Two Bootstrap Strategies for a 𝑘-Problem up to Location-Scale with Dependent Samples

Table 1

Percentages of rejection, as estimated from 1000 replicates, of the null hypothesis of the equality of 𝐾=2 distributions up to location/scale under Normal and asymmetric Normal dependence structures when 𝑛=100; 𝐹₁ is the Normal distribution; 𝐹₂ is either the 𝒩, 𝑡₉, 𝑡₆, 𝑡₃, or 𝒟ℰ distribution.

Copula	𝐹2	Bootstrap method I			Bootstrap method II
		𝑆𝑛	𝑇^𝒢1𝑛	𝑇^𝒢2𝑛	𝑆𝑛	𝑇^𝒢1𝑛	𝑇^𝒢2𝑛
𝐶^𝑁, 𝜏=−1/2	𝒩	3.4	3.5	3.3	1.5	6.5	3.0
	𝑡9	6.3	5.5	5.7	3.5	8.4	5.8
	𝑡6	10.8	8.4	11.3	5.4	15.4	10.5
	𝑡3	41.9	29.2	45.9	37.1	49.3	46.6
	𝒟ℰ	36.6	20.9	38.8	21.3	28.5	35.2
𝐶^𝑁, 𝜏=1/2	𝒩	4.6	4.8	4.9	0.4	5.6	3.4
	𝑡9	7.4	8.7	8.0	3.1	13.1	6.6
	𝑡6	13.2	12.6	14.7	5.1	20.9	13.1
	𝑡3	48.6	41.1	54.0	41.8	63.8	55.9
	𝒟ℰ	53.6	39.5	57.1	28.7	47.3	51.9
𝐾^𝑁, 𝜏=−1/2	𝒩	4.9	4.4	5.2	1.5	5.9	4.3
	𝑡9	6.8	5.3	7.6	2.4	8.3	6.9
	𝑡6	10.8	9.2	12.1	5.8	14.9	10.6
	𝑡3	37.6	26.8	40.1	33.1	48.4	44.0
	𝒟ℰ	38.4	21.6	40.4	22.7	30.6	35.1
𝐾^𝑁, 𝜏=1/2	𝒩	4.1	4.5	5.2	0.7	5.2	2.9
	𝑡9	6.5	7.2	7.5	1.7	10.4	5.8
	𝑡6	13.4	13.0	15.5	6.2	21.5	14.1
	𝑡3	49.7	40.9	54.2	42.9	61.3	57.0
	𝒟ℰ	48.8	35.2	51.3	27.5	44.0	45.5