Alternative Distributions for X = (X1, X2) Testing Normality of X = (X1, X2) Testing U(X) = (X1 − X2, X1 + X2) FA mvShapiro Royston H R6 FA mvShapiro Royston H N (0,1)*N (0,1) 5 5 5 5 5 5 5 Exponential*Exponential 100 100 100 100 100 90 89 χ² (2)*χ² (2) 100 100 100 100 100 90 89 χ² (5)* χ² (5) 92 99 97 94 93 48 46 χ² (10)* χ² (10) 62 80 72 66 62 25 23 Lognormal (0,0.5)*Lognormal (0,0.5) 96 99 98 97 97 67 62 Lognormal (0,0.25)* Lognormal (0,0.25) 48 62 57 50 49 20 18 Gamma (5,1)*Gamma (5,1) 63 80 73 66 63 26 24 t (2)*t (2) 96 98 98 98 96 84 75 t (5)*t (5) 45 51 55 52 46 26 26 Beta (1, 1)*Beta (1,1) 51 93 79 63 52 4 3 Beta (1,2)*Beta (1,2) 73 97 89 81 75 12 10 Beta (2,2)*Beta (2,2) 6 23 13 6 6 3 2 Logistic (0,1)*Logistic (0,1) 24 27 30 27 24 12 14 Halfnormal*Halfnormal 93 100 97 95 93 35 33 Weibull (1)*Weibull (1) 100 100 100 100 100 90 89 Weibull (1.5)*Weibull (1.5) 89 98 95 92 89 38 36 Pearson II (0) 26 49 34 35 26 49 33 Pearson II (1) 6 11 7 6 6 11 7 Pearson VII (4) 30 37 40 49 30 38 39 Pearson VII (5) 43 26 28 35 43 25 28 N (0, 1)*Exponential 99 99 98 98 99 43 45 N (0, 1)*Beta (1, 1) 33 47 40 23 32 5 32 N (0, 1)*Beta (1, 2) 51 61 55 38 52 9 54 N (0, 1)*Halfnormal 75 80 75 68 75 17 42 N(0, 1)*Gamma(5, 1) 41 47 43 34 41 14 3 N(0, 1)*t(2) 81 82 84 81 81 51 7 N(0, 1)*t(5) 28 30 33 30 29 14 9 N(0, 1)*Chisq(2) 99 100 99 97 99 42 7 N(0, 1)*Chisq(5) 73 79 74 60 74 21 1 N(0, 1)*Weibull(1) 99 99 99 99 99 44 47 N(0, 1)*Weibull(1.5) 68 74 69 62 68 18 47 NMIX(.5, 2,0,0) 10 4 3 7 11 19 14 NMIX(.5, 2, 0,0.9) 66 53 56 67 67 62 61 NMIX(.75, 2, .9, 0) 86 85 69 84 86 69 71 NMIX(.75, 2, 0, .9) 51 6 15 47 53 65 63