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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 β c 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 NT θ = (βT , gT )T , Ei = (0, . . . , In , 0, . . . , 0)T , • Testing statistics of outlying subject Ti = ˆeT Ei(In − ¯ CD[i](β) = RTi Hβ,iRi Hβ,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 β Fw(β) = DT (ˆe)WxX(XT WxX)−1XT WxD(ˆe) Partial influence matrix for nonparametric components g Fw(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)−1XW Fr(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) + ij ρ(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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