Journal of Probability and Statistics / 2011 / Article / Tab 2

Research Article

Simultaneous Inference on All Linear Combinations of Means with Heteroscedastic Errors

Table 2

Comparison of interval widths between Scheffรฉโ€™s and generalized Scheffรฉโ€™s methods. Their interval widths differ in quantities: ๐‘„ 1 = โˆš ๐ผ โ‹… ๐น ๐›ผ , ๐ผ , ๐‘ โˆ’ ๐ผ โ‹… M S E in Scheffรฉโ€™s method and ๐‘„ 2 = ๎” ๐น ๐›ผ , ฬ‚ ๐œˆ 1 , ฬ‚ ๐œˆ 2 โ‹… โˆ‘ ๐‘† 2 ๐‘– in the generalized Scheffรฉโ€™s method ( ๐›ผ = 0 . 0 5 ).

Sample sizeEqual variancesUnequal variances
(0.1, 0.1, 0.1, 0.1)(1, 1, 1, 1)(0.3, 0.3, 0.1, 0.1)(3, 3, 1, 1)

Balanced ๐‘„ 1 ๐‘„ 2 ๐‘„ 1 ๐‘„ 2 ๐‘„ 1 ๐‘„ 2 ๐‘„ 1 ๐‘„ 2
โ€ƒ(5, 5, 5, 5) 0.343 0.370 3.422 3.680 0.754 0.909 7.598 9.182
โ€ƒ(10, 10, 10, 10) 0.322 0.331 3.229 3.323 0.718 0.813 7.1668.105
โ€ƒ(20, 20, 20, 20) 0.315 0.319 3.153 3.194 0.703 0.778 7.0327.780
โ€ƒ(50, 50, 50, 50) 0.310 0.312 3.105 3.1200.694 0.759 6.945 7.595
Unbalanced
โ€ƒ(5, 5, 10, 10) 0.329 0.350 3.284 3.490 0.602 0.905 6.052 9.125
โ€ƒ(5, 5, 20, 20) 0.318 0.340 3.196 3.423 0.489 0.905 4.894 9.093
โ€ƒ(10, 10, 20, 20) 0.318 0.326 3.173 3.250 0.597 0.812 5.951 8.092
โ€ƒ(10, 10, 50, 50) 0.312 0.321 3.128 3.218 0.466 0.810 4.669 8.138

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