An Application of the Closed Testing Principle to Enhance One-Sided Confidence Regions for a Multivariate Location Parameter

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1 An Application of the Closed Testing Principle to Enhance One-Sided Confidence Regions for a Multivariate Location Parameter Universität Bern Institut für mathematische Statistik und Versicherungslehre MCP 2007, Vienna, July 10, 2007 Research supported by the Swiss National Science Foundation

2 Outline Introduction Direct Derivation of Confidence Regions for ϑ Enhanced Confidence Regions for ϑ Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 2

3 One-Sided Confidence Regions X 1,..., X n i. i. d. random vectors in R p X i P ϑ, ϑ Θ unknown P ϑ (at least directionally) symmetric w. r. t. ϑ Problem: Find a 1 α confidence region for ϑ that is as strict as possible in specific directions possibly unbounded in irrelevant directions (e. g. a cone or an orthant). Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 3

4 Connection with One-Sided Location Tests Let ϕ α be a non-randomized level α test for H 0 : ϑ Θ 0 (γ) vs. H 1 : ϑ Θ Θ 0 (γ). (E. g. Θ 0 (γ) = γ + (, 0] p ) Inversion of ϕ α C 1 α (X 1,..., X n ) = {γ : ϕ α ((X 1,..., X n ), γ) = 0}, and P ϑ (C 1 α (X) γ) 1 α ϑ Θ 0 (γ) γ Θ. C 1 α is a 1 α confidence region for the meta-parameter γ. Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 4

5 Outline Introduction Direct Derivation of Confidence Regions for ϑ Enhanced Confidence Regions for ϑ Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 5

6 Direct Derivation of Confidence Regions for ϑ Assume that γ Θ 0 (γ), γ Θ. Then P ϑ (C 1 α (X) ϑ) 1 α ϑ Θ. C 1 α is also a 1 α confidence region for the location parameter ϑ. Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 6

7 Direct Derivation of Confidence Regions for ϑ Assume that γ Θ 0 (γ), γ Θ. Then P ϑ (C 1 α (X) ϑ) 1 α ϑ Θ. C 1 α is also a 1 α confidence region for the location parameter ϑ. Problems: conservative unpleasant shape to be illustrated... Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 6

8 Min and Max Tests Min Test Reject H 0 : j {1,..., p} : ϑ j γ j in favor of H 1 : ϑ > γ at the level α if and only if ϕ j,α ((X 1j,..., X nj ), γ j ) = 1, j 1,..., p. Bonferroni Max Test Reject H 0 : ϑ γ in favor of H 1 : j {1,..., p} : ϑ j > γ j at the level α if and only if j 1,..., p : ϕ j,α/p ((X 1j,..., X nj ), γ j ) = 1. Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 7

9 Example Two variables of the pulmonary function data by Randles (1989) (slightly modified from Merchant et al., 1975). FEV FVC Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 8

10 Pulmonary Function Data, Wilcoxon Min Test FEV3 γ2 Θ 0 (γ) FEV C 1 α (X) γ 1 FVC FVC C 1 α (X) = c 1 α (X) Θ 0 (0) Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 9

11 P. F. Data, Wilcoxon Bonferroni Max Test FEV3 γ2 Θ 0 (γ) FEV C 1 α (X) γ 1 FVC FVC C 1 α (X) = c 1 α (X) Θ 0 (0) Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 10

12 P. F. Data, Sign Test by Larocque/Labarre (2004) FEV3 γ2 Θ 0 (γ) FEV C 1 α (X) γ 1 FVC FVC C 1 α (X) c 1 α (X) Θ 0 (0) (outside a sufficiently large ball) Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 11

13 Back to the Drawbacks of the Direct Approach If confidence regions for γ are directly used as confidence regions for ϑ, they are usually conservative and similar in shape to Θ 0, rather than to Θ 1 = Θ Θ 0. Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 12

14 Outline Introduction Direct Derivation of Confidence Regions for ϑ Enhanced Confidence Regions for ϑ Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 13

15 Enhanced Confidence Regions for ϑ (1) Temptation: C 1 α (X) = Θ 1 (γ) γ C 1 α (X) Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 14

16 Enhanced Confidence Regions for ϑ (1) Temptation: C 1 α (X) = Θ 1 (γ) γ C 1 α (X) liberal multiple testing problem! Solution: Reduce the set of possible meta-parameters in advance. Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 14

17 Enhanced Confidence Regions for ϑ (2) Let C 1 α : X P(R p ) be a 1 α confidence region for γ based on (Θ 0 (γ)) γ R p. Let Θ 0 (γ) = γ + Θ 0 (0), γ R p, closed, Θ 1 (γ) = R p Θ 0 (γ). Assume that Θ 0 (γ) Θ 0 (γ + (δ,..., δ) T ), γ R p, δ > 0. With γ i = (i,..., i) T R p, i I = [l, ), define C 1 α (X) := Θ 1 (γ i ). i I:γ i C 1 α (X) i i Then ( ) P ϑ C1 α (X) ϑ 1 α ϑ R p. Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 15

18 Idea of the Proof (Θ 0 (γ i )) i I is closed under (finite and infinite) intersections. Apply the closed testing principle (Marcus, Peritz, and Gabriel, 1976). Translate to confidence regions. Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 16

19 Pulmonary Function Data, Wilcoxon Min Test FEV C 1 α (X) FVC Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 17

20 Pulmonary Function Data, Wilcoxon Min Test FEV ( l l) C 1 α (X) FVC Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 17

21 Pulmonary Function Data, Wilcoxon Min Test FEV ( l l) C 1 α (X) C1 α (X) FVC Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 17

22 Enhanced Confidence Regions: Properties C1 α (X) = c 1 α (X) + Θ 1 (0) (by definition) C1 α (X) C 1 α (X) under suitable conditions (Θ 1 (0) convex cone, translation invariance, and a monotonicity property) Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 18

23 Enhanced Confidence Regions: Properties C1 α (X) = c 1 α (X) + Θ 1 (0) (by definition) C1 α (X) C 1 α (X) under suitable conditions (Θ 1 (0) convex cone, translation invariance, and a monotonicity property) Disadvantage: Restricted set of possible confidence regions (search on the diagonal) Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 18

24 Summary A confidence region C 1 α (X) obtained by inversion of a test for a composite null hypothesis is for a meta-parameter. Even if C 1 α (X) may also be a confidence region for the parameter ϑ itself, it is not very useful. The proposed method based on the closed testing principle yields a confidence region C 1 α (X) with a more useful shape. C1 α (X) is also less conservative than C 1 α (X) under suitable conditions. Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 19

25 References Larocque, D., Labarre, M. (2004). A conditionally distribution-free multivariate sign test for one-sided alternatives. Journal of the American Statistical Association, 99, Marcus, R., Peritz, E., Gabriel, K. R. (1976). On closed testing procedures with special reference to ordered analysis of variance. Biometrika, 63, Merchant, J. A., Halprin, G. M., Hudson, A. R., Kilburn, K. H., McKenzie, W. N., Hurst, D. J., Bermazohn, P. (1975). Responses to cotton dust. Archives of Environmental Health, 30, Randles, R. H. (1989). A distribution-free multivariate sign test based on interdirections. Journal of the American Statistical Association, 84, Vock, M. (2007). Enhanced one-sided confidence regions for a multivariate location parameter, submitted. Introduction Direct Confidence Regions for ϑ Enhanced Confidence Regions for ϑ 20

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