Integral Least-Squares Inferences for Semiparametric Models with Functional Data

Abstract

The inferences for semiparametric models with functional data are investigated. We propose an integral least-squares technique for estimating the parametric components, and the asymptotic normality of the resulting integral least-squares estimator is studied. For the nonparametric components, a local integral least-squares estimation method is proposed, and the asymptotic normality of the resulting estimator is also established. Based on these results, the confidence intervals for the parametric component and the nonparametric component are constructed. At last, some simulation studies and a real data analysis are undertaken to assess the finite sample performance of the proposed estimation method.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Hui-Shen Shen

1, College of Mathematics and Statistics, Hechi University, Yizhou, Guangxi 546300, China

Received 13 February 2014; Revised 18 June 2014; Accepted 19 June 2014; 7 July 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In the recent literature, there has been increased interest in regression modeling for functional data, where both the predictor and response are random functions. Compared with the discrete multivariate analysis, functional data analysis can take into account the smoothness of the high dimensional covariates and can suggest some new approaches to the problems that have not been discovered before. Examples of functional data can be found in different application fields such as biomedicine, economics, and archaeology (see Ramsay and Silverman [1]). Furthermore, the statistical analysis for the regression model with functional data also has been considered by many authors. For example, Ramsay and Silverman [2] studied the linear regression model with functional data. Ait-Saïdi et al. [3] proposed a cross-validated estimation procedure for the single-functional index model. Ferraty et al. [4] and Chen et al. [5] considered the inferences for single and multiple index functional regression models by using the functional projection pursuit regression technology. In addition, Ferraty and Vieu [6] and Rachdi and Vieu [7] considered the nonparametric regression modeling for functional data. More works for the functional data analysis can be found in [8-10] and among others.

However, the linear functional model, which assumes that the model satisfies the linear relationship between the response and the covariates, may be too restrictive. Then, the semiparametric model with functional data is a useful extension of functional linear regression models and functional nonparametric regression models. More specifically, let X ( t ) , Z ( t ) , and Y ( t ) be continuous random functions of index t ; then, the semiparametric regression model with functional data has the following structure: [figure omitted; refer to PDF] where Y ( t ) is the response variable, X ( t ) is the p × 1 covariate vector, Z ( t ) is the q × 1 covariate vector, β = ( _{β 1} , ... , _{β q} ^{) T} is a vector of unknown parameters, θ ( t ) = ( _{θ 1} ( t ) , ... , _{θ p} ( t ) ^{) T} is a vector of unknown function of t , and ... ( t ) is a zero-mean stochastic process. Here, without loss of generality, we assume that index t ranges over a nondegenerate compact interval such as [ 0,1 ] .

Because the samples of response and covariate are functions of index t , the Euclidean distance cannot measure the distance between Y ( t ) and X ( t ) . Hence, the ordinary least-squares method cannot be implemented directly. Recently, the studies for such semiparametric regression model with functional data have been considered by many papers (see Aneiros-Pérez and Vieu [11], Shin [12], and Lian [13]). In this paper, we provide additional positive results of the inferences for semiparametric models with functional data and extend the application literature of the classical least-squares technology. More specifically, we propose an integral least-squares method for estimating the parametric components and the nonparametric component. Furthermore, the asymptotic normalities of the integral profile least-squares estimators are studied. Some simulation studies and a real data application imply that the proposed method is workable.

The rest of this paper is organized as follows. In Section 2, we introduce the integral least-squares based estimation procedure for the parametric components and the nonparametric components. The asymptotic distributions of these estimators are also derived under some regularity conditions. In Section 3, some simulations and a real data analysis are carried out to assess the performance of the proposed estimation method. The technical proofs of all asymptotic results are provided in the Appendix.

2. Estimation and Asymptotic Distributions

Suppose { _{X i} ( t ) , _{Z i} ( t ) , _{Y i} ( t ) ^{} i = 1 n} is a random sample of size n . From (1), we have that [figure omitted; refer to PDF] In this paper, we assume that, for given t , ( _{Y i} ( t ) , _{X i} ( t ) , _{Z i} ( t ) ) and _{... i} ( t ) are i.i.d. for different i and E { _{[straight epsilon] i} ( t ) |" _{X i} ( t ) , _{Z i} ( t ) } = 0 .

Now, we in turn model (2); for t in a small neighborhood of _{t 0} , _{θ k} ( t ) can be locally approximated by a linear function [figure omitted; refer to PDF] For given β , applying the integral least-squares method, we can get the weighted local integral least-squares estimator of { _{θ 1} ( t ) , ... , _{θ p} ( t ) } by minimizing [figure omitted; refer to PDF] where _{K h} ( · ) = ^{h - 1} K ( · / h ) , K ( · ) is a kernel function, h is a bandwidth, and _{X i k} ( s ) denote the k th component of _{X i} ( s ) . Let [figure omitted; refer to PDF] Then, the solution to (4) is given by [figure omitted; refer to PDF] where _{I p} is p × p identity matrix and _{0 p} is p × p zero matrix. Substituting (6) into (2), and by a simple calculation, we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Applying the integral least-squares technology to linear model (7), we can get the integral least-squares estimator of β , say β ^ , by minimizing [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] If the matrix Γ ^ is invertible, β ^ can be given by [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] where ^{A [ecedil]7; 2} = A ^{A T} , μ ( t ) = ^{Ψ - 1} ( t ) Φ ( t ) , Φ ( t ) = E { X ( t ) ^{Z T} ( t ) |" t } , and Ψ ( t ) = E { X ( t ) ^{X T} ( t ) |" t } . The following result states the asymptotic normality of β ^ .

Theorem 1.

Suppose that conditions ( C 1 ) - ( C 5 ) in the Appendix hold; then, one has [figure omitted; refer to PDF] where Σ = ^{Γ - 1} B ^{Γ - 1} .

In order to construct the confidence interval of β by Theorem 1, we give the estimator of Σ , say Σ ^ = ^{Γ ^ - 1} B ^ ( β ^ ) ^{Γ ^ - 1} , where Γ ^ is defined in (11) and [figure omitted; refer to PDF] Invoking || β ^ - β || = _{O p} ( ^{n - 1 / 2} ) , with the similar argument to Lemma A.6, we can prove that Σ ^ is a consistent estimator of Σ . Thus, by Theorem 1, we have [figure omitted; refer to PDF] where _{I q} is an identity matrix of order q . Therefore, the confidence region of β can be constructed by using (17).

Furthermore, substituting β ^ into (6), we can get the integral least-squares estimator of θ ( t ) as [figure omitted; refer to PDF] We state the asymptotic normality of θ ^ ( t ) in the following theorem.

Theorem 2.

Suppose that conditions ( C 1 ) - ( C 5 ) in the Appendix hold. For given _{t 0} , then one has [figure omitted; refer to PDF] where b ( _{t 0} ) = ( 1 / 2 ) ^{θ [variant prime][variant prime]} ( _{t 0} ) ^{h 2 ∫ 0 1} ... ^{t 2} K ( t ) d t , V ( _{t 0} ) = ^{Ψ - 1} ( _{t 0} ) ^{σ 2} ( _{t 0} ) ^{∫ 0 1} ... ^{K 2} ( t ) d t .

Let [figure omitted; refer to PDF] Using the law of large numbers, with the similar argument to Lemma A.2 and Proposition 4.1 of Xue and Zhu [14] and Lemma 1 of Wu et al. [15], it can be shown that ^{σ ^ 2} ( _{t 0} ) , Ψ ^ ( _{t 0} ) , and b ^ ( _{t 0} ) are consistent estimators of ^{σ 2} ( _{t 0} ) , Ψ ( _{t 0} ) , and b ( _{t 0} ) , respectively. Finally, we can obtain the estimator V ^ ( _{t 0} ) by substituting ^{Ψ - 1} ( _{t 0} ) and ^{σ 2} ( _{t 0} ) in V ( _{t 0} ) with ^{Ψ ^ - 1} ( _{t 0} ) and ^{σ ^ 2} ( _{t 0} ) , respectively. It can be shown that b ^ ( _{t 0} ) and V ^ ( _{t 0} ) are the consistent estimators of b ( _{t 0} ) and V ( _{t 0} ) , respectively. By Theorem 2, we have that [figure omitted; refer to PDF] where _{I p} is the unit matrix of order p .

Using (21), a pointwise confidence interval for _{θ r} ( _{t 0} ) can be given by [figure omitted; refer to PDF] where _{b ^ r} ( _{t 0} ) is the r th component of b ^ ( _{t 0} ) , _{ν ^ r} ( _{t 0} ) is the ( r , r ) th element of V ^ ( _{t 0} ) , and _{Z α / 2} is the 1 - α / 2 quantile value of the standard normal distribution.

3. Numerical Results

In this section, we conduct several simulation experiments to illustrate the finite sample performances of the proposed method and consider a real data set analysis for further illustration.

3.1. Simulation Studies

To evaluate the performance of the proposed method, we consider the following model: [figure omitted; refer to PDF] where β = 2.5 and θ ( t ) = sin ( 2 π t ) . To perform the simulation, we generated n = 30,50,100 samples, respectively. The covariates X ( t ) and Z ( t ) are generated according to the model [figure omitted; refer to PDF] where _{X 01} ~ U ( 5,7 ) , _{X 02} ~ N ( 0,1.5 ) , _{Z 01} ~ N ( 0,5 ) , and _{Z 02} ~ N ( 0,1 ) , respectively.

We use the Epanechnikov kernel function K ( t ) = 0.75 ( 1 - ^{t 2} _{) +} and use the cross-validation method to determine bandwidth h . Let _{θ ^ - i} ( · ) and _{β ^ - i} be the integral least-squares estimators of θ ( · ) and β , respectively, which are computed with all of the measurements but not the i th observation. Define the integral least-squares cross-validation function [figure omitted; refer to PDF] The cross-validation bandwidth is the one that minimizes (25); that is, [figure omitted; refer to PDF]

For the parametric component β , the average and standard deviation of the estimator β ^ , based on 1000 simulations, are reported in Table 1. In addition, the average length and coverage probability of the confidence interval ( _{β ^ L} , _{β ^ U} ) , with a nominal level 1 - α = 95 % , are computed with 1000 simulation runs. The results are also summarized in Table 1.

Table 1: The estimators and the 95 % confidence intervals for β .

n	β ^	SD	( β L , β U )	Length	Coverage
30	2.472	0.166	(2.326, 2.618)	0.292	0.937
50	2.491	0.136	(2.389, 2.594)	0.205	0.945
100	2.502	0.096	(2.441, 2.562)	0.121	0.951

For the nonparametric component θ ( t ) , the average pointwise confidence intervals, based on 1000 simulations, with a nominal level 1 - α = 95 % are presented in Figure 1, and the corresponding coverage probabilities are presented in Figure 2.

Figure 1: The pointwise confidence intervals for θ ( t ) .

[figure omitted; refer to PDF]

Figure 2: The corresponding coverage probabilities.

[figure omitted; refer to PDF]

Table 1 shows that, for the parametric component, our method can give a shorter confidence interval and the corresponding coverage probability is close to real nominal level. Figures 1 and 2 show that the average interval length decreases as the sample size increases, while the corresponding coverage probability increases. In addition, we can see that, for the nonparametric component, the proposed estimation method works well except for boundary points.

3.2. Application to Spectrometric Curves Data

In this section, we present an application of the proposed estimation method to spectrometric curves data. This original data comes from a quality control problem in the food industry. This data set concerns a sample of finely chopped meat, and each food sample contains finely chopped pure meat with different fat, protein, and moisture (water) contents. The sample size of this data set is n = 240 , and, for each food sample, the functional data consist of 100 channel spectrum of absorbances, which were recorded on the Tecator Infratec Food and Feed Analyzer working in the wavelength range 850-1050 nm by the near infrared transmission (NIT) principle. Because of the fineness of the grid, we can consider each subject as a continuous curve. Thus, each spectrometric analysis can be summarized by some continuous curves giving the observed absorbance as function of the wavelength. More details of the data can be found in Ferraty and Vieu [16].

The aim is to find the relationship between the percentage of fat content Y ( t ) and the corresponding percentages of protein content _{X 1} ( t ) , the moisture content _{X 2} ( t ) , and the spectrometric curve t . The results, obtained by Aneiros-Pérez and Vieu [11], indicate that there is a strong linear relationship between the fat content and the protein and moisture contents, but the spectrometric curve t has a functional effect on the fat content. Hence, we consider the following semiparametric model: [figure omitted; refer to PDF]

We computed the estimators of the parametric components _{β 1} and _{β 2} and the nonparametric component θ ( · ) by using the proposed integral least-squares method. The results for the parametric components are reported in Table 2, and the results for the nonparametric components are reported in Figure 3, where the solid curve is the estimator of θ ( · ) and the dashed curve is the pointwise confidence interval of θ ( · ) . From Table 2, we can see that there is a significant negative correlation relationship between the fat content and the protein and moisture contents. In addition, Figure 3 indicates that the baseline function θ ( · ) really varies over the spectrometric curve t . This finding basically agrees with what was discovered by Aneiros-Pérez and Vieu [11].

Table 2: The estimators and the 95 % confidence intervals for β .

β	Estimator	Confidence interval	Interval length
β 1	- 1.758	( - 1.907 , - 1.609 )	0.298
β 2	- 0.661	( - 0.743 , - 0.579 )	0.164

Figure 3: The estimator and 95 % pointwise confidence interval for θ ( t ) .

[figure omitted; refer to PDF]

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11101119), the Higher-Education Reform Project of Guangxi (Grant no. 2014JGA209), and the Training Program for Excellent Young Teachers in Guangxi Universities.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

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[2] J. O. Ramsay, B. W. Silveman Functional Data Analysis , Springer, New York, NY, USA, 1997.

[3] A. Ait-Saïdi, F. Ferraty, P. Vieu, "Cross-validated estimations in the single-functional index model," Statistics , vol. 42, no. 6, pp. 475-494, 2008.

[4] F. Ferraty, A. Goia, E. Salinelli, P. Vieu, "Functional projection pursuit regression," TEST , vol. 22, no. 2, pp. 293-320, 2013.

[5] D. Chen, P. Hall, H. Müller, "Single and multiple index functional regression models with nonparametric link," The Annals of Statistics , vol. 39, no. 3, pp. 1720-1747, 2011.

[6] F. Ferraty, P. Vieu, "Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination," Journal of Nonparametric Statistics , vol. 16, no. 1-2, pp. 111-125, 2004.

[7] M. Rachdi, P. Vieu, "Nonparametric regression for functional data: automatic smoothing parameter selection," Journal of Statistical Planning and Inference , vol. 137, no. 9, pp. 2784-2801, 2007.

[8] M. Escabias, A. Aguilera, M. Valderrama, "Functional PLS logit regression model," Computational Statistics & Data Analysis , vol. 51, no. 10, pp. 4891-4902, 2007.

[9] F. Yao, H. Müller, J. Wang, "Functional linear regression analysis for longitudinal data," The Annals of Statistics , vol. 33, no. 6, pp. 2873-2903, 2005.

[10] J. T. Zhang, J. Chen, "Statistical inferences for functional data," The Annals of Statistics , vol. 35, no. 3, pp. 1052-1079, 2007.

[11] G. Aneiros-Pérez, P. Vieu, "Semi-functional partial linear regression," Statistics and Probability Letters , vol. 76, no. 11, pp. 1102-1110, 2006.

[12] H. Shin, "Partial functional linear regression," Journal of Statistical Planning and Inference , vol. 139, no. 10, pp. 3405-3418, 2009.

[13] H. Lian, "Functional partial linear model," Journal of Nonparametric Statistics , vol. 23, no. 1, pp. 115-128, 2011.

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[15] C. O. Wu, C. T. Chiang, D. R. Hoover, "Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data," Journal of the American Statistical Association , vol. 93, no. 444, pp. 1388-1402, 1998.

[16] F. Ferraty, P. Vieu Nonparametric Functional Data Analysis , Springer, New York, NY, USA, 2006.

Appendix

Appendix

Proof of Theorems

For convenience and simplicity, let c denote a positive constant which may be different value at each appearance throughout this paper. Before we prove our main theorems, we list some regularity conditions which are used in this paper.

( C 1 ): The bandwidth satisfies h = c ^{n - 1 / 5} , for some constant c > 0 .

( C 2 ): The kernel K ( · ) is a symmetric probability density function, and ∫ ... ^{t 4} K ( t ) d t < ∞ .

( C 3 ): θ ( t ) , Φ ( t ) , and Ψ ( t ) are twice continuously differentiable on ( 0,1 ) .

( C 4 ): _{sup ... 0 ...4; t ...4; 1} E ( ... ( t ^{) 4} |" t ) < ∞ , _{sup ... 0 ...4; t ...4; 1} E ( _{X r} ( t ^{) 4} |" t ) < ∞ , and E ( ... ( t ^{) 4} |" t ) and E ( _{X r} ( t ^{) 4} |" t ) are continuous at t , r = 1 , ... , p , where _{X r} ( t ) is the r th component of X ( t ) .

( C 5 ): For given t , Ψ ( t ) is positive definite matrix.

The proofs of Theorems 1 and 2 rely on the following lemmas.

Lemma A.1.

Let _{a 1} , _{a 2} , ... , _{a n} and _{b 1} , _{b 2} , ... , _{b n} ( _{b 1} ...5; _{b 2} ...5; , ... , ...5; _{b n} ) be two sequences of real numbers. Let _{S k} = ^{∑ i = 1 k} ... _{a i} ; then, one has [figure omitted; refer to PDF]

Lemma A.2.

Let _{e i} , i = 1 , ... , n , be a sequence of multi-independent random variate with E ( _{e i} ) = 0 and E ( ^{e i 2} ) < c < ∞ . Then, one has [figure omitted; refer to PDF] Further, let ( _{j 1} , _{j 2} , ... , _{j n} ) be a permutation of ( 1,2 , ... , n ) . Then, we have [figure omitted; refer to PDF]

Lemma A.3.

Let ( _{X 1} , _{Y 1} ) , ... , ( _{X n} , _{Y n} ) be i.i.d. random vectors, where _{Y i} is scalar random variables. Further, assume that E | _{Y 1} ^{| s} < ∞ , _{sup ... x} ∫ ... | y ^{| s} f ( x , y ) d y < ∞ , where f ( · , · ) denotes the joint density of ( X , Y ) . Let K ( · ) be a bounded positive function with a bounded support, satisfying a Lipschitz condition. Then, [figure omitted; refer to PDF] provided that ^{n 2 δ - 1} h [arrow right] ∞ for some δ < 1 - ^{s - 1} .

Lemma A.4.

Suppose that conditions ( C 1 ) - ( C 5 ) hold. Then, one has [figure omitted; refer to PDF] where μ ^ ( t ) , g ^ ( t ) are defined in (9) and _{C n} = ^{{ log ... ( 1 / h ) / n h } 1 / 2} + ^{h 2} .

Proof.

Let [figure omitted; refer to PDF] then, a simple calculation yields [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] together with (A.6), (A.7), by using Lemma A.3, we obtain [figure omitted; refer to PDF] uniformly for t ∈ ( 0,1 ) , where [ecedil]7; is the Kronecker product. By using the same argument, we have [figure omitted; refer to PDF] uniformly for t ∈ ( 0,1 ) . Combining (A.9), (A.10), and (9) yields [figure omitted; refer to PDF] uniformly for t ∈ ( 0,1 ) . With the similar proof to (A.11), we have [figure omitted; refer to PDF] Combining (A.12) with (5), we can prove that [figure omitted; refer to PDF] uniformly for t ∈ ( 0,1 ) .

Invoking (A.9) and (A.13), it is easy to show that [figure omitted; refer to PDF] This completes the proof of Lemma A.4.

Lemma A.5.

Suppose that conditions ( C 1 ) - ( C 5 ) hold. Then, one has that [figure omitted; refer to PDF] where B is defined by (14) and _{η ^ i} ( β ) = ^{∫ 0 1} ... _{Z i} ( t ) [ _{Y i} ( t ) - _{Z i} ( t ^{) T} β ] d t .

Proof.

By (8), a simple calculation yields [figure omitted; refer to PDF] where μ ( t ) = Ψ ( t ^{) - 1} Φ ( t ) . Then, we have [figure omitted; refer to PDF] Hence, we get [figure omitted; refer to PDF] Note that _{J i 1} , i = 1 , ... , n , are i.i.d. It is easy to show that E ( _{J i 1} ) = 0 , Var ... ( _{J i 1} ) = B + o ( 1 ) . Using the central limit theorem, we have [figure omitted; refer to PDF] Hence, to prove this lemma, we only need to prove _{J ν} [arrow right] P 0 , ν = 2,3 , 4 . Now, we deal with _{J 2} . Let _{b r s} ( t ) be the ( r , s ) component of [ μ ( t ) - μ ^ ( t ) ^{] T} and let _{a i s} ( t ) be the s th component of _{X i} ( t ) _{... i} ( t ) , s = 1 , ... , p . Moreover, let [figure omitted; refer to PDF] Let _{J 2 , r} be the r th component of _{J 2} . By Lemmas A.1-A.4, we obtain [figure omitted; refer to PDF] That is, _{J 2} [arrow right] P 0 . By Lemma A.4, a simple calculation yields [figure omitted; refer to PDF] Together with E { [ _{Z i} ( t ) - μ ( t ^{) T} _{X i} ( t ) ] _{X i} ( t ^{) T} } = 0 , and similar to the proof of _{J 2} , we have _{J 3} [arrow right] P 0 . In addition, by Lemma A.4, we have || _{J 4} || ...4; _{O p} ( n ^{C n 2} ) = _{o p} ( 1 ) . This completes the proof of Lemma A.5.

Lemma A.6.

Suppose that conditions ( C 1 ) - ( C 5 ) hold; then, one has [figure omitted; refer to PDF] where Γ ^ and Γ are defined in (11) and (13), respectively.

Proof.

Combining (8) and (11), a simple calculation yields [figure omitted; refer to PDF] By the law of large numbers, we can derive that _{A 1} [arrow right] P Γ . We now show _{A 2} [arrow right] P 0 . From Lemma A.4, we have that [figure omitted; refer to PDF] Together with E { [ _{Z i} ( t ) - μ ( t ^{) T} _{X i} ( t ) ] _{X i} ( t ^{) T} } = 0 , using the similar argument to Lemma A.5, we can prove _{A 2} [arrow right] P 0 . Similarly, we can prove that _{A 3} [arrow right] P 0 . In addition, by (A.25), it is easy to show that || _{A 4} || = _{O p} ( ^{C n 2} ) . This completes the proof.

Proof of Theorem 1.

From (12), we can derive that [figure omitted; refer to PDF] Together with Lemmas A.5 and A.6, it is easy to show that [figure omitted; refer to PDF] This completes the proof of Theorem 1.

Proof of Theorem 2.

For t in a small neighborhood of _{t 0} , such that | t - _{t 0} | ...4; h , by using Taylor expansion, we can get [figure omitted; refer to PDF] Together with (5) and (1), we can derive that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Invoking (18), (A.9), and (A.29), it is easy to show that [figure omitted; refer to PDF]

Let _{Δ n} ( _{t 0} ) = ^{n - 1} _{D x ...} ( _{t 0} ) ; we can prove that E { _{Δ n} ( _{t 0} ) } = 0 and Var ... { _{Δ n} ( _{t 0} ) } = ( n h ^{) - 1} Δ ( _{t 0} ) , where [figure omitted; refer to PDF] By the central limit theorem, we have that [figure omitted; refer to PDF] Combining this with (A.9), we can prove that [figure omitted; refer to PDF] Using the similar argument to (A.9) in Lemma A.4, we can get [figure omitted; refer to PDF] Together with (A.9) and β ^ - β = _{O p} ( ^{n - 1 / 2} ) , it is easy to prove that [figure omitted; refer to PDF] Invoking (A.31), (A.34), and (A.36), a simple calculation yields [figure omitted; refer to PDF] This completes the proof of Theorem 2.