D-Optimal Designs for Binary and Weighted Linear Regression Models: One Design Variable
Abstract
:1. Introduction
2. Models
Two-Parameter Models
3. On the Number of Support Points
4. Determination of Support Points
4.1. A Conjecture Proved under a Sufficient Condition
- Case 1: , ;
- Case 2: , , .
- Case 3: , , , , ;
- Case 4: , , , , ;
- Case 5: , , , .
- , . We show that .Since , . Moreover,The right-hand side of the equation is always positive, because and . Therefore, .
- , . We show .Since , . MoreoverThis equation is always positive on the right-hand side because and . Therefore, .
- ,Now, , is the only possible solution to Equations (10) and (11). Thus, . Thus, , identify 2 max TP’s of . Moreover, they are TP’s at which has a common value of zero since . From the property of , the only possibility is that they are the values of z say , ) at which the two maximal TP’s of , i.e., , . Hence, the Equivalence Theorem is satisfied on and on any subset which contains , . Hence, the two-point design
- Because , and , and .
- Since is increasing over , and since , then . Therefore, .
- Since is increasing over , and then . Therefore, .Hence, the two-point design
- ,Clearly, . We want negative. First, since . Second
- First, . Secondly, by above. Thus, . Consider now . Because , . We assumed . If is an increasing function, . Hence, . Thus, and the two-point design
- ,This is the complementary to case 4 with andThus, the two point design
- This is the complementary of case 6 withBecause and because of andSince andThus, the two-point design
4.2. Examples of
5. Group III and Group IV Weight Functions Results
5.1. Group III Weight Functions
5.2. Group IV Other Weight Functions
- Implications for support points of the D-optimal design on are
6. Results
- The function is first concave increasing then convex increasing;
- The function is increasing (this also guarantees that is closed convex). In some cases, the ratio is also increasing. (Note: shows that induced design space and the widest possible design space.)
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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1- Binary Weight Functions: Group I , | ||||
Name | ||||
(widest possible design space) | ||||
Logistic | ||||
Skewed Logistic | ||||
Generalised | ||||
Complementary log-log | ||||
Probit | ||||
2-Binary Weight Functions: Group II , | ||||
Double reciprocal | ||||
Double exponential | ||||
3-Density Weight functions: Group III | ||||
Beta | ||||
Gamma | ||||
Normal | ||||
4-Other Weight functions: Group IV | ||||
Name | ||
---|---|---|
Logistic | ||
Skewed logistic | ||
Generalised | ||
Complementary log-log | ||
Probit |
Name | ||||
Beta | ||||
Beta | ||||
Gamma | ||||
Normal |
Name | ||||
Logistic | ||||
Skewed logistic | ||||
Generalised | ||||
Complementary l. | ||||
Probit |
Name | |||||
Beta | z | ||||
Beta | |||||
Gamma | |||||
Normal |
, | ↓ | ↓ | ||||
, | ↑ | ↑ | ||||
, | ↑ | ↑ | ||||
↑ | ↑ | |||||
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Gündüz, N.; Torsney, B. D-Optimal Designs for Binary and Weighted Linear Regression Models: One Design Variable. Mathematics 2023, 11, 2075. https://doi.org/10.3390/math11092075
Gündüz N, Torsney B. D-Optimal Designs for Binary and Weighted Linear Regression Models: One Design Variable. Mathematics. 2023; 11(9):2075. https://doi.org/10.3390/math11092075
Chicago/Turabian StyleGündüz, Necla, and Bernard Torsney. 2023. "D-Optimal Designs for Binary and Weighted Linear Regression Models: One Design Variable" Mathematics 11, no. 9: 2075. https://doi.org/10.3390/math11092075
APA StyleGündüz, N., & Torsney, B. (2023). D-Optimal Designs for Binary and Weighted Linear Regression Models: One Design Variable. Mathematics, 11(9), 2075. https://doi.org/10.3390/math11092075