Journal of Applied Mathematics

Volume 2014 (2014), Article ID 585146, 14 pages

http://dx.doi.org/10.1155/2014/585146

Research Article

A Least Squares Method for Variance Estimation in Heteroscedastic Nonparametric Regression

Yuejin Zhou,1,2 Yebin Cheng,² and Tiejun Tong³

¹School of Science, Anhui University of Science and Technology, Huainan 232001, China

²School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai 20043, China

³Department of Mathematics, Hong Kong Baptist University, Hong Kong

Received 28 March 2014; Revised 7 June 2014; Accepted 9 June 2014; Published 3 July 2014

Academic Editor: Li Ma

Copyright © 2014 Yuejin Zhou et al. 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.

Abstract.

Interest in variance estimation in nonparametric regression has grown greatly in the past several decades. Among the existing methods, the least squares estimator in Tong and Wang (2005) is shown to have nice statistical properties and is also easy to implement. Nevertheless, their method only applies to regression models with homoscedastic errors. In this paper, we propose two least squares estimators for the error variance in heteroscedastic nonparametric regression: the intercept estimator and the slope estimator. Both estimators are shown to be consistent and their asymptotic properties are investigated. Finally, we demonstrate through simulation studies that the proposed estimators perform better than the existing competitor in various settings.

1. Introduction

Consider the nonparametric regression model

where are observations, are design points with , is an unknown mean function, and are independent random errors with mean zero and variance , respectively. In the special case when are all the same, model (1) reduces to a homoscedastic nonparametric regression. In this paper, we are interested in estimating the variance in the situation when are not all the same but known constants. Note that such a setting can arise in various situations. As an illustration, we consider a regression model with repeated observations on design points , respectively, where the measurement errors are normal. If in practice we only report the average values on each design point, we have the new model as , where with .

Needless to say, an accurate estimate of variance is important in nonparametric regression. For instance, it is required in constructing confidence bands, in choosing the amount of smoothing, in testing the goodness of fit, and in estimating the detection limits of immunoassay [1–8]. In the past several decades, researchers have proposed many methods for estimating , especially when the regression model is homoscedastic. Among the existing methods, one popular class is referred to as difference-based estimators. The first-order difference-based estimator was proposed in Rice [9],

Assume that is a Lipschitz continuous function and . Note that as . Therefore, is an asymptotically unbiased estimator of . Since then, many difference-based estimators have been proposed in the literature. For instance, Gasser et al. [10] proposed a second-order difference-based estimator. Hall et al. [11] proposed an th-order difference-based estimator with a finite number. Other significant works include Dette et al. [12], Müller et al. [13], Tong et al. [14], Du and Schick [15], and Wang et al. [16], among others. Furthermore, Brown and Levine [17], Wang et al. [18], and Cai and Wang [19] considered the difference-based kernel and wavelet estimators for the variance function in nonparametric regression. Note that the difference-based estimators do not require an estimate of the mean function and so are popular in practice.

As a variation of the difference-based estimation, Tong and wang [20] proposed a least squares estimator of . Let the lag- Rice estimator be

For the equally spaced design with , it can be shown that for any , where and . This indicates that the lag- Rice estimators are always positively biased estimators of , especially when the sample size is small. To reduce bias, Tong and Wang regressed on using a simple linear regression and then estimate as the intercept. The least squares estimator achieves the asymptotically optimal rate that is usually possessed by residual-based estimators only. In addition, Tong et al. [21] established the asymptotic normality and also demonstrated the efficiency of the least squares estimator. We also note that Park et al. [22] investigated the least squares method in small sample nonparametric regression via a local quadratic approximation to determine the regressor and weights.

The aforementioned methods have significantly advanced our understanding on the difference-based estimation of the error variance. Nevertheless, most of the above methods, including the least squares method, only applied to nonparametric regression models with homoscedastic errors. In practice, it is not uncommon that the errors may have different variances. In such situations, we note that the bias term of the least squares estimator in Tong and Wang [20] will be significantly enlarged; for more details, see Sections 2 and 3. Inspired by this, we propose two adaptive least squares estimators for the residual variance in heteroscedastic nonparametric regression.

The remainder of this paper is organized as follows. In Section 2, we propose two least squares estimators for the error variance: the intercept estimator and the slope estimator. In Section 3, we investigate the asymptotic properties of the proposed estimators and present some theoretical results including the asymptotic normalities of the estimators. In Section 4, we conduct simulation studies to evaluate the proposed estimators and compare them with the existing competitor in the literature. We then conclude the paper in Section 5 with a brief discussion and provide the technical proofs in Section 6.

2. Methodology

For model (1), without loss of generality, we assume that . In matrix notation, the model is written as

where , , and . The covariance matrix of is , where

When , namely, for all , it reduced to the homoscedastic setting in Tong and Wang [20]. In this paper, we assume that the values are not all the same.

For this setting, one naive approach is to apply the transformation . Through this transformation the errors become homogeneous. Nevertheless, meanwhile, it makes the transformed mean function no longer a Lipschitz continuous function. Specifically, if and , the difference will not be negligible when tends to be zero. As a consequence, the difference-based methods fail to apply in such situations.

To advance the research project, we reconsider the lag- Rice estimator defined in Tong and Wang [20]. Suppose that has a bounded first derivative. For model (4), the expectation of the lag- Rice estimator is

where , , and . Note that when . Therefore, for model (4) with heteroscedastic errors, it is not guaranteed that is an asymptotically unbiased estimator of .

In what follows, we develop two new estimators for : the first method estimates as the intercept and the second method estimates as the slope. For the first method, we let and . Then, for any , we have

Now treating as the response variable and as the independent variable, we fit the following simple linear regression and estimate as the fitted intercept,

where are the random errors and is the total number of pairs used for the fit. Note that involves pairs of difference; we assign weights , where , to the response variable . We then fit the linear model (8) using the weight least squares that minimizes the weighted sum of squares . Specifically, the estimated error variance is

where , , and . Let and

The quadratic form of can be represented as , where is a symmetric matrix with for , for , and otherwise.

For the second method, we fit the linear regression with two independent variables and and with no intercept term. Specifically, we fit

where are the random errors associated with the linear regression. We then estimate as the fitted slope . For ease of notation, let . By minimizing the weighted sum of squares , we have the second estimator of as

Let and

It is easy to verify that has the quadratic form , where is a symmetric matrix with for , for , and otherwise.

3. Main Results

This section investigates the statistical properties of the proposed least squares estimators. Note that in (9) and in (12) are two similar estimators, except that (9) treats as i.i.d. random errors and (12) treats as i.i.d. random errors. For simplicity, in what follows, we present the asymptotic results for only. To evaluate the achievement of the proposed estimators, we will also investigate the behavior of in Tong and Wang [20] under the new model (4). Recall that for , we have

where , , and is a symmetric matrix with for , for , and otherwise.

Theorem 1. For the equally spaced design, the estimator in (9) is an unbiased estimator of when is a linear function, regardless of the choice of and . Under the same setting, however, the estimator in (14) does not preserve the unbiasedness property. More specifically, the bias term of has the expression

Theorem 2. Assume that has a bounded second derivative and with . When , for any with and the equally spaced design, then

where

As a comparison, the bias and variance of are

where

Theorem 3. Assume that has a bounded second derivative and with . When with , for any with and the equally spaced design, then

As a comparison,

Theorem 4. Assume that has a bounded second derivative and with . For with and any with , then

where , , and denotes convergence in distribution.

The proofs of the theorems are given in Section 6, respectively. Theorems 1 and 2 indicate that is an unbiased or asymptotically unbiased estimator of whereas is not. The comparison on the asymptotic variances, or equivalently on and , will be presented in Section 4. Furthermore, when the heteroscedasticity level is high, Theorem 3 shows that the bias term of is getting more severe so that it does not remain to be a consistent estimator. The asymptotic normality in Theorem 4 can be used to construct confidence intervals for . When , an approximate confidence interval for is

where is the upper -th percentile of the standard normal distribution. When are from normal distribution with variance , we have so that the confidence interval is fully specified. In general, we need to give an estimate for the unknown .

4. Simulation Studies

In this section, we conduct simulation studies to evaluate the finite sample performance of the proposed estimators, and . Their performance will also be compared with the estimator . Let for . Throughout the simulations, we choose the bandwidth , as suggested in Tong and Wang [20].

Our first simulation study considers only one value being different from the others. Specifically, for a given location , we let and for any , where is a constant. Note that is satisfied. In this study, we let . To investigate the behavior of the estimators along with the variance pattern, we consider the mean function and and and , respectively. Given the and values, we then simulate independently from . With 1000 repetitions, we plot the relative mean squared errors, , along with the location for in Figure 1. It is evident that our estimators and perform better than in most locations. To check the behavior near the boundary, we also plot the values of and along with the location for (chosen ) and (chosen ) in Figure 2. Combining Figures 1 and 2, we recommend the use of the new estimators when no significant different variance appears in the boundaries.

(a)

(b)

(c)

(d)

Figure 1: Plots of the relative MSEs of the estimators versus the location for . The dotted, dashed, and solid lines correspond to , , and , respectively.

(a)

(b)

(c)

(d)

Figure 2: Plots of in and in versus the location , under various . The dashed and solid lines correspond to and , respectively.

Our second simulation study is to investigate the average improvement of and over when one or more variances are different from the others. To proceed, we consider three mean functions,

two standard deviations, and , and three sample sizes, , , and , respectively. In total, there are 18 combinations. The values corresponding to and are , , and , respectively. We then randomly sample one location or five locations from the set without replacement. For , the choice of the values follows the previous study. For with the five locations , we let for and for . This results in . For each combination setting, we repeat the simulation 1000 times and report the relative MSEs in Table 1 for and in Table 2 for . From the simulation results, we observe that and have smaller relative MSEs than in all the settings. In addition, we note that the performances of and are almost identical.

Table 1: The relative MSEs of , , and for scenario (i) under various simulation settings.

30
0.5

8.592259
8.588712
8.971043

9.105655
9.109716
9.59135

27.52861
27.93159
29.50932

2

8.568082
8.564596
8.922776

8.577039
8.57355
8.937509

8.655637
8.653321
9.055993

100
0.5

24.68565
24.67358
25.16409

24.63531
24.63209
25.13371

25.1026
25.10747
25.67267

2

24.64412
24.63515
25.1233

24.62441
24.61774
25.1069

24.62372
24.61822
25.11145

500
0.5

47.85348
47.85206
48.00883

47.86474
47.8618
48.01908

47.88284
47.87881
48.03735

2

47.85314
47.85136
48.00809

47.85561
47.85345
48.01027

47.85829
47.85587
48.01283

Table 2: The relative MSEs of , , and for scenario (ii) under various simulation settings.

30
0.5

5.356385
5.358625
5.76456

5.825295
5.844773
6.410941

24.08339
24.74582
27.14087

2

5.291982
5.292094
5.659067

5.284907
5.285037
5.659967

5.367277
5.370764
5.809319

100
0.5

14.70915
14.70799
15.1483

14.6843
14.68265
15.1374

15.01971
15.02443
15.58368

2

14.71244
14.71141
15.14792

14.70282
14.70165
15.13869

14.69388
14.6925
15.13526

500
0.5

42.08678
42.08626
42.35964

42.09082
42.0904
42.36408

42.10065
42.10021
42.37591

2

42.0839
42.08347
42.3567

42.08482
42.08442
42.35764

42.08672
42.08632
42.35965

5. Conclusion

In this paper, we have proposed two least squares estimators for the error variance in heteroscedastic nonparametric regression: the intercept estimator and the slope estimator. Both estimators are shown to be consistent and their asymptotic properties are investigated, including the consistency and the asymptotic normalities. Simulation studies indicate that the proposed estimators perform better than the existing competitor in most settings. In the boundaries, however, we note that the proposed estimators behave not as well as expected when significantly different variances appear in the boundaries of design points. As a practical rule, we have suggested adopting the boundaries as and . Further research may be necessary in this direction.

6. Proofs

This section provides the technical proofs of the theorems in Section 3. To prove the theorems, we first establish two lemmas. For ease of notation, let and .

Lemma 5. Assume that has a bounded second derivative. When , for any with and the equally spaced design, then

(a), ;(b);(c);(d);(e).

Proof. For simplicity, we prove only for . Let and . By the definition of and , we have

First, we consider the upper bound of . We know

Thus, we have

Next, we consider the lower bound of . By the definition, we can know

Let and . Then, and are monotonically increasing about , and

Note that is a monotonically increasing function of for with and . Let be the unique integer such that and . Therefore, we have

Let with . It is easy to verify that . Then, . Let . Since for , then for . Then,

Consequently, we obtain and . Note that

So, we can get .

Note that

where . By , for any , we have

Let , . We know

For any , we have

We know

Note that

Thus,

We know

Note that

Therefore, we have

Lemma 6. Assume that has a bounded second derivative. When with , for any with and the equally spaced design, then

(i), ;(ii);(iii);(iv);(v).

Proof. (i) Here we only consider the proof of . For , it is similar with in Lemma 5 to verify that and . Thus, we can get

(ii) According to , we have

(iii) By part in Lemma 5, we know

For any , we have

(iv) By in Lemma 5, we have

(v) Similar to in Lemma 5, it is easy to get

Lemma 7 (see [23]). Let be entries of a real symmetric matrix , let be i.i.d. random variables, and . Assume that and ; then

where and are the Euclidean norm and the spectral norm of the matrix , respectively.

6.1. Proof of Theorem 1

Let . It is easy to verify that

By combining (50) and (51), we have

This shows that is an unbiased estimator of . In what follows we consider . Therefore,

For , we have

By (54) and (55), we get

6.2. Proof of Theorem 2

It is easy to verify that . This leads to

Thus,

By Lemma 5, for any , we have

In what follows, we calculate . Note that

For , we have

where . For , by Lemma 5, for any , we have

Finally, we consider ,

Combining (61), (62), and (63), we know

Note that, for and any with , we have

Therefore, we get

where

Let , . Then, we have

By (68) and and in Lemma 5, we can get

where and . This completes the proof of the theorem.

6.3. Proof of Theorem 3

By lemma 2, we know

According to (63), for , we have

Note that, under the condition and , it can be shown that

In addition, by Cauchy-Schwarz inequality, we know

Thus,

When and satisfy and , namely, and , then

Next, we consider the order of the bias and variance of . By (55), we have

Note that

By in Lemma 6, then

Consequently, it shows that

Thus, we get

By (68) and in Lemma 6, for and , it is similar with (74) to get

This completes the proof of the theorem.

6.4. Proof of Theorem 4

By Theorem 1, we know

Note that the first term corresponds to the bias term. By Lemma 6, we know . Thus, for any with ,

For the second term, by Lemma 6, we have

Thus, for any with , we have

Now we consider the third term. Let , , and , then , and are i.i.d. random variables, . So we have

where is a real symmetric matrix with for , for , and otherwise.

We know that the Euclidean norm of the matrix can be denoted as

where . By the definition of spectral norm, we have

where is the conjugate transpose of the matrix . Since is a real symmetric matrix, . Then, .

Let be eigenvalues of matrix and ordered to be nonincreasing in absolute value. Then, are eigenvalues of the matrix . Thus, is the spectral norm of matrix . Namely, . Let

It is well known that

Let and . Then, combining (87) and (90), we have

Note that . By Cauchy-Schwarz inequality, we have

Note also that

Consequently, for , we have

Note that . Then, by Lemma 7, we obtain

That is,

where

Hence, we get

where , . Combining (83), (85), and (98), by Slutsky’s theorem, we have

Note that and . Then, when and , namely, and , we have and

where . By Slutsky’s theorem, we obtain

This proves the theorem.

Conflict of Interests

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

Acknowledgments

Yebin Cheng’s research was supported in part by National Natural Science Foundation of China Grant no. 11271241 and Shanghai Leading Academic Discipline Project no. 863. Tiejun Tong’s research was supported in part by Hong Kong Research Grant HKBU202711 and Hong Kong Baptist University FRG Grants FRG2/10-11/020 and FRG2/11-12/110. Yuejin Zhou’s research was supported in part by Doctoral Innovation Foundation of SHUFE CXJJ-2011-442. The authors thank the editor, the associate editor, and the three reviewers for their constructive comments that substantially improved an earlier draft.

References

R. J. Carroll, “The effects of variance function estimation on prediction and calibration: an example,” in Statistical Decision Theory and Related Topics, vol. 2, pp. 273–280, 1986.

R. J. Carroll and D. Ruppert, Transforming and Weighting in Regression, Chapman & Hall, London, UK, 1988.

J. Kay, “On the choice of the regularisation parameter in image restoration,” in Pattern Recognition, vol. 301 of Lecture Notes in Computer Science, pp. 587–596, 1988.

M. J. Buckley, G. K. Eagleson, and B. W. Silverman, “The estimation of residual variance in nonparametric regression,” Biometrika, vol. 75, no. 2, pp. 189–199, 1988.

R. L. Eubank and C. H. Spiegelman, “Testing the goodness of fit of a linear model via nonparametric regression techniques,” Journal of the American Statistical Association, vol. 85, no. 410, pp. 387–392, 1990.

T. Gasser, A. Kneip, and W. Köhler, “A flexible and fast method for automatic smoothing,” Journal of the American Statistical Association, vol. 86, no. 415, pp. 643–652, 1991.

W. Härdle and A. Tsybakov, “Local polynomial estimators of the volatility function in nonparametric autoregression,” Journal of Econometrics, vol. 81, no. 1, pp. 223–242, 1997.

K. B. Kulasekera and C. Gallagher, “Variance estimation in nonparametric multiple regression,” Communications in Statistics A: Theory and Methods, vol. 31, no. 8, pp. 1373–1383, 2002.

J. A. Rice, “Bandwidth choice for nonparametric regression,” The Annals of Statistics, vol. 12, no. 4, pp. 1215–1230, 1984.

T. Gasser, L. Sroka, and C. Jennen-Steinmetz, “Residual variance and residual pattern in nonlinear regression,” Biometrika, vol. 73, no. 3, pp. 625–633, 1986.

P. Hall, J. W. Kay, and D. M. Titterington, “Asymptotically optimal difference-based estimation of variance in nonparametric regression,” Biometrika, vol. 77, no. 3, pp. 521–528, 1990.

H. Dette, A. Munk, and T. Wagner, “Estimating the variance in nonparametric regression—what is a reasonable choice?” Journal of the Royal Statistical Society B. Statistical Methodology, vol. 60, no. 4, pp. 751–764, 1998.

U. U. Müller, A. Schick, and W. Wefelmeyer, “Estimating the error variance in nonparametric regression by a covariate-matched -statistic,” Statistics, vol. 37, no. 3, pp. 179–188, 2003.

T. Tong, A. Liu, and Y. Wang, “Relative errors of difference-based variance estimators in nonparametric regression,” Communications in Statistics A: Theory and Methods, vol. 37, no. 18–20, pp. 2890–2902, 2008.

J. Du and A. Schick, “A covariate-matched estimator of the error variance in nonparametric regression,” Journal of Nonparametric Statistics, vol. 21, no. 3, pp. 263–285, 2009.

L. Wang, L. D. Brown, and T. T. Cai, “A difference based approach to the semiparametric partial linear model,” Electronic Journal of Statistics, vol. 5, pp. 619–641, 2011.

L. D. Brown and M. Levine, “Variance estimation in nonparametric regression via the difference sequence method,” The Annals of Statistics, vol. 35, no. 5, pp. 2219–2232, 2007.

L. Wang, L. D. Brown, T. T. Cai, and M. Levine, “Effect of mean on variance function estimation in nonparametric regression,” The Annals of Statistics, vol. 36, no. 2, pp. 646–664, 2008.

T. T. Cai and L. Wang, “Adaptive variance function estimation in heteroscedastic nonparametric regression,” The Annals of Statistics, vol. 36, no. 5, pp. 2025–2054, 2008.

T. Tong and Y. Wang, “Estimating residual variance in nonparametric regression using least squares,” Biometrika, vol. 92, no. 4, pp. 821–830, 2005.

T. Tong, Y. Ma, and Y. Wang, “Optimal variance estimation without estimating the mean function,” Bernoulli, vol. 19, no. 5, pp. 1839–1854, 2013.

C. G. Park, I. Kim, and Y.-S. Lee, “Error variance estimation via least squares for small sample nonparametric regression,” Journal of Statistical Planning and Inference, vol. 142, no. 8, pp. 2369–2385, 2012.

R. J. Bhansali, L. Giraitis, and P. S. Kokoszka, “Approximations and limit theory for quadratic forms of linear processes,” Stochastic Processes and Their Applications, vol. 117, no. 1, pp. 71–95, 2007.