Partially Observed Functional Data with Measurement Error

3 Estimation Methods

3.2 Partially Observed Functional Data with Measurement Error

Letθ_n = #_∞

k=m[E[Y U_k]]²/λ²_k. Then assumption (A6) indicates thatθ_n → 0.

Denote υ = #m

j=1V_ij withV_ij = φ_{j Mi}, (C_MiMi −C_MiOiC_O⁻¹

iO_iC_OiMi)φ_{j Mi}. Based on the above assumptions, Theorem 1 gives the converge rate for the estimatorγˆ^NMEin theL²sense.

Theorem 1 Suppose that (A1)–(A7) are satisfied. Then

ˆγ^{N ME}−γ ²=Op(n⁻¹mρ⁻²+ιn+θn+υ), withιn=n⁻¹#m

j=1δ⁻_j².

Theorem1indicates that the approximation error rate ofγˆ^NMEforγis controlled by four terms. The first term depends on sample sizen, tuning parameterm, ridge parameterρ, which is of the higher order than the one given in Hall and Horowitz (2007) that is mainly due to functional curves observed on the part of the domain.

The second term is related to the spacings between adjacent eigenvalues, and its effect on convergence rate ofγ is also emphasized in Hall and Horowitz (2007).

The third term is related to the convergence ofγ inL²sense, which is also shown in Yao et al. (2005b) to get approximation error rate for functional coefficient. The fourth term is introduced by approximatingU_{ij Mi} withU˜_{ij Mi}.

Note that in practice, the ridge parameterρincluded in the regularized estimation of thejth score of theith functional observation is chosen by generalized cross- validation based on the set of samples observed on the entire domain (see Kraus 2015).

3.2 Partially Observed Functional Data with Measurement

Step 1: Estimate the mean and covariance functions by local linear smoothers.

Step 2: Estimate eigenvalues {λ_j} and eigenfunctions {φ_j} by

Tcˆ^WME_X (s, t ) φˆ_j^WME(s)ds= ˆλ^WME_j φˆ_j^WME(t ).

Step 3: Estimate FPC scores {Uij}by principal component analysis via condi- tional expectation (PACE):U˜ij =E[Uij|Zi].

Step 4: Based on obtained estimatorsUˆ˜_ij andφˆ_j^WME, we get estimatorγ^WMEfor X_iOi observed with measurement error.

We first calculate estimators for the mean and the covariance function ofXin the scenario (9), denoted asμˆ^WMEandcˆ_X^WME, that are required to derive estimators for the FPC scoresU_ij =

(X_i(t )−μ(t ))φ_j(t )dt. For simplicity of presentation, we suppress notation on “WME” unless otherwise stated in this subsection.

Let K(·) be a nonnegative univariate kernel function that is assumed to be a symmetric probability density function (pdf) with compact support supp(K) = [−1,1], andh_μ,h_c be the bandwidths for obtaining estimators ofμ,c_X. Assume that the second derivatives ofμ, c_X on T,T², respectively, exist. We use local linear smoothers for the mean functionμ(Yao et al.2005a,b; Kneip and Liebl2020) defined asμ(t )ˆ = ˆβ0, where

(βˆ₀,βˆ₁)=argmin

β0,β1

n i=1

Ni

l=1

K t_il−t

h_μ

[Z_il−β₀−β₁(t−t_il)]². (10)

LetGˆ_ilk = (Z_il − ˆμ(t_il))(Z_ik − ˆμ(t_ik))be the raw covariance points. The local linear smoother for the covariance functionc_Xis defined ascˆ_X= ˆ˜β₀, where

(βˆ˜₀,βˆ˜₁,βˆ˜₂)=arg min

˜ β0,β˜1,β˜2

n i=1

1≤l,k≤Ni

K

t_il−t hc

K

t_ik−s hc

× [ ˆG_ilk− ˜β₀− ˜β₁(t_il−t )− ˜β₂(t_ik−s)]². (11) Similar to the technique introduced in Yao et al. (2005a), the points Gˆ_ill, l = 1· · ·, N_i are not included in (11). Let T1 = [inf{L_i ∈ T, i = 1,· · ·, n} +

|T|/4,sup{R_i ∈ T, i = 1,· · ·, n} − |T|/4]with|T|being the length ofT. The estimator ofσ_X² is defined asσˆ_X² ifσˆ_X² >0, otherwiseσˆ_X² =0 with

ˆ σ_X² =2

T1

(VˆX(t )− ˜G(t ))dt/|T|,

whereVˆX(t )is the local linear estimator using the points{ ˆGill},G(t )˜ is the estimate ˆ

cX(s, t ) restricted to s = t (Staniswalis and Lee 1998; Yao et al. 2005a). The estimators of{λj, φj}j≥1 are the corresponding solutions of the eigen-equations

Tcˆ_X(s, t )φˆ_j(s)ds= ˆλ_jφˆ_j(t ).

Based on the K-L expansion ofXi, model (9) can be rewritten as Z_il =μ(t_il)+^∞

j=1

U_ijφ_j(t_il)+ε_il, t_il ∈O_i, i=1· · ·, n, l =1· · · , N_i.

LetXi = (X_i(t_i1),· · · , X_i(t_iNi))^T,Zi = (Z_i1,· · · , Z_iNi)^T,μ_i = (μ(t_i1),· · ·, μ(t_iNi))^T, φ_ij = (φ_j(t_i1),· · ·, φ_j(t_iNi))^T. Assume that U_ij and ε_il are jointly Gaussian. Following Yao et al. (2005a), the best prediction ofU_ij of theith subject given the observations(Z_il, t_il), l=1,· · ·, N_iis obtained as

U˜_ij =λ_jφ_ij^TΣ_Z⁻¹

i (Zi−μ_i),

whereΣZ_i = cov(Zi,Zi)= cov(Xi,Xi)+σ_X²IN_i with identity matrixIN_i. That is, the(u, v)th element ofΣ_Zi is(Σ_Zi)_u,v = c_X(t_iu, t_iv)+σ_X²I_uv withI_uv = 1 ifu =v, and 0 otherwise. Then the estimator ofU_ij is given through substituting μ, λ_j, φ_jwithμ,ˆ λˆ_j,φˆ_j as

Uˆ_ij^WME= ˆλjφˆ_ij^TΣˆ_Z⁻¹

i (Zi − ˆμi), (12)

where the(u, v)th entry ofΣˆ_Zi is(Σˆ_Zi)_u,v = ˆc_X(t_iu, t_iv)+ ˆσ_X²I_uv. ReplacingUˆ_ij in (2) withUˆ_ij^WME, we then get the estimatorγˆ^WMEofγ from (3)

ˆ

γ^WME(t )= m j=1

ˆ γ_jφˆ_j,

whereγˆjis thejth entry ofγˆ withUˆ_ij^WMEin (2).

Next, we give some theoretical results forγˆ^WME(t ). We assume the following regularity conditions which are similar to the assumptions in Kneip and Liebl (2020), Yao et al. (2005b).

(B1) The observational points{t_il, l=1,· · · , N_i}givenO_i for theith subject are i.i.d. random variables with pdff_t|Oi(u) > 0 for allu ∈ O_i ⊆ T and zero else. For the marginal pdff_tof observation timest_ij,f_t(u) >0 for allu∈T. (B2) LetN =min{N_i, i =1,· · ·, n}.N #n^r with 0 < r <∞, wherea_n #b_n means that there exists a constant 0 < L < ∞ such that a_n/b_n → L as n→ ∞.

(B3) h_μ→0,h_c→0,nN h_μ→ ∞,nMh_c→ ∞asn→ ∞withM=N²−N. (B4) Kis a second order kernel with compact support[−1,1].

(B5) LetG_ilk =(Z_il−μ(t_il))(Z_ik−μ(t_ik)). Definef_Zt,f_{t t},f_{Gt t} as the joint pdf of(Z_il, t_il)onR×T,(t_il1, t_il2)onT²,(G_ilk, t_il, t_ik)onR×T², respectively.

All of the second derivatives off_Zt,f_{t t},f_{Gt t} are uniformly continuous and bounded. Moreover,f_tis uniformly continuous and bounded onT.

(B6) LetΛ=diag{λ₁,· · ·, λ_m},Ξ =(λ₁φ_i1,· · ·, λ_mφ_im)^T,Υ =Λ−Ξ Σ_Z⁻¹

i Ξ^T

and ς_n ≡ trace(Υ ). Denoter_μ = h²_μ +1/9

nN h_μ +1/√

n, r_c = h²_c + 1/9

nMh²_c+1/√

n.υ_n≡mr_μ→0,τ_n≡r_c(#_m

j=1δ⁻_j¹)→0.

Theorem 2 Under the regularity conditions (A3), (A6), (B1)–(B6), we have that

ˆγWME−γ ²=O_p(υ_n+τ_n+ς_n+θ_n).

Theorem2gives the rate of convergence of the estimatorγˆ^WMEin theL²sense.

The rate of convergence ofγˆ^WMEdepends on the sample size and bandwidths which is common for estimating curves or surface by local linear smoothers for functional data analysis (see Li and Hsing2010). Related results of Theorem2 can also be found in Yao et al. (2005b). The terms υ_n, τ_n are related to rates of convergence of estimators for the mean and covariance function by using local linear smoothers.

The termς_nis introduced by approximatingU_ij withU˜_ij.