk are disjoint, certain properties of the array {pjk ; j, k = 1,2, · · ·} are established, including a recurrence generating the array in terms of {p 1n ; n = 1,2, · · ·}, and asymptotic results for {pnn ; n = 1,2, · · ·}. Some examples are considered.