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DS-optimal designs for RCR models with heteroscedastic errors
Mateusz Wilk1,2, Aleksander Zaigrajew1
1WMiI UMK, 2IP PAN
DS-optimal designs for RCR models with heteroscedastic errors - - PowerPoint PPT Presentation
DS-optimal designs for RCR models with heteroscedastic errors - - PowerPoint PPT Presentation
DS-optimal designs for RCR models with heteroscedastic errors Mateusz Wilk 1 , 2 , Aleksander Zaigrajew 1 1 WMiI UMK, 2 IP PAN 28 11 2016 Model y = 1 + 2 x + e ( x ) , x [ a , b ] R = [ 1 2 ] T E [ ] = 0 cov [ ] =
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Estimator
ˆ β0 =
- F TV −1F
−1 F TV −1Y F = 1 x1 1 x2 . . . 1 xn Y = y1 y2 . . . yn V = cov[Y ]
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Design
ζ ∼ x(1) x(2) . . . x(k) ω1 ω2 . . . ωk
- ω1, ω2, . . . , ωk ∈ (0, 1),
k
- i=1
ωi = 1 x(1), x(2), . . . , x(k) are distinct
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Information matrix
M0(ζ) =
b
- a
[1 x]T[1 x]/g(x)dζ(x) M(ζ) = (M0(ζ)−1 + n∆)−1 M(ζ) ≈ 1 n
- cov[ ˆ
β0] −1 = 1 nF TV −1F
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Lemma 1. If ag(b) bg(a) and g(x) (b − x)g(a) + (x − a)g(b) b − a abg(x) bx(b − x)g(a) + ax(x − a)g(b) b − a
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- r if ag(b) bg(a) and
g(x) (b − x)g(a) + (x − a)g(b) b − a abg(x) bx(b − x)g(a) + ax(x − a)g(b) b − a
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then for any given k-point design ζ there exists 2-point design ζ∗ =
- a
b 1 − ω∗ ω∗
- such that M0(ζ∗) M0(ζ), and
M(ζ∗) M(ζ).
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- Example. If a, b > 0 then the function
g(x) = cx, c > 0, satisfies all conditions from Lemma 1.
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DS-optimality
A design ζ∗ is said to be DS-optimal for the BLUE of β0, if P
- β0 − ˆ
β0(ζ∗) < ǫ
- → max ∀ε > 0
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DS-optimality
◮ A design is DS-optimal if and only if it
is simultaneously D- and E- optimal, i.e. |M(ζ)| → max and λmin[M(ζ)] → max.
◮ If a design is DS-optimal, then it is also
A-optimal, i.e. tr M(ζ)−1 → min.
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Lemma 2. For the FCR model (∆ = 0) condition (1 + a2)/g(a) = (1 + b2)/g(b) is necessary and sufficient for the existence of the DS-optimal design ζ∗ among all 2-point designs with the support at endpoints. It has the form ζ∗ = a b 1/2 1/2
- .
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- Example. The function
g(x) = cx, c > 0, satisfies the above condition if and only if ab = 1.
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Lemma 3. For the RCR model (∆ = diag(δ1, δ2)) the DS-optimal design ζ∗ among all 2-point designs with the support at endpoints exists only for two cases: (i) a = −b and g(b) = g(−b), here ζ∗ = −b b 1/2 1/2
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(ii) a2 = b2 and n(δ1 − δ2) = (1 + a2)g(b) − (1 + b2)g(a) b2 − a2 , here ζ∗ =
- a
b 1 − ω∗ ω∗
- ,
where ω∗ :=
- (1 + a2)/g(a)
- (1 + a2)/g(a) +
- (1 + b2)/g(b)
.
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- Example. For the linear function
g(x) = cx, c > 0, (i) does not hold, while if a2 = b2 then the DS-optimal design exists if and only if n(δ1 − δ2) = c(1 − ab)/(a + b).
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Literature
◮ Cheng, J., Yue R.-X., Liu, X. (2013), Optimal
designs for random coefficient regression models with heteroscedastic errors, Communications in Statistics-Theory and Methods 42, 2798-2809.
◮ Liski, E.P., Luoma, A., Zaigraev, A. (1999),
Distance optimality design criterion in linear models, Metrika 49, 193-211.
◮ Liski, E.P., Mandal, N.K., Shah, K.R., Sinha,