Kraus (2015), Proposition 1) - 5 Real Data Analysis

5 Real Data Analysis

Lemma 1 Kraus (2015), Proposition 1)

(a) Let E X ²<∞and assumption (A1) be satisfied. Then E(|| ˆμNME−μ||²)= O(n⁻¹)forn→ ∞.

(b) Let E X ⁴ < ∞ and observation pattern (A2) holds. Then E(|| ˆCNME

X −

C_X||²_S)=O(n⁻¹)forn→ ∞(here|| · ||Sdenotes the Hilbert–Schmidt norm).

Lemma 2 (Kneip and Liebl (2020), Theorem 4.1) Under the assumptions (B1)–

(B5), we have that

(a) sup_t_∈T| ˆμWME(t )−μ(t )| =O_p(r_μ)withr_μ=h²_μ+1/9

nN h_μ+1/√ n.

(b) sup_{(s,t )}_∈T2|ˆcWME(s, t )−c_X(s, t )| =O_p(r_μ+r_c)withr_c =h²_c+1/9

nMh²_c+ 1/√

Proof of Theorem1The following results can be derived from the theory developed by Bhatia et al. (1983):

sup_j_≥₁|ˆλNME

j −λj| ≤ ˆCNME

X −CX ,

sup_j_≥₁δj ˆφjNME

−φj ≤8^1/2 ˆCNME

X −CX . (13)

Therefore, we obtain from Lemma1, sup_j_≥₁|ˆλNME

j −λ_j| =O_p(n⁻^1/2), sup_j_≥₁δj ˆφNME

j −φj =Op(n⁻^1/2). (14)

Note that,

T(γˆNME_(s)₋_{γ (s))}²_ds

⎧⎨

⎩

m−1

j=1

⎡

⎣n⁻¹#n

i=1[YiUˆNME

ij ]

ˆ λNME

φˆNME

j (s)−E[Y Uj] λj

φj(s)

⎤

⎦

⎫⎬

⎭

⎧⎨

⎩ ∞ j=m

E[Y U_j] λ_j φ_j(s)

⎫⎬

⎭

⎧⎨

⎩

m−1

j=1

⎡

⎣n⁻¹#n

i=1[Y_iUˆNME

ij ]

ˆ λNME

ˆ φNME

j (s)−E[Y U_j] λ_j φ_j(s)

⎤

⎦

⎫⎬

⎭

⎧⎨

⎩ ∞ j=m

E[Y U_j] λ_j φ_j(s)

⎫⎬

⎭ds

:=A₁(n)+A₂(n)+A₃(n). (15)

For simplicity, we suppress the notation on “NME.” Assumption (A6) implies thatA₂(n) → 0 asm→ ∞. ForA₃(n), Cauchy–Schwarz inequality implies that A²₃(n) ≤A²₁(n)×A²₂(n)→^p 0. Combing the result (14), and the formula (15), we see that the result of the theorem follows if we can get the convergence rate ofUˆ_ij of the trajectories per subject withUˆ_ij = ˆU_{ij O}_i+ ˆU_{ij M}^(α)

Denote the estimates of U_{ij M}_i, C_O_i_O_i, C_O_i_M_i, φ_{j M}_i as Uˆ_{ij M}_i₍₋_i), Cˆ_O_i_O_i₍₋_i), Cˆ_O_i_M_i₍₋_i),φˆ_{j M}_i₍₋_i) with deleting the ith curves X_i(t ). Let ξ˜_{ij M}^(ρ)

i = (C^(ρ)_O

iO_i)⁻¹ C_O_i_M_iφ_{j M}_i withC_O^(ρ)

iO_i =C_O_i_O_i+ρFOi,U˜_{ij M}^(ρ)

i = ˜ξ_{ij M}^(ρ)

i, X_iO_i, and the notation ξ˜_{ij M}_iU˜_{ij M}_i are corresponded to the symbolsξ˜_{ij M}^(ρ)

i,U˜_{ij M}^(ρ)

i withρ=0. Since EUˆ_{ij M}^(ρ)

i − ˜U_{ij M}_i²= EUˆ_{ij M}^(ρ)

i− ˜U_{ij M}^(ρ)

i+ ˜U_{ij M}^(ρ)

i− ˜U_{ij M}_i²

= 2EUˆ_{ij M}^(ρ)

i− ˜U_{ij M}^(ρ)

²+2U˜_{ij M}^(ρ)

i− ˜U_{ij M}_i²

≤ 4EUˆ_{ij M}^(ρ)

i− ˆU_{ij M}^(ρ)

i(−i)

²+4EUˆ_{ij M}^(ρ)

i(−i)− ˜U_{ij M}^(ρ)

² +2U˜_{ij M}^(ρ)

i− ˜U_{ij M}_i², (16)

we then analyze the terms E ˆU_{ij M}^(ρ)

i− ˆU_{ij M}^(ρ)

i(−i) 2, E ˆU_{ij M}^(ρ)

i(−i)− ˜U_{ij M}^(ρ)

2, ˜U_{ij M}^(ρ)

i−

U˜_{ij M}_i ²in turn. Letξˆ_{ij M}^(ρ)

i(−i)=(Cˆ_O^(ρ)

iOi(−i))⁻¹Cˆ_O_i_M_i₍₋_i)φˆ_{j M}_i₍₋_i). Then E ˆU_{ij M}^(ρ)

i(−i)− ˜U_{ij M}^(ρ)

= Eˆξ_{ij M}^(ρ)

i(−i)− ˜ξ_{ij M}^(ρ)

i, X_iO_i²

= E{E[ˆξ_{ij M}^(ρ)

i(−i)− ˜ξ_{ij M}^(ρ)

i, XiO_i²|{XkO_i, k=i}]}

= E||C_O^1/2

iOi((Cˆ_O^(ρ)

iO_i(−i))⁻¹Cˆ_O_i_M_i₍₋_i)φˆ_{j M}_i₍₋_i)−(C_O^(ρ)

iOi)⁻¹C_O_i_M_iφ_{j M}_i)||²

≤ 4

E||C_O^1/2

iOi(Cˆ_O^(ρ)

iOi(−i))⁻¹(Cˆ_O_i_M_i₍₋_i)−C_O_i_M_i)(φˆ_{j M}_i₍₋_i)−φ_{j M}_i)||² +E||C_O^1/2

iO_i(Cˆ^(ρ)_O

iO_i(−i))⁻¹C_O_i_M_i(φˆ_{j M}_i₍₋_i)−φ_{j M}_i)||² +E||C_O^1/2

iO_i(Cˆ^(ρ)_O

iO_i(−i))⁻¹(Cˆ_O_i_M_i₍₋_i)−CO_iM_i)φj M_i||² +E||C_O^1/2

iOi((Cˆ_O^(ρ)

iOi(−i))⁻¹−(C_O^(ρ)

iOi)⁻¹)C_O_i_M_iφ_{j M}_i||²

:=B₁+B₂+B₃+B₄. (17)

LetFm= {^λ₂^m <λˆ_m< ³₂λ_m}. Suppose the eventFmholds. Otherwise, we have P(|ˆλm−λm| ≥ ^λ₂^m)≤P( ˆCNME

X −CX ≥ ^λ₂^m)→ 0 from assumption (A4). We have the following results for termsB1toB4with the equality

Cˆ_O^(ρ)

iO_i(−i)

₋1

− C_O^(ρ)

iO_i

₋1

=(Cˆ_O_i_O_i₍₋_i)−C_O_i_O_i)

C_O^(ρ)

iO_i

₋1 Cˆ_O^(ρ)

iO_i(−i)

₋1

. For the termB₁,

B₁≤E1 C_O^1/2

iOi

²₂·

ˆ C_O^(ρ)

iOi(−i)

₋₁ ²

∞·Cˆ_O_i_M_i_(−i)−C_O_i_M_i²₂·φˆ_{j M}_i_(−i)−φ_{j M}_i² 2

=O n⁻²δ⁻_j²

·O(ρ⁻²).

Denote · _∞as the operator norm. For the termB2, under the assumption (A7), E C_O^1/2

iO_i 2

∞<∞and the result (14), it is clear that B₂≤E

1C_O^1/2

iOi

∞·

Cˆ_O^(ρ)

iOi(−i)

₋1

C_O_i_M_i ²

·φˆ_{j M}_i₍₋_i)−φ_{j M}_i² 2

≤

r_M²

iOij k

(λO_iO_ik+ρ)²·O

n⁻¹δ_j⁻² =O

n⁻¹δ⁻_j²

For the termB₃, B₃≤E

3 C_O^1/2

iO_i 2 2· (Cˆ_O^(ρ)

iO_i(−i))⁻¹ ²_∞· ˆC_O_i_M_i₍₋_i)−C_O_i_M_i ²₂· φ_{j M}_i ² 4

=O(n⁻¹ρ⁻²).

Note that _(λ^ρλ^{Oi Oi k}

Oi Oi k+ρ)² <1. Under the assumption (A7), we have that

B₄≤E3 C_O^1/2

iOi·(C_O^(ρ)

iOi)⁻¹·(Cˆ^(ρ)_O

iOi(−i))⁻¹·C_O_i_M_i ²₂· ˆC_O_i_O_i₍₋_i)−C_O_i_O_i ²₂· φ_{j M}_i ²4

≤

⎧⎨

⎩

ρλ_O_i_O_i_k

(λ_O_i_O_i_k+ρ)²· r_O_i_M_i_{j k}² (λ_O_i_O_i_k+ρ)²·ρ⁻¹

⎫⎬

⎭·O(n⁻¹)

=O(n⁻¹)·O(ρ⁻¹).

These results combined with (17) indicate E ˆU_{ij M}^(ρ)

i(−i)− ˜U_{ij M}^(ρ)

2=O

n⁻¹ρ⁻²+n⁻¹δ⁻_j²

. (18)

We then analyze E ˆU_{ij M}^(ρ)

i− ˆU_{ij M}^(ρ)

i(−i) 2, E ˆU_{ij M}^(ρ)

i− ˆU_{ij M}^(ρ)

i(−i) = Eˆξ_{ij M}^(ρ)

i− ˆξ_{ij M}^(ρ)

i(−i), X_iO_i

≤ {E ˆξ_{ij M}^(ρ)

i− ˆξ_{ij M}^(ρ)

i(−i)

2}^1/2{E X_iO_i ²}^1/2

≤ L{E ˆξ_{ij M}^(ρ)

i− ˆξ_{ij M}^(ρ)

i(−i)

2}^1/2, (19)

where the last inequality holds from the finite second moment ofXthat is bounded by constantL. We also have,

E ˆξ_{ij M}^(ρ)

i− ˆξ_{ij M}^(ρ)

i(−i)

2= E (Cˆ_O^(ρ)

iO_i)⁻¹Cˆ_O_i_M_i−(Cˆ^(ρ)_O

iOi(−i))⁻¹Cˆ_O_i_M_i₍₋_i)

φˆ_{j M}_i₍₋_i) ²

= E 3

(Cˆ^(ρ)_O

iO_i)⁻¹−(Cˆ_O^(ρ)

iO_i(−i))⁻¹ Cˆ_O_i_M_i

+(Cˆ^(ρ)_O

iOi(−i))⁻¹(Cˆ_O_i_M_i− ˆC_O_i_M_i₍₋_i))

4φˆ_{j M}_i₍₋_i) ²

≤ 2 E

(Cˆ^(ρ)_O

iO_i)⁻¹−(Cˆ_O^(ρ)

iOi(−i))⁻¹

Cˆ_O_i_M_i ²

+E (Cˆ_O^(ρ)

iOi(−i))⁻¹(Cˆ_O_i_M_i− ˆC_O_i_M_i₍₋_i)) ²}. (20) Note that

E ˆCO_iM_i− ˆC_O_i_M_i₍₋_i) ²=O(n⁻²),

E (Cˆ_O^(ρ)

iOi)⁻¹−(Cˆ_O^(ρ)

iO_i(−i))⁻¹

CˆO_iM_i 2=O(n⁻²),

E (Cˆ_O^(ρ)

iO_i(−i))⁻¹(CˆO_iM_i− ˆCO_iM_i(−i)) ²=O(n⁻²ρ⁻²).

Combining formulas (19) and (20), we deduce that

E ˆU_{ij M}^(ρ)

i− ˆU_{ij M}^(ρ)

i(−i)

2=O(n⁻²ρ⁻²). (21)

On the other hand,

E ˜U_{ij M}^(ρ)

i− ˜U_{ij M}_i ²=O(ρ), (22)

var(U˜_{ij M}_i−U_{ij M}_i)= φ_{j M}_i, C_M_i_M_iφ_{j M}_i − φ_{j M}_i, C_M_i_O_iC_O⁻¹

iOiC_O_i_M_iφ_{j M}_i

:=V_ij. (23)

Therefore, withnρ³→0 and the formulas (16), (18), (21)–(23), we have that E ˆU_{ij M}^(ρ)

i−U_{ij M}_i ²=O

n⁻¹ρ⁻²+n⁻¹δ_j⁻²+V_ij

. Then the results are proved withnρ³→0.

Proof of Theorem 2 Let U˜i = (U˜i1,· · · ,U˜im)^T, Ui = (Ui1,· · · , Uim)^T. The covariance matrix of U˜i is var(Ui) = ΞΣ⁻_Z¹

iΞ^T with Ξ = cov(U˜i,Zi) =

(λ₁φ_i1,· · ·, λ_mφ_im)^T. Moreover, var(U˜i−Ui)=Λ−ΞΣZ_iΞ^T. Combining these results with formulas (14), (12) and the results of Lemma2, the result of Theorem 3 is obtained by replacingUˆ_ij^NMEwithUˆ_ij^WMEin (15) with assumptions (B1)–(B6).

References

Besse, P., & Ramsay, J. O. (1986). Principal components analysis of sampled functions.Psychome- trika, 51(2), 285–311.

Bhatia, R., Davis, C., & McIntosh, A. (1983). Perturbation of spectral subspaces and solution of linear operator equations.Linear Algebra and its Applications, 52, 45–67.

Cardot, H., Ferraty, F., & Sarda, P. (1999). Functional linear model.Statistics & Probability Letters, 45(1), 11–22.

Che, M., Kong, L., Bell, R. C., & Yuan, Y. (2017). Trajectory modeling of gestational weight: A functional principal component analysis approach.PloS One, 12(10), e0186761.

Crambes, C., Kneip, A., & Sarda, P. (2009). Smoothing splines estimators for functional linear regression.The Annals of Statistics, 37(1), 35–72.

Delaigle, A., & Hall, P. (2016). Approximating fragmented functional data by segments of Markov chains.Biometrika, 103(4), 779–799.

Goldberg, Y., Ritov, Y., & Mandelbaum, A. (2014). Predicting the continuation of a function with applications to call center data.Journal of Statistical Planning and Inference, 147, 53–65.

Hall, P., & Horowitz, J. L. (2007). Methodology and convergence rates for functional linear regression.The Annals of Statistics, 35(1), 70–91.

Hall, P., Müller, H.-G., & Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis.The Annals of Statistics, 34(3), 1493–1517.

Hansen, P. C. (1990). The discrete Picard condition for discrete ill-posed problems.BIT Numerical Mathematics, 30(4), 658–672.

Horváth, L., & Kokoszka, P. (2012).Inference for functional data with applications, vol. 200.

Berlin: Springer.

James, G. M., Hastie, T. J., & Sugar, C. A. (2000). Principal component models for sparse functional data.Biometrika, 87(3), 587–602.

Kneip, A., & Liebl, D. (2020). On the optimal reconstruction of partially observed functional data.

Annals of Statistics, 48(3), 1692–1717.

Kraus, D. (2015). Components and completion of partially observed functional data.Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(4), 777–801.

Li, Y., & Hsing, T. (2010). Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data.The Annals of Statistics, 38(6), 3321–3351.

Liebl, D. (2013). Modeling and forecasting electricity spot prices: A functional data perspective.

The Annals of Applied Statistics, 7(3), 1562–1592.

Liebl, D., & Rameseder, S. (2019). Partially observed functional data: The case of systematically missing parts.Computational Statistics and Data Analysis, 131, 104–115.

Marx, B. D. & Eilers, P. H. (1999). Generalized linear regression on sampled signals and curves: a p-spline approach.Technometrics, 41(1), 1–13.

Morris, J. S. (2015). Functional regression.Annual Review of Statistics and Its Application, 2, 321–359.

Ramsay, J. (2005). Functional data analysis. In B. S. Everitt & D. C. Howell (Eds.)Encyclopedia of Statistics in Behavioral Science(Vol. 2. pp. 675–678). Chichester: John Wiley & Sons Ltd.

Ramsay, J. O., & Dalzell, C. (1991). Some tools for functional data analysis.Journal of the Royal Statistical Society: Series B (Methodological), 53(3), 539–561.

Reiss, P. T., Goldsmith, J., Shang, H. L., & Ogden, R. T. (2017). Methods for scalar-on-function regression.International Statistical Review, 85(2), 228–249.

Rice, J. A., & Silverman, B. W. (1991). Estimating the mean and covariance structure non- parametrically when the data are curves.Journal of the Royal Statistical Society: Series B (Methodological), 53(1), 233–243.

Riesz, F., & Nagy, S. (1955). B.(1990). functional analysis.Dover Publications, Inc., New York.

First published in, 3(6), 35.

Shang, H. L. (2014). A survey of functional principal component analysis.AStA Advances in Statistical Analysis, 98(2), 121–142.

Shin, H. (2009). Partial functional linear regression.Journal of Statistical Planning and Inference, 139(10), 3405–3418.

Staniswalis, J. G., & Lee, J. J. (1998). Nonparametric regression analysis of longitudinal data.

Journal of the American Statistical Association, 93(444), 1403–1418.

Wang, Y., Kong, L., Jiang, B., Zhou, X., Yu, S., Zhang, L., & Heo, G. (2019). Wavelet-based lasso in functional linear quantile regression.Journal of Statistical Computation and Simulation, 89(6), 1111–1130.

Yao, F., Müller, H.-G., & Wang, J.-L. (2005a). Functional data analysis for sparse longitudinal data.Journal of the American Statistical Association, 100(470), 577–590.

Yao, F., Müller, H.-G., & Wang, J.-L. (2005b). Functional linear regression analysis for longitudinal data.The Annals of Statistics, 33(6), 2873–2903.

Yu, D., Kong, L., & Mizera, I. (2016). Partial functional linear quantile regression for neuroimaging data analysis.Neurocomputing, 195, 74–87.

Zhao, Y., Ogden, R. T., & Reiss, P. T. (2012). Wavelet-based lasso in functional linear regression.

Journal of computational and graphical statistics, 21(3), 600–617.

Zhu, H., Li, R., & Kong, L. (2012). Multivariate varying coefficient model for functional responses.

Annals of Statistics, 40(5), 2634–2666.

Semiparametric Varying-Coefficient

Dalam dokumen (ICSA Book Series in Statistics) Wenqing He, Liqun Wang, Jiahua Chen, Chunfang Devon Lin - Advances and Innovations in Statistics and Data Science-Springer (2022) (Halaman 165-171)