A parametric test of the negativity of the substitution matrix
提出一个Wald检验来检验替代矩阵的负定性，该统计量的渐近分布是卡方分布的混合，并利用Kodde和Palm的上下界临界值简化计算，最后应用于Barten和Geyskens的实证结果。
Abstract The negativity of the substitution matrix implies that its latent roots are non‐positive. When inequality restrictions are tested, standard test statistics such as a likelihood ratio or a Wald test are not X 2 ‐distributed in large samples. We propose a Wald test for testing the negativity of the substitution matrix. The asymptotic distribution of the statistic is a mixture of X 2 ‐distributions. The Wald test provides an exact critical value for a given significance level. The problems involved in computing the exact critical value can be avoided by using the upper and lower bound critical values derived by Kodde and Palm (1986). Finally the methods are applied to the empirical results obtained by Barten and Geyskens (1975).