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