Some Empirical Distributions of BivariateT2and Homoscedasticity CriterionMUnder Unequal Variance and Leptokurtosis

Abstract Computer generated pseudo random numbers were used to simulate drawing 1000 pairs of samples of N 1, N 2 = 5, 10, 20 from bivariate populations normal (O, σ i 2 I) having σ2/σ1 = 1, 1.6 or 3.2, and from circular bivariate symmetrical leptokurtic populations with zero means, equal variances and β2 − 3 = 3.2 or 6.2. Results suggest that the null distribution of T 2 for pairs of bivariate normal samples with N 1 = N 2 ≥ 10 is rather robust against variance inequality but that this robustness does not extend to disparate sample sizes, and that upper tail frequencies of the distribution of bivariate T 2 for N 1, N 2 ≥ 10 are not substantially affected by moderate degrees of symmetrical leptokurtosis. In simulations of sampling from circular normal populations with scale parameters in the ratio of 1.6:1 and 3.2:1 respectively, the proportion of M exceeding the null 5% point ranged from 9% and 49% for N 1 = N 2 = 5 to 60% and 100% for N 1 = N 2 = 20. In simulations of homoscedastic leptokurtic sampling, the proportions of M exceeding the null normal 5% point for N 1 = N 2 = 5 and for N 1 = N 2 = 20 were 8% and 17% for β2 − 3 = 3.2, and 22% and 42% for β2 − 3 = 6.2.