International Journal of Mathematics and Mathematical Sciences
Volume 2016 (2016), Article ID 2601601, 16 pages
http://dx.doi.org/10.1155/2016/2601601

Research Article

Asymptotic Theory in Model Diagnostic for General Multivariate Spatial Regression

Wayan Somayasa,¹ Gusti N. Adhi Wibawa,¹ La Hamimu,² and La Ode Ngkoimani²

¹Department of Mathematics, Haluoleo University, Kendari, Indonesia
²Department of Geological Engineering, Haluoleo University, Kendari, Indonesia

Received 25 March 2016; Revised 13 July 2016; Accepted 28 July 2016

Academic Editor: Andrei I. Volodin

Copyright © 2016 Wayan Somayasa 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.

We establish an asymptotic approach for checking the appropriateness of an assumed multivariate spatial regression model by considering the set-indexed partial sums process of the least squares residuals of the vector of observations. In this work, we assume that the components of the observation, whose mean is generated by a certain basis, are correlated. By this reason we need more effort in deriving the results. To get the limit process we apply the multivariate analog of the well-known Prohorov’s theorem. To test the hypothesis we define tests which are given by Kolmogorov-Smirnov (KS) and Cramér-von Mises (CvM) functionals of the partial sums processes. The calibration of the probability distribution of the tests is conducted by proposing bootstrap resampling technique based on the residuals. We studied the finite sample size performance of the KS and CvM tests by simulation. The application of the proposed test procedure to real data is also discussed.

1. Introduction

As mentioned in the literatures of model checks for multivariate regression, the appropriateness of an assumed model is mostly verified by analyzing the least squares residual of the observations; see, for example, Zellner [1], Christensen [2], pp. 1–22, Anderson [3], pp. 187–191, and Johnson and Wichern [4], pp. 395–398. A common feature of these works is the comparison between the length of the matrix of the residuals under the null hypothesis and that of the residuals under a proposed alternative.

Instead of considering the residuals directly MacNeill [5] and MacNeill [6] proposed a method in model check for univariate polynomial regression based on the partial sums process of the residuals. These popular approaches are generalized to the spatial case by MacNeill and Jandhyala [7] for ordinary partial sums and Xie and MacNeill [8] for set-indexed partial sums process of the residuals. Bischoff and Somayasa [9] and Somayasa et al. [10] derived the limit process in the spatial case by a geometric method generalizing a univariate approach due to Bischoff [11] and Bischoff [12]. These results can be used to establish asymptotic test of Cramér-von Mises and Kolmogorov-Smirnov type for model checks and change-point problems. Model checks for univariate regression with random design using the empirical process of the explanatory variable marked by the residuals was established in Stute [13] and Stute et al. [14]. In the papers mentioned above the limit processes were explicitly expressed as complicated functions of the univariate Brownian motion (sheet).

The purpose of the present article is to study the application of set-indexed partial sums technique to simultaneously check the goodness-of-fit of a multivariate spatial linear regression defined on high-dimensional compact rectangle. In contrast to the normal multivariate model studied in the standard literatures such as in Christensen [2], Anderson [3], and Johnson and Wichern [4] or in the references of model selection such as in Bedrick and Tsai [15] and Fujikoshi and Satoh [16], in this paper we will consider a multivariate regression model in which the components of the mean of the response vector are assumed to lie in different spaces and the underlying distribution model of the vector of random errors is unknown.

To see the problem in more detail let be fixed. Let a -dimensional random vector be observed independently over an experimental design given by a regular lattice:

where is the -dimensional unit cube. Let be the true but unknown -valued regression function on which represents the mean function of the observations. Let and be the observation and the corresponding mean in the experimental condition . Under the null hypothesis we assume that follows a multivariate linear model. That is, we assume a model

where, for , is a -dimensional vector of unknown parameters; is a -dimensional vector of known regression functions whose components are assumed to be square integrable with respect to the Lebesgue measure on , that is, , for all and . is the mutually independent -dimensional vector of random errors defined on a common probability space . We assume that, for all , , and . Let be the -dimensional pyramidal array of random observations and let be the -dimensional pyramidal array of random errors taking values in the Euclidean space . Then under the observations can be represented by

where . Under the alternative a multivariate nonparametric regression model

is assumed, where . By applying the similar argument as in Christensen [2] and Johnson and Wichern [4], the -dimensional array of the least squares residuals of the observations is given by the following component-wise projection:

with , for , and . Thereby, for , we define as the subspace of spanned by the arrays . It is worth mentioning that the Euclidean space is furnished with the inner product denoted by and defined by

for every , and .

Next we define the set-indexed partial sums operator. Let be the family of convex subset of , and let be the Lebesgue pseudometric on defined by , for . Let be the set of continuous functions on under . We embed the array of the residual into a -dimensional stochastic process indexed by by using the component-wise set-indexed partial sums operator

such that, for any ,

where, for , . Let us call this process the -dimensional set-indexed least squares residual partial sums process. The space is furnished with the uniform topology induced by the metric defined by

for and .

We notice that, in the works of Bischoff and Somayasa [9], Bischoff and Gegg [17], and Somayasa and Adhi Wibawa [18], the limit process of the partial sums process of the least squares residuals has been investigated by applying the existing geometric method of Bischoff [11, 12]. However, the method becomes not applicable anymore in deriving the limit process of as the dimension of varies. Therefore, in this work, we attempt to adopt the vectorial analog of Prohorov’s theorem; see, for example, Theorem in Billingsley [19] for obtaining the limit process. For our result we need to extend the ordinary partial sums formula to -dimensional case defined on as follows. Let and be the set of all -combinations of the set , with . For a chosen value of , we denote the th element of by a -tuple , for , where is the number of elements of which is clearly given by

For example, let . Then, for , we denote the elements of as , , and . In a similar way, we denote the elements of which consists of , respectively, by , , and . Finally the element of is sufficiently written by . Hence the -dimensional ordinary partial sums operator transforms any -dimensional array to a continuous function on defined by

for every , where for positive integers we define a notation

It is clear that the partial sums process of the residuals obtained using (11) is a special case of (8) since for every it holds

It is worth noting that the extension of the study from univariate to multivariate model and also the expansion of the dimension of the lattice points are strongly motivated by the prediction problem in mining industry and geosciences. As for an example recently Tahir [20] presented data provided by PT Antam Tbk (a mining industry in Southeast Sulawesi). The data consist of a joint measurement of the percentage of several chemical elements and substances such as Ni, Co, Fe, MgO, , and CaO which are recorded in every point of a three-dimensional lattice defined over the exploration region of the company. Hence, by the inherent existence of the correlation among the variables, the statistical analysis for the involved variables must be conducted simultaneously.

There have been many methods proposed in the literatures for testing . Most of them have been derived for the case of under normally distributed random error. Generalized likelihood ratio test which has been leading to Wilk’s lambda statistic or variant of it can be found in Zellner [1], Christensen [2], pp. 1–22, Anderson [3], pp. 187–191, and Johnson and Wichern [4], pp. 395–398. Mardia and Goodall [21] derived the maximum likelihood estimation procedure for the parameters of the general normal spatial multivariate model with stationary observations. This approach can be straightforwardly extended for obtaining the associated likelihood ratio test in model check for the model. Unfortunately, in the practice especially when dealing with mining data, normal distribution is sometimes found to be not suitable for describing the distribution model of the observations, so that the test procedures mentioned above are consequently no longer applicable. Asymptotic method established in Arnold [22] for multivariate regression with can be generalized in such a way that it is valid for the general model. As a topic in statistics it must be well known. However, we cannot find literatures where the topic has been studied.

The rest of the paper is organized as follows. In Section 2 we show that when is true converges weakly to a projection of the -dimensional set-indexed Brownian sheet. The limit process is shown to be useful for testing asymptotically based on the Kolmogorov-Smirnov (KS-test) and Cramér-von Mises (CvM-test) functionals of the set-indexed -dimensional least squares residual partial sum processes, defined, respectively, by

For both tests the rejection of is for large value of the KS and CvM statistics, respectively. Under localized alternative the above sequence of random processes converges weakly to the above limit process with an additional deterministic trend (see Section 3). In Section 4, we define a consistent estimator for . In Section 5 we investigate a residual based bootstrap method for the calibration of the tests. Monte Carlo simulation for the purpose of studying the finite sample behavior of the KS and CvM tests is reported in Section 6. Application of the test procedure in real data is presented in Section 7. The paper is closed in Section 8 with conclusion and some remarks for future research. Auxiliary results needed for deriving the limit process are presented in the appendix. We note that all convergence results derived throughout this paper which hold for simultaneously go to infinity, that is, for , for all ; otherwise they will be stated in some way. The notion of convergence in distribution and convergence in probability will be conventionally denoted by and , respectively.

2. The Limit of VR under

Let be the one-dimensional set-indexed Brownian sheet having sample path in . We refer the reader to Pyke [23], Bass and Pyke [24], and Alexander and Pyke [25] for the definition and the existence of . Let us consider a subspace of which is closely related to , defined by

Under the inner product and the norm defined by

it is clear that and are isometric. For our result we need to define subspaces and associated with the regression functions , where and , for .

Now we are ready to state the limit process of the sequence of -dimensional set-indexed residual partial sums processes for the model specified under .

Theorem 1. For , let be an orthonormal basis (ONB) of . We assume that , for . Let be the -dimensional set-indexed Brownian sheet with the covariance function , for , where is the identity matrix. Suppose are in , where is the space of continuous functions on (see Definition A.4 for the definition of ). Then under it holds that

where

Thereby is a projector such that, for every and ,

For , we set for , and stands for the integral in the sense of Riemann-Stieltjes. Moreover, the limit process is a centered Gaussian process with the covariance function given by

Proof. By applying the linear property of and Lemma C.2, we have under ,

It can be shown by extending the uniform central limit theorem studied in Pyke [23], Bass and Pyke [24], and Alexander and Pyke [25] to its vectorial analog that the term on the right-hand side of (21) converges weakly to . Therefore we only need to show that the second term satisfies the weak convergence:

where is a -dimensional centered Gaussian process with the covariance matrix given by

By Prohorov’s theorem it is sufficient to show that, for any finite collection of convex sets in and real numbers , with , it holds that

where the left-hand side has the covariance which can be expressed as

where , for and . Let and be fixed. Then by a simultaneous application of the definition of (see Lemma C.3), (11), the definition of the Riemann-Stieljes integral (cf. Stroock [26], pp. 7–17), and the independence of the array , we further get

By recalling Lemma C.3 the last expression clearly converges to

Hence the matrix converges element wise to the symmetric matrix which can be represented as , for a matrix defined by

with , for . Thereby denotes the Hadarmard product defined, for example, in Magnus and Neudecker [27], pp. 53-54. Since , we successfully have shown that converges to the following general linear combination:

which is actually the covariance of . Next we observe that, by applying the definition of and the definition of the Riemann-Stieltjes integral, we can also write as follows:

where

Let , , and be the Euclidean norm of . Then by considering the stochastic independence of the array of the -vector of the random errors, it holds that

in which by the well-known bounded convergence theorem (cf. Athreya and Lahiri [28], pp. 57-58) the last term converges to zero. Thus the Lindeberg condition is satisfied. Therefore by the Lindeberg-Levy multivariate central limit theorem studied, for example, in van der Vaart [29], pp. 16, it can be concluded that converges in distribution to , where has the -variate normal distribution with mean zero and the covariance given by (29).
The tightness of can be shown as follows. By the definition can also be expressed as

Since is tight, hence by recalling Lemma in Billingsley [19], pp. 38–40, it is sufficient to show that the mapping is continuous on for every . Proposition C.4 finishes the proof.

Corollary 2. By Theorem 1 and the well-known continuous mapping theorem (cf. Theorem in Billingsley [19]) the distribution of the statistics and can be approximated, respectively, by those of

Let and be the th quantile of the distributions of and , respectively. When is used, will be rejected at level if and only if . Likewise if is used, then will be rejected at level if and only if .

3. The Limit of VR under

The test procedures derived above are consistent in the sense of Definition in Lehmann and Romano [30]. That is, the probability of rejection of under the competing alternative converges to 1. As an immediate consequence we cannot observe the performance of the tests when the model moves away from . Therefore, to be able to investigate the behavior of the tests, we consider the localized model defined as follows:

When is true we get the similar least squares residuals as given in Section 1. Therefore, observing Model (35), the test problem will not be altered.

In the following theorem, we present the limit process of the -dimensional set-indexed partial sums process of the residuals under associated with Model (35).

Theorem 3. For , let be an ONB of with , for . If and (see Definition A.3 for the notion of ), then, observing (35), we have under that

where , with .

Proof. Considering the linearity of and Lemma C.2, when is true we have

Since, for , is in , it can be shown that converges uniformly to , for every . Also the last two terms on the right-hand side of the preceding equation converge in distribution to by Theorem 1. Thus to the rest we only need to show that converges to . By the definition of component-wise projection we have

The right-hand side of the last expression is obtained directly from the definition of the ordinary partial sums (11) and the definition of the Riemann-Stieltjes integral on . Since converges uniformly to and has bounded variation on for all and , then the last expression clearly converges component-wise to , which can be written as . We are done.

Corollary 4. By Theorem 3 the power function of the KS and CvM tests at a level can now be approximated by computing the probabilities of the form

for a fixed , respectively. In Section 5 we investigate the empirical power functions of the KS and CvM tests by simulation.

4. Estimating the Population Covariance Matrix

If the covariance matrix is unknown, as it usually is, it is impossible to use and in practice. What we propose to do is to employ a consistent estimate of . We need some further notations for expressing the residuals of the model. For , let , , and be -dimensional column vectors defined by

Furthermore, let , , and be -dimensional matrices whose th column is given, respectively, by the column vectors , , and , . Then Model (35) can also be represented as follows:

where, for , , with being the identity matrix in .

Associated with the subspace we define the design matrix as an element of whose th column is given by the -dimensional column vector:

We denote the column space of by for the sake of brevity. We also define the column-wise projection of any matrix in into the product space by

A reasonable estimator of the covariance matrix is denoted by , defined by

where

constitutes the component-wise orthogonal projector into the orthogonal complement of the product space .

Zellner [1] and Arnold [22] investigated the consistency of toward in the case of the multivariate regression model with . Some difficulties appear when the situation is extended to the case of , since it involves the problem of finding the limit of matrices with the components given by inner products of two vectors.

Theorem 5. Suppose the localized model (41) is observed. If is true, then, under the conditions of Theorem 1, we have .

Proof. If is true, it can be easily shown that

For technical reason we assume without lost of generality that is an orthogonal matrix, for . Hence we further get the representation

Since are independent and identically distributed random matrices with mean , by the well-known weak law of large numbers, we get

Note that in the practice we consider the polynomial regression model. Hence, for every , the design matrix satisfies the so-called Huber condition (cf. Pruscha [31], pp. 115–117). By this reason, for the rest of the terms, we can immediately apply the technique proposed in Arnold [22] to show , for all . Therefore, we finally get the following component-wise convergence:

where is the -zero matrix.

Remark 6. Since , without altering the convergence result presented in Theorem 1, the population variance-covariance matrix can be directly replaced by the consistence estimator .

5. Calibration of the Tests

The limits of the test statistics are not distribution-free and we need therefore calibration for the distribution of the statistical tests. For the calibration we adapted the idea of residual based bootstrap for multivariate regression studied in Shao and Tu [32] for approximating the distributions of and .

For fixed , let be the empirical distribution function of the vectors of least squares residuals centered at zero vector, where . Let be an array of independent and identically distributed random vectors sampled from and let be the ordinary LSE of , where . Then we generate the array of -dimensional bootstrap observations which is denoted in this paper by through the model:

Based on this model we get the array of -dimensional bootstrap least squares residuals which is given by the component-wise projection of the bootstrap observations:

Hence, the bootstrap analog of and is

where

The question regarding the consistency of the bootstrap approximation of the -dimensional processes for is summarized in the following theorem.

Theorem 7. Let be an ONB of  , for . Suppose the conditions of Theorem 1 are fulfilled. Then under it holds that

where is defined in Theorem 1.

Proof. We notice that are independent and identically distributed with and . Hence, the invariance principle implies that converges in distribution to . Hence under we have and can be written as

where it can be shown easily that

with being the collection of the terms converging in probability to . Then by recalling Theorem 5 and the linearity of we only need to show that

The proof is established by imitating the steps of proving convergence result of Theorem 1. We are done.

6. Simulation Study

In this section, we report on a simulation study designed to investigate the finite sample size behavior of the KS and CvM tests. We simulate a multivariate model with four components defined on the unit rectangle . The hypothesis under study is against , where , , , and . Thereby we define , , , , , , and , for . The samples are generated from a localized model under the experimental design given by , where is defined as

for constants , , , and determined prior to the generation of the samples. For fixed and and and the vector of random errors is generated independently from the -variate normal distribution with mean zero and variance-covariance matrix given by

however, we assume in the computation that is unknown. It is therefore estimated using defined in Section 4. It is important to note that for computational reason we restricted the index set to the Vapnick Chervonenkis Classes (VCC) of subsets of which is given by the family of closed rectangle with the point as the essential point. That is, the family .

When , , , and are set simultaneously to zero then we get the samples which coincide to the model specified under . Conversely, when at least one of them takes a nonzero value then we obtain the samples which can be regarded as from the alternative whose corresponding samples are generated by assigning nonzero values to either one of the constants, , , , and , or the combinations of them.

Table 1 presents the empirical probabilities of rejection of for and some selected values of , , , and . The empirical powers of the and tests are denoted by and , respectively. The notations and stand, respectively, for the standard deviation of the samples. The critical values of the statistics and are 6.0742 and 7.9910 which are approximated by simulation. For the values of and fluctuate around as it should be. This means that, independent of the selected number of the lattice points, both tests attain the specified level of significance.

Table 1: The empirical probabilities of rejection of the KS and CvM tests for several selected values of , , , and . The sample sizes are selected to and . The simulation results are based on 10000 runs.

Sample size

0
0
0
0
4.90%
1.31299
4.60%
2.52127

10
0
0
0
10.60%
1.52902
11.28%
4.11538

20
0
0
0
32.80%
1.71455
33.40%
6.26419

40
0
0
0
88.64%
1.86015
87.22%
10.37546

0
50
0
0
5.03%
1.28540
5.33%
2.63926

0
200
0
0
4.43%
1.22630
5.90%
2.57881

0
300
0
0
4.60%
1.18783
7.03%
2.52180

0
500
0
0
4.63%
0.90516
11.00%
2.32439

0
0
50
0
4.83%
1.27888
5.03%
2.59347

0
0
100
0
5.17%
1.29476
5.20%
2.68005

0
0
200
0
6.03%
1.28627
5.23%
2.53931

0
0
500
0
8.23%
1.27694
7.73%
2.71275

0
0
0
50
5.00%
1.27241
5.03%
2.69462

0
0
0
100
4.77%
1.23645
4.63%
2.56912

0
0
0
300
4.77%
1.27318
5.07%
2.53470

0
0
0
500
4.97%
1.26322
4.97%
2.55223

10
50
0
0
11.34%
1.55144
12.22%
3.82300

10
100
0
0
11.12%
1.49017
12.70%
3.81009

10
200
0
0
11.20%
1.52118
12.06%
3.71303

10
500
0
0
12.46%
1.10222
22.44%
2.87591

20
50
50
50
32.78%
1.76263
33.62%
5.75683

20
50
100
50
31.80%
1.71323
32.52%
5.98971

20
50
200
50
28.16%
1.68594
29.16%
5.79669

20
50
500
50
27.08%
1.52512
28.86%
4.92925

50
50
50
20
98.60%
1.86506
97.60%
13.43552

50
50
50
40
98.60%
1.83424
97.95%
13.10968

50
50
50
60
98.55%
1.91241
98.25%
13.49794

0
0
0
0
4.75%
1.26388
4.55%
2.52770

10
0
0
0
11.35%
1.48925
11.40%
3.70839

20
0
0
0
33.65%
1.75109
33.40%
5.82381

30
0
0
0
65.70%
1.82802
64.90%
8.31756

0
50
0
0
4.70%
1.26907
4.95%
2.60854

0
200
0
0
4.75%
1.23471
5.90%
2.53725

0
300
0
0
5.35%
1.19808
7.35%
2.62291

0
0
50
0
0.0485
1.27867
5.15%
2.57206

0
0
100
0
4.35%
1.23893
4.45%
2.40576

0
0
200
0
5.80%
1.27524
5.50%
2.62313

0
0
500
0
8.45%
1.28708
6.50%
2.64556

0
0
0
50
5.00%
1.28525
4.85%
2.71581

0
0
0
100
4.50%
1.25700
4.70%
2.57227

0
0
0
500
4.70%
1.25102
5.10%
2.51076

20
50
0
0
32.10%
1.70264
32.30%
5.72887

20
100
0
0
33.55%
1.74878
32.50%
5.76105

20
200
0
0
33.30%
1.72065
34.35%
5.71531

20
500
0
0
33.90%
1.61848
46.30%
5.77714

30
50
100
0
62.55%
1.78912
61.80%
8.08037

30
50
200
0
60.50%
1.79929
58.60%
7.76765

30
50
500
0
56.35%
1.78203
54.00%
7.76498

20
20
20
20
32.55%
1.70930
33.05%
5.60510

30
50
50
50
63.10%
1.86549
61.65%
8.63149

50
50
50
50
97.75%
1.88146
97.10%
13.27029

50
100
50
50
97.60%
1.89668
96.90%
13.30456

Furthermore, Figure 1 exhibits the graphs of the empirical power function of the KS and CvM tests for associated with hypothesis specified above against , for . For the four cases we generate the error vectors independently from the same 4-variate normal distribution mentioned above. In the clockwise direction the left-top panel presents the graphs of the power function for testing against , the right-top panel is for against , the right-bottom is for against , and left-bottom is for against . The common characteristic of the tests is that the power gets larger as the the model moves away from . The KS tests represented by smooth line tend to have slightly larger power. However, somewhat unexpectedly, in the second case, the CvM test has much larger power.

Figure 1: The empirical power functions of the KS test (smooth line) and CvM test (dotted line) for the 4-variate model using -point regular lattice. The simulation is under 10000 runs.

7. Example of Application

In this example, the proposed method is applied to a mining data studied in Tahir [20]. As introduced in Section 1, the data consist of a simultaneous measurement of the percentages of Nickel (Ni), Cobalt (Co), Ferum (Fe), and other substances like Calcium-Monoxide (CaO), Silicon-Dioxide (), and Magnesium-Monoxide (MgO). The sample was obtained by drilling bores set according to a three-dimensional lattice of size with equidistance rows running west to east, equidistance columns running south to north, and equidistance depths from the surface of the earth to the bottom. To simplify the computation of the test statistics we consider the experimental design as a two-dimensional lattice of size by taking the average value of the samples measured in the same position. We further assume that the exploration region is given by a closed rectangle so that by suitable rescaling it can be transformed into a closed unit rectangle . Table 2 exhibits, respectively, the pairs scatter plot and Pearson’s correlation coefficient among the percentages of Ni, CaO, Co, the logarithm of the percentages of (), MgO (logMgO), and Fe (logFe). By this reason a multivariate analysis must be conducted in the statistical modelling taking into account the unknown covariance matrix of the vector of the variables. Furthermore, based on the individual scatter plot of the samples which are not presented in this work, it can be inferred that polynomials of lower order seem to be adequate to approximate the population model. More precisely, let be the vector of observations representing the observed percentages of CaO, , logMgO, Co, Ni, and logFe, respectively. We aim to test the hypothesis

for some unknown constants , and , with , and . For this case we have , , and , with , , , , , and .

Table 2: Pearson’s correlation matrix of the percentages of CaO, , , Ni, , and Co observed over a regular lattice of size . Source of data: Tahir [20].

CaO

Ni

Co

1.0000

0.3949

0.4045

Ni
−0.1285

−0.1166

Co
−0.0665

We obtained the values and with the associated simulated values of and , respectively. We notice that in the computation we consider the VCC as the index sets instead of . Hence when using the KS test as well as CvM test the hypothesis will be not rejected for almost all commonly used values of . There exists a significant evidence that the assumed model is appropriate for describing the functional relationship between the experimental conditional and the percentages of those elements.

In the practice some computational difficulties appear for testing using our proposed method. First, to the knowledge of the authors, the analytical formula for computing the critical and values of the tests have been not yet available in the literatures; therefore we need to approximate them by simulation using computer. Second, although the test procedures are established for a much larger family of sets , in the application the computation is always restricted to the VCC of subsets of like that of or .

8. Concluding Remark

In this article we have developed an asymptotic method for checking the validity of a general multivariate spatial regression model by considering the multidimensional set-indexed partial sums of the residuals. For the calibration of the distribution of the test statistics we propose the residual based bootstrap for multivariate regression. It is shown by applying imitation technique that the residual bootstrap resampling technique is consistent. In a simulation study the finite sample size behavior of the KS and CvM statistics is investigated in greater detail. For the first-order model CvM test has much larger power, whereas for constant, second-order, and third-order models the powers of the two tests are almost the same.

Other possibilities of tests for multidimensional case can be obtained by incorporating a sampling technique according to an arbitrary experimental design. Sometimes because of technical, economic, or ecological reason, practitioners will not or cannot sample the observations equidistantly. One possible approach is to sample according to a continuous probability measure; see, for example, the sampling method proposed in Bischoff [11]. Under this concern we get the so-called weighted KS and CvM tests which can be viewed as generalization of the KS and CvM tests studied in this paper.

Instead of considering the least squares residuals of the observations we can also define a test by directly investigating the partial sums of the observations. The limit process will be given by a type of signal plus a noise which is given by the multidimensional set-indexed Brownian sheet. Observing the limit process we can formulate likelihood ratio test based on the Cameron-Matrin-Girsanov density formula of the limit process. Establishing such type of test will be of our concern in our future research project.

Appendix

A. Function of Bounded Variation on I

Definition A.1. Let be a real-valued function with variables. For , let be a real-value function defined on , given by

for . Furthermore, for and , is defined on recursively starting from the last components of and . More precisely,

Let be permutation of ; then it holds that

This means that the operation of does not depend on the order. By this reason we write by ignoring the brackets. The reader is referred to Yeh [33] and Elstrodt [34], pp. 44-45.

Definition A.2 (see Yeh [33]). Let be a collection of rectangles on the unit interval with , for . The Cartesian product which consists of rectangles is called a nonoverlapping finite exact cover of . The family of all nonoverlapping finite exact cover of is denoted by .

Definition A.3 (see Yeh [33]). For , with , let be the element of defined by . Let be a real-valued function on . Operator acting on a function is defined by

The variation of over the finite exact cover is defined by

Accordingly, the total variation of over is defined by

Furthermore, function is said to have bounded variation in the sense of Vitaly on if there exists a real number such that for some real number . The class of such functions is denoted by .

Definition A.4 (see Yeh [33]). Let be a variable in . For fixed , let be a -dimensional unit closed rectangle constructed in the following way. We choose components of the variable . For each choice from all possible elements of the set , we set each with or and let the remaining variables satisfy . Then for each we get unit closed rectangles . For convention we denote the collection of all of closed rectangles by and the th element of will be denoted by . Function is said to have bounded variation in the sense of Hardy on , if and only if for each and there exists a real number such that , where, for and , is a function with variables obtained from the function by setting the selected variables with or , whereas the remaining variables lie in the interval . The class of such functions will be denoted by .

B. Integration by Parts on I

For family defined in Definition A.4, let , where, for , the family is a collection of different points in . As an example, for , we have . For each and , let be the number of ’s appearing in . Next, let and be defined on . If is Riemann-Stieltjes integrable with respect to on , we denote the integral by . For , it is understood that is defined as the product of and at that point of (see Yeh [33]).

Theorem B.1 (integration by parts (see Yeh [33])). Let be Riemann-Stieltjes integrable with respect to on each member of . Then is Riemann-Stieltjes integrable with respect to on , and we have the formula

Moreover, if have bounded variation in the sense of Hardy on and is continuous on , then we have the inequality

C. Some Property of the Partial Sums Operator

Lemma C.1 (see Bischoff and Somayasa [9]). For every one-dimensional pyramidal array , it holds that , where is the subspace defined in (15). Furthermore, for any arrays and , we have

where is the one-dimensional component of the partial sums operator .

Proof. Associated with we can construct a step function defined by

where , for . For any , it holds that

Hence, having as the density. By the definition of the inner product , we further get

Lemma C.2 (see Bischoff and Somayasa [9]). For any in it holds that, for ,

where

Furthermore, by the definition of the component-wise projection, we finally get

for every -dimensional array .

Proof. For fixed , let be an ONB of . Then by Lemma C.1 the corresponding ONB of is given by the set

Hence, by the linearity of and by Lemma C.1, we get

Lemma C.3 (see Bischoff and Somayasa [9]). Let be an orthonormal set in obtained by the Gram-Schmidt procedure from the step functions:

for , and . Then is an ONB of . The projection of any function into with respect to this basis is given by

Moreover, if, for and , is continuous on and is an ONB of , then as . Consequently, it also holds that , for .

Proof. Since , by the linearity of it follows that builds a basis for whenever the set is a basis of . Furthermore, if is continuous on , it can be shown that as . Hence, by the definition of the Gram-Schmidt process and also by the continuity of , we can further show that as . The last assertion is immediately obtained from the definition of and .

Proposition C.4. The -dimensional projection is continuous uniformly on the space of continuous function for all .

Proof. Let and be any functions in . Then, by the definition and the inequality presented in Theorem B.1, for any we have

for some constant defined by

Hence we get

Given any positive small number , there exists a small number , such that, for any , if , then

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors wish to thank the Ministry of Research, Technology and Higher Education (RISTEK-DIKTI) for the financial support. They also thank Karlsruher Institut für Technologie (KIT) Institut für Stochastik for hospitality. Special thanks are addressed to Professor Andrei I. Volodin for his constructive comments for the improvement of the paper.

References

A. Zellner, “An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias,” Journal of the American Statistical Association, vol. 57, no. 298, pp. 348–368, 1962.

R. Christensen, Advanced Linear Modeling: Multivariate, Time Series, and Spatial Data; Nonparametric Regression and Response Surface Maximization, Springer, New York, NY, USA, 2001.

D. W. Anderson, An Introduction to Multivariate Statistical Analysis, John Wiley & Sons, New York, NY, USA, 3rd edition, 2003.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, Prentice Hall, New York, NY, USA, 3rd edition, 2007.

I. B. MacNeill, “Properties of partial sums of polynomial regression residuals with applications to test for change of regression at unknown times,” The Annals of Statistics, vol. 6, no. 2, pp. 422–433, 1978.

I. B. MacNeill, “Limit processes for sequences of partial sums of regression residuals,” The Annals of Probability, vol. 6, no. 4, pp. 695–698, 1978.

I. B. MacNeill and V. K. Jandhyala, “Change-point methods for spatial data,” in Multivariate Environmental Statistics, G. P. Patil and C. R. Rao, Eds., pp. 298–306, Elevier Science, Berlin, Germany, 1993.

L. Xie and I. B. MacNeill, “Spatial residual processes and boundary detection,” South African Statistical Journal, vol. 40, no. 1, pp. 33–53, 2006.

W. Bischoff and W. Somayasa, “The limit of the partial sums process of spatial least squares residuals,” Journal of Multivariate Analysis, vol. 100, no. 10, pp. 2167–2177, 2009.

W. Somayasa, Ruslan, E. Cahyono, and L. O. Ngkoimani, “Checking adequateness of spatial regressions using set-indexed partial sums technique,” Far East Journal of Mathematical Sciences, vol. 96, no. 8, pp. 933–966, 2015.

W. Bischoff, “A functional central limit theorem for regression models,” The Annals of Statistics, vol. 26, no. 4, pp. 1398–1410, 1998.

W. Bischoff, “The structure of residual partial sums limit processes of linear regression models,” Theory of Stochastic Processes, vol. 2, pp. 23–28, 2002.

W. Stute, “Nonparametric model checks for regression,” The Annals of Statistics, vol. 25, no. 2, pp. 613–641, 1997.

W. Stute, W. González Manteiga, and M. Presedo Quindimil, “Bootstrap approximations in model checks for regression,” Journal of the American Statistical Association, vol. 93, no. 441, pp. 141–149, 1998.

E. J. Bedrick and C.-L. Tsai, “Model selection for multivariate regression in small samples,” Biometrics, vol. 50, no. 1, pp. 226–231, 1994.

Y. Fujikoshi and K. Satoh, “Modified AIC and in multivariate linear regression,” Biometrika, vol. 84, no. 3, pp. 707–716, 1997.

W. Bischoff and A. Gegg, “Partial sum process to check regression models with multiple correlated response: with an application for testing a change-point in profile data,” Journal of Multivariate Analysis, vol. 102, no. 2, pp. 281–291, 2011.

W. Somayasa and G. N. Adhi Wibawa, “Asymptotic model-check for multivariate spatial regression with correlated responses,” Far East Journal of Mathematical Sciences, vol. 98, no. 5, pp. 613–639, 2015.

P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, NY, USA, 1968.

M. Tahir, “Prediction of the amount of nickel deposit based on the results of drilling bores on several points (case study: south mining region of PT. Aneka Tambang Tbk., Pomalaa, Southeast Sulawesi),” Research Report, Halu Oleo University, Kendari, Indonesia, 2010.

K. V. Mardia and C. R. Goodall, “Spatial-temporal analysis of multivariate environmental monitoring data,” in Multivariate Environmental Statistics, G. P. Patil and C. R. Rao, Eds., pp. 347–386, North-Holland, Amsterdam, The Netherlands, 1993.

S. Arnold, “The asymptotic validity of invariant procedures for the repeated measures model and multivariate linear model,” Journal of Multivariate Analysis, vol. 15, no. 3, pp. 325–335, 1984.

R. Pyke, “A uniform central limit theorem for partial sum processes indexed by sets,” in London Mathematical Society Lecture Note Series, vol. 79, pp. 219–240, 1983.

R. F. Bass and R. Pyke, “Functional law of the iterated logarithm and uniform central limit theorem for partial-sum processes indexed by sets,” The Annals of Probability, vol. 12, no. 1, pp. 13–34, 1984.

K. S. Alexander and R. Pyke, “A uniform central limit theorem for set-indexed partial-sum processes with finite variance,” The Annals of Probability, vol. 14, no. 2, pp. 582–597, 1986.

D. W. Stroock, A Concise Introduction to the Theory of Integration, Birkhäuser, Berlin, Germany, 3rd edition, 1999.

J. N. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley & Sons, New York, NY, USA, 3rd edition, 2007.

K. B. Athreya and S. N. Lahiri, Measure Theory and Probability Theory, Springer, New York, NY, USA, 2006.

A. W. van der Vaart, Asymptotic Statistics, vol. 3, Cambridge University Press, London, UK, 1998.

E. L. Lehmann and J. P. Romano, Testing Statitical Hypotheses, Springer, New York, NY, USA, 3rd edition, 2005.

H. Pruscha, Vorlesungen über Mathematische Statistik, B.G. Teubner, Stuttgart, Germany, 2000.

J. Shao and D. S. Tu, The Jackknife and Bootstrap, Springer, New York, NY, USA, 1995.

J. Yeh, “Cameron-Martin translation theorems in the Wiener space of functions of two variables,” Transactions of the American Mathematical Society, vol. 107, no. 3, pp. 409–420, 1963.

J. Elstrodt, Maß- und Integrationstheorie, vol. 7 of Korregierte und Aktualisierte Auflage, Springer, Berlin, Germany, 2011.