Download Appendix B: Ratio Constancy Proof for F

Appendix B:
Ratio Constancy Proof for F-Optimal Designs under the Uniform
Prior
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Ratio Constancy of F-Optimal Designs under the Uniform Prior
The F-optimal ratio constancy proof is very much like the D-optimal case. When β 1 ~ U(c, d ) , the
Bayes risk is

1
min ∫ R(δ , β )π (β )dβ = min 
2
δ ∈'
 ( d − c)( x 1 − x 2 )
 (e − cx 2 − e − dx 2 ) (e − cx 2 − e − dx2 ) 
  .

+
n2 x2
n1 x1
 

(1)
When β 1 ~ U(ac, ad ) , the Bayes risk is

1
min ∫ R(δ , β )π (β )dβ = min 
δ ∈'
 ( ad − ac)( x 1* − x 1* ) 2

(
) (
*
*
 e − cx1* − e − dx1*
e − cx1 − e − dx1

+

n 2 x 1*
n 1 x 1*

)  .
(2)


Let x 1 and x2 be the solution that minimizes expression (1) when β 1 ~ U(c, d ) . Let x 1* and x *2 be
the solution that minimizes expression (2) when β 1 ~ U(ac, ad ) . Multiplying the right hand side of
a2
(2) by 2 gives
a

1
min ∫ R(δ , β )π (β )dβ = min 
δ ∈'
 (d − c)( ax 1* − ax *2 ) 2

By grouping (ax *i ) , it is obvious that x *i =
(
) (
*
*
 e − cx*2 − e − dx*2
e − cx1 − e − dx1

+

n 2 ax *2
n 1ax 1*

)  .
(3)


xi
and these designs are ratio constant both in the
a
selection of EC’s. The following steps verify that equality of ECs holds among F-optimal Bayesian
designs based on the uniform prior.
 c+d 
 xi

2 
q i = e
=e
 c+ d  *
a
x
 2  i
 c + d   xi 

 a 
2   a
= e
 c+d 
 xi

2 
= e
= q *i
(4)
Note that while it is not shown here, it can be verified using calculus techniques that the allocation
percentages remain the same for designs based on β 1 ~ U(c, d ) and β 1 ~ U(ac, ad ) .
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