Robust Inference in Semiparametric Models
East China Normal University, Shanghai, China
• Semiparametric Models for Longitudinal Data
• Influence Diagnostics of Semiparametric Models
1. Case Deletion and Subject Deletion Analysis
• Robust Estimation of Semiparametric Models
{(xij, tij, yij), i = 1, . . . , m, j = 1, . . . , ni}.yij = XTijβ0 + g(tij) + eij,• β0 ∈ Rp , g is a smooth function ,
• eij are random errors, and are independent between subjects
eij = ZTijbi + Ui(tij) + ij, n =
• ni = 1: usually partialy linear model
• In dependent data ni = 1
1. Speckman (1988),Hardle, Liang and Gao (2000):kernel
2. Heckman (1986), He and Shi (1996), Eubank (1999):spline
• Longitudinal data and mixed models
1. Zeger and Diggle (1994): application to CD4
2. Moyeed and Diggle (1994): convergence rates
3. Zhang, Lin et al (1998), Lin and Zhang (1999), Ruppert,
Wand and Carroll(2003): penalized likelihood
4. Lin and Carroll (2000,2001a,b,2002):GEE
1. Cook and Wesiberg (1982), Wei (1998), Banerjee and Frees
(1997), Fung, Zhu, Wei and He (2002):Case Deletion
2. Cook (1986), Lu, Ko and Chang (1997), Lesaffre and
Verbeke (1998), Zhu, He and He (2003):Local Influence
Influence Diagnostics of Semiparametric Model
L(β, g) = l(β, g) −g (t)dt = l(β, g) − gT Kg
l(β, g) = − log |V | − (Y − Xβ − N g)T V −1(Y − Xβ − N g),• Penalized likelihood estimation
H = I − ΣV −1 + ΣV −1 ¯
g N + λK)−1N T Wg dcdT• Cook distance of parameter βc WxX (X T WxX )−1X T Wxdc• Partial influence for nonparametric part g
DFITij = |dTc N(ˆg(ij) − ˆg)|/sij
where s2ij is the cth diagonal element of N(0, Ir)C−1(0, Ir)T NTθ = (βT , gT )T , Ei = (0, . . . , In , 0, . . . , 0)T ,• Testing statistics of outlying subject
Ti = ˆeT Ei(In − ¯
CD[i](β) = RTi Hβ,iRiHβ,i = ETi WxX(XT WxX)−1XT WxEi
CD[i](g) = RTi WgNS−1NTi (NiS−1NTi )−1NiS−1NT WgRi• Perturbation penalized likelihood L(θ, ω)
θω is the estimate of θ under perturbation
F = ∂2L(ˆθω)
Fw = DT (ˆe)V −1 ¯
e1, · · · , ˆem)
Partial influence matrix for parametric components βFw(β) = DT (ˆe)WxX(XT WxX)−1XT WxD(ˆe)
Partial influence matrix for nonparametric components gFw(g) = DT (ˆe)WgN(NT WgN + λK)−1NT WgD(ˆe)
Fe = DT (ΣV −1ˆe)V −1 ¯
Fe(g) = DT (ΣV −1ˆe)WgNS−1NT WgD(ΣV −1ˆe)
Fe(β) = DT (ΣV −1ˆe)WxX(XT WxX)−1XT WxD(ΣV −1ˆe)
e)WgNS−1NT WgD(ˆe)
e)WxX(XT WxX)−1XT WxD(ˆe)
H1)11, · · · , 1Tm(V −1
Fr(β) = 1TWxX(XT W X)−1XWFr(g) = 1T WgX(XT W X)−1XT W• Data source Zhang, Lin et al (1998)
• 34 women in one menstrual cycle, y =log progesterone level,
x = age, BMI, t = days within one cycle
yij = β1AGEi + β2BMIi + g(tij) + bi + Ui(tij) + ijρ(yij − xTijβ − g(tij)) = minimum• Choice of ρ depends on interest
2. median ρ(r) = |r|ρ(r) = (τ I(r > 0) + (1 − τ )I(r < 0))|r|• Choice of smoothing method for g: kernel; local polynomial;
π(t) = (B1(t), . . . , BN (t))T
Order of polynomials l + 1; Knots: 0 = s0 < s1 · · · < sk = 1
sup |g(t) − αT π(t)| = O(k−r)
where r is an order of smoothness of g. • Advantages: Local smoothing but global representation, Good
ρ(yij − xTijβ − π(tij)T α) = minimum
Let θ = (βT , αT )T and zTij = (xTij, π(tTij))T . The problem is reducedto a linear model problem. ψ(yij − zTijθ)zij = 0
2. ψ is not everywhere differentiable: Hunter and Lange (2000)
• View as a model selection problem
ij θ. This is a useful information-type
criterion for N within a reasonable range
• If k = kn → ∞ and k/n → 0, the estimate of (βT , g) is
• If kn ≈ n1/(2r+1), we have
ij ) − g(tij ))2 = Op(n−2r/(2r+1)),√n(ˆβ − β) → N(0,A−1BA−1)
• A and B can be estimated consistently
yij = xijβ + cos(πtij) + wi(tij) + ij• tij random sample from U(0, 1), xij = 5t2ij + N(0, 0.5)
• wi(t) stationary Gaussian process with γ(t) = 0.4 exp(−η|t|) for
• ZD-estimator as in Moyeed and Diggle (1994): uses kernel
• Estimates of βj(j = 1, 2):
• Non-significance of AGE and BMI in both mean and median

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