. Author manuscript; available in PMC: 2016 Jan 1.

Published in final edited form as: J R Stat Soc Ser C Appl Stat. 2015 Jan 1;64(1):49–73. doi: 10.1111/rssc.12067

Table 5.

Robustness of the tests for H₂₀. Empirical sizes and powers of the tests $T_{a 1}^{(2)}, T_{a 2}^{(2)}, T_{m 1}^{(2)} and T_{m 2}^{(2)}$ for testing H₂₀ at the nominal level 0.05 for n = 500 and h = 0.2 when 50% of the marks are missing. The bandwidths are h₁ = 0.1 and h₂ = h. Each entry is based on 500 Gaussian multipliers samples and 500 repetitions.

Size/Power

Model

(α, β, γ)

T_{a 1}^{(2)}

T_{a 2}^{(2)}

T_{m 1}^{(2)}

T_{m 2}^{(2)}

r_k(w) is misspecified

(−0.69, 0, 0.3)

5.0

3.8

5.4

6.2

(−0.6, 0.6, 0.3)

24.0

25.2

34.2

36.0

(−1.2, 1.2, 0.3)

60.8

66.6

72.8

78.4

(−1.5, 1.5, 0.3)

76.6

82.0

85.8

88.8

g_k (a|t, v, z) is misspecified

(−0.69, 0, 0.3)

4.8

6.6

6.0

5.8

(−0.6, 0.6, 0.3)

17.2

18.0

28.4

28.2

(−1.2, 1.2, 0.3)

44.8

47.2

56.4

61.0

(−1.5, 1.5, 0.3)

58.0

60.4

68.6

73.2

r_k (w) and g_k (a|t, v, z) are misspecified

(−0.69, 0, 0.3)

4.0

4.8

4.4

(−0.6, 0.6, 0.3)

16.6

19.6

26.8

26.6

(−1.2, 1.2, 0.3)

43.2

46.6

55.6

60.6

(−1.5, 1.5, 0.3)

53.8

58.8

67.4

71.4

missing-at-random assumption is violated

(−0.69, 0, 0.3)

6.8

6.0

7.6

7.8

(−0.6, 0.6, 0.3)

28.6

33.6

39.6

42.0

(−1.2, 1.2, 0.3)

61.8

67.0

74.0

78.4

(−1.5, 1.5, 0.3)

77.4

81.6

85.4

89.2