makalah

Regional Conference on Applied and Engineering Mathematics 2010Vol. 1, No. 33, pp. 185-190

Examining the Properties of the Estimators of the Spatial Unilateral Autoregressive Model by Bootstrapping

Norhashidah Awanga and Latifah Rahayub

a,bSchool of Mathematical Sciences,Universiti Sains Malaysia, 11800 USM, Penang, Malaysia

Email : [email protected]

ABSTRACT

Many methods and procedures have been developed and proposed to overcome the estimation problem in spatial modeling, for example, by defining classes of models called separable models and the unilateral models. A special type of spatial models that received much attention is the spatial unilateral autoregressive models denoted as AR(p1,1) model. Several procedures have been proposed to estimate the parameters of this model. These include the Yule-Walker, the least squares and the maximum likelihood methods. In this paper, we examine the properties of these estimators for the first-order model, i.e. the AR(1,1) model. The examination is done by measuring the accuracy of these estimators for AR(1,1) model based on the value of the standard error of the estimates. To achieve this objective, bootstrap method based on resampling the residuals is used to obtain the standard error of the estimates.

Keywords: Bootstrap, spatial modeling, spatial autoregressive model, standard error.

1.0 INTRODUCTION

Spatial models are used to model the events or observations which are referenced by location coordinates or observed over a lattice. There are several spatial models that have been proposed to study the correlation of spatial data. Two classes of spatial models that have attracted many researchers are the separable models and the unilateral models. A special characteristic of separable models is that it has a product correlation structure, which in turn simplifies the estimation, whereas, the unilateral models can be analyzed using extension of time series theory in some special cases.

Of the unilateral models, the spatial unilateral autoregressive model received much attention. This model is denoted as AR (p1,1) and is defined by

ijjpipjijiij YYYY εααα ++++= −−−− 1,1,1,01,110 11 (1)

where ijY is an associated random variable and ijε is an

independent and identically random variable with ( ) 0=ijE ε

and ( ) 2σε =ijVar at site labeled (i,j) and ijα s are parameters

to be estimated for processes on rectangular grid of size m×n.

Many procedures have been developed to estimate the parameters of this model. Tjøstheim (1978) discussed about the extension of the Yule-Walker method in time series analysis to spatial series. However, Guyon (1982), Basu and Reinsel (1992) and Ha and Newton (1993) showed that the autocovariance function used in this Yule-Walker estimate is asymptotically biased and proposed the unbiased version of the estimate. Basu (1991) discussed about two methods of the conditional least squares procedure for estimating the parameters of the AR(1,1) model. Awang (2005) introduced an alternative method using the maximum likelihood method with certain modification at the border to obtain the estimate of the parameters of the AR(p1,1) model.

In this paper, we examine the properties of the estimators of the first-order spatial unilateral autoregressive model using the bootstrap method. Bootstrap is a computer-based method which helps to simplify the theoretical analysis of the estimator. Our specific objective is to compare the accuracy of the Yule-Walker, the unbiased version of Yule-Walker, the least squares and the maximum likelihood estimators of the AR(1,1) model defined by

),()1,1(11)1,(01),1(10 jijijijiij YYYY εααα +++= −−−− . (2)

The standard error of the estimators is used as the measure of accuracy. To achieve this objective, bootstrap method based on resampling the residuals is used to obtain the standard error of the estimates.

The next section reviews the estimation methods for the AR (p1,1) model, namely the Yule Walker method, the maximum likelihood method and the least squares method. Section 3 briefly discussed about the bootstrap method while in Section 4, the methodology applied in our analysis is given in detail. Section 5 presents the results of the analysis and in Section 6 the numerical example, some conclusions and direction for future research are given in section 7.

2.0 ESTIMATING THE PARAMETERS OF THE SPATIAL UNILATERAL AUTOREGRESSIVE MODEL

2.1 Yule Walker Method


Tjøstheim (1978) considered spatial Yule-Walker estimators to estimate the parameters of the spatial AR (p1, p2) models. The biased sample autocovariance function at lag (s,t) and (s,-t) for s ≥ 0 and t ≥ 0 is defined by

∑∑−

=

−

=++=

sm

i

tn

jtjsiijYY

mntsR

1 1,

1),(ˆ (3)

and

∑ ∑−

= +=−+=−

sm

i

n

tjtjsiijYY

mntsR

1 1,

1),(ˆ

(4)

For a spatial unilateral AR(p1,p2) model, the spatial analogue of the one-dimensional Yule-Walker equation as in time series is given as,

∑∑= =

−−=1 2

0 0

),(),(p

k

p

lkl ltksRtsR α )0,0( ≥≥ ts . (5)

If we define

,)',,,,,,,,(21211 ,,01,010,10 ppppp αααααα =α

( ) '),(,),,0(,),1,(,,),1,0(),0,(,),0,1( 21211 ppRpRpRRpRR =r ,

and

−−

−−−−

=

)0,0(),2(),1(

),2()0,0()0,1(

),1()0,1()0,0(

2121

21

21

RppRppR

ppRRR

ppRRR

R ,

then, the spatial Yule Walker estimator of α is given by

1ˆ −=α R r . (6)

The Yule-Walker equations are then be solved with the R̂ ’s replacing the R’s. The resulting estimators, denote by α̂ , are called the Yule-Walker estimators. Guyon (1982) compares

),(ˆ tsR with the unbiased estimator of ),(),( , tjsiijYYEts ++=γ

R (s,t) = mn ),(ˆ tsR / (m-s) (n-t), and claimed that the unbiased estimator is preferred.

In the Yule-Walker and unbiased Yule-Walker methods, for AR (1,1) model as defined in equation (2), the matrices α, r, and R take the form

( ) ',, 110110 ααα=α , ( ) ')1,1(),1,0(),0,1( RRR=r and

−

−=

)0,0()0,1()1,0(

)0,1()0,0()1,1(

)1,0()1,1()0,0(

RRR

RRR

RRR

R .

Then, the spatial Yule-Walker estimator of α is given by equation (6). To obtain the unbiased Yule-Walker estimate, the divisor mn is replaced by (m-s)(n-t).

2.2 Least Squares Method

Two types of conditional least squares estimation may be used to estimates the parameters of the spatial unilateral AR model (see Basu, 1991). For Type 1, for estimating the parameters of the AR(1,1) model, we assume that the unobserved border values are all zeros, that is

( ) 0Yb == −− 010000,10,1 ,,,,,,,,' mnn YYYYYY . The least

squares estimate of ( )110110 ,,' ααα=α is given as

( ) YX'XX''α 1−=ˆ , (7)

where

( ) ',,,,,,,,,,,, 212222111211 mnmmnn YYYYYYYYY =Y

and X is a matrix of dimension (mn) ×3 given as,


=

−−−−

−−

−

−−

−−

−

1,11,,1

1,11,2,1

1,1

1,21,32

213122

21

1,11,21

112112

11

1,1

11

00

00

00

00

00

000

nmnmnm

mmm

m

nnn

nnn

n

YYY

YYY

Y

YYY

YYY

Y

YYY

YYY

Y

Y

Y

X

In Type 2, the conditional least squares estimates are obtained by conditioning on the given observed border,

( ) ',,,,,,,, 2111211'

mnmmn YYYYYY =oY . Then, the

estimator is given as,

( ) ( )1o1

ooo Y'XX'Xα −=ˆ (8)

where

( ) ),,,,,,,,,( 2442332 mnmnn YYYYYY ='1Y ,

and 0X is a matrix of dimension (m-2)(n-1) × 3 defined as,

=

−−−−

−−

−−

−−

1,11,,1

1,112,1

1,31,43

314132

1,21,32

213122

nmnmnm

mmm

nnn

nnn

YYY

YYY

YYY

YYY

YYY

YYY

0X

2.3 Maximum Likelihood Method

The spatial autoregressive models can be estimated by the method of maximum likelihood (see Awang, 2005 for details). For AR(1,1) model, the estimate of the parameters can be obtained by solving the equation

=

−

11

01

10

1

11'1111

'0111

'10

01'1101

'0101

'10

10'1110

'0110

'10

11

01

10

'

'

'

ˆ

ˆ

ˆ

ZY

ZY

ZY

ZZZZZZ

ZZZZZZ

ZZZZZZ

ααα

. (9)

Here, YWZYWZ 01011010 , == and YWZ 1111 = . The matrices

110110 and, WWW are the N×N weight matrices given as

===

0D0000

0000D0

00000D

000000

W

D00000

000D00

0000D0

00000D

W

0I0000

0000I0

00000I

000000

W

1 0,

1 0,

1 0

I is an n×n identity matrix and D is an n×n matrix defined by

010000

000010

000001

000000

.

3.0 BOOTSTRAP

The bootstrap is a general methodology for answering how accurate an estimator is. This method was first introduced by Efron (1979) as a method to estimate the standard error of an arbitrary estimator. For independent data, a bootstrap sample is a random sample of size n taken with replacement from

data y , that is nyyy ,,, 21 and assigns probability mass

of n1 on each observation. These steps are repeated B times.

Here, the non-parametric bootstrap is used to estimate the standard error and confidence intervals of the estimators. In the parametric bootstrap, the process of resampling is

performed on the observed data nyyy ,,, 21 which come

from a parametric model. For dependent data, the procedure has different context of sampling and there is no obvious way

to estimate a general joint density for nyyy ,,, 21 .

In this paper, we discuss about the bootstrap methods for examining the properties of the estimators of the spatial AR(1,1) model. We apply the non-parametric bootstrap of the residuals for estimating the standard error of the estimates. This method is an extension of the bootstrap method used in time series analysis (refer to Efron and Tibshirani, 1986).

4.0 METHODOLOGY


The following steps are applied to obtain the bootstrap estimate of the standard error of the AR(1,1) parameters:

1. The grid size, m x n and value of parameters are chosen. The parameters must satisfy the stationary conditions stated in Basu and Reinsel (1993). This model is stationary if and only if

1) ,1,, 110110 <ααα

2) ,1 1101102

01 αααα +>−

3) ( ) ( ) 041 2110110

2211

201

210 >+>−−+ αααααα .

2. The series { }ijε , i =1, 2, …, m and j = 1, 2, …,n will then be generated from independent standard normal with

mean 0 and variance 1. The border values of { }ijY will

then be determined by assuming that the cells bordering the lattice have fixed values of zeroes. Then the remaining values of { }ijY are obtained recursively from equation (2).

3. The estimates of the parameters will be obtained by the estimation procedures as discussed in section 2.

4. The estimate of residuals { }ijε̂ will be obtained from

ijijij YY ˆˆ −=ε , where 1,1111,01,110 ˆˆˆˆ

−−−− ++= jijijiij YYYY ααα.

5. A bootstrap sample { }*ijY will be created from { }*

ijε ,

where { }*ijε is obtained by sampling with replacement

from the residual { }ijε̂ .

6. The parameter will be reestimated from { }*ijY by the

same estimation procedure as in step (3). The estimated will be denoted as ( )*

1,*10

*

1,, pααα = .

7. The steps 1-7 will be repeated B times (B = 10, 50, 100, 200, 500, 1000 and 1500).

8. Finally, the standard deviation of the estimate is

calculated by * *

ˆ

( )

1s Bαα α−

= −� .

5.0 RESULTS

This aims of this research are to examine the properties of the bootstrap estimates for the parameter of AR(1,1) model. Comparison is made among the estimates obtained from the Yule-Walker method, the least squares and the maximum likelihood methods. The comparison is made based on the size of standard error of the estimates. The programmes to perform the bootstrapping are written in S-Plus.

The results show that, for B = 10, B = 50 and B = 100, the standard errors are large but as B increases, the standards errors become smaller. For B = 200, the standard errors become more stable and when B = 500, 1000 and B = 1500, the standard errors are not too different from the standard errors of B = 200. Therefore, we decide to present the results for B = 200 and B = 500 because of the stability of the standard error value and the distribution of the standard errors are more symmetric when B equals 500 and larger.

The bootstrap replication are performed for the grid size (10×6), (8×8), (10×8), (15×6), (10×10), (20×5), (20×10), (15×15), (20×15), (20×16), (30×12), (25×15), and (20×20), and the α-values chosen are (i) 2.0,2.0,3.0 110110 === ααα ;

(ii) 1.0,3.0,5.0 110110 === ααα ;

(iii) 6.0,7.0,8.0 110110 −=== ααα . The grid sizes are chosen in such a away that its represent the small, moderate and large grid sizes and these grids have been used by previous researchers (Basu and Reinsel, 1992). Additional grid sizes such as (10×10), (20×10), (15×15) and (20×15) are added to represent moderate and square grid sizes. Three set of α-values selected satisfy all stationary conditions and these values also have been used by Basu and Reinsel (1992).

Figure 1 displays the bootstrap estimate of the standard errors for α’ = (0.3, 0.2, 0.2) and B = 200. We can see that the Yule-


Walker estimate, especially for 11α has smallest standard errors, followed by the maximum likelihood estimate; whereas the least squares estimate has largest standard errors. Generally, the standard error is large for small grid sizes but as the grid size increases, the standard error become smaller.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

50 100 150 200 250 300 350 400 450

grid size

std

. de

v

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

50 100 150 200 250 300 350 400 450grid size

std

.de

v

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

50 100 150 200 250 300 350 400 450

grid size

std

.de

v

Figure 1: Bootstrap estimate of the standard error for 10α ,

01α and 11α with B = 200 and α’ (0.3, 0.2, 0.2), ( :

Biased yw, : unbiased yw, = MLE, × : least squares )

Figures 2 is the plot of bootstrap estimate for standard errors with B = 200 and α’ = (0.5, 0.3, 0.1). It can also be seen that the Yule-Walker estimates, especially for 11α has smallest standard errors and the least square estimates has largest standard errors. Again, as the grid size increases, the standard error getting smaller.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

50 100 150 200 250 300 350 400 450

grid size

std

.dev

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

50 100 150 200 250 300 350 400 450

grid size

std

dev

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

50 100 150 200 250 300 350 400 450

grid size

std

.dev

Figure 2: Bootstrap estimate of the standard error for 10α , 01α

and 11α with B = 200 and α’= (0.5, 0.3, 0.1), ( :

Biased yw, : unbiased yw, = MLE, ×: least squares)

Figure 3 is the plot of bootstrap estimate standard errors with

B = 200 for α’ = (0.8, 0.7, -0.6), we can see that for 10α , the

maximum likelihood estimates has smallest standard error and unbiased Yule-Walker estimates has largest standard

error, but for 01α the maximum likelihood estimates has

smallest standard errors and the least squares estimates has

largest standard errors. For 11α , the Yule-Walker estimates

has smallest standard error and the unbiased Yule-Walker estimates has largest standard error.

0

0 .02

0 .04

0 .06

0 .08

0 .1

0 .12

0 .14

0 .16

50 100 150 200 250 300 350 400 450

g rid s ize

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

50 100 150 200 250 300 350 400 450

grid size

std

. de

v

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

50 100 150 200 250 300 350 400 450

grid size

std

.dev

Figure 3: Bootstrap estimate of the standard error for

10α , 01α and 11α with B = 200 and α’= (0.8,

0.7, -0.6), ( : Biased yw, : unbiased yw, = MLE, ×: least square )

The results also show that the standard errors for B = 500 are not too different from the standard errors for B = 200 for all parameters and grid sizes. However, the distribution of the bootstrap estimate for B = 500 is more symmetric than the distribution of the bootstrap estimate when B = 200, indicating that the distribution of the estimate will be more symmetric if B increases (refer to the histogram plots in figures 4 and 5).

From the results above, we conclude that when the grid size increases, the standard error becomes smaller and if the number of bootstrap replications is very small (less than 100), the standard error is large. When the number of replications is more than 100, the standard error is stable (does not change much) and we conclude that B = 200 is enough for estimating the standard error. However, when B = 200 and less, the distribution of the bootstrap estimate is slightly skewed, and symmetric distribution can only be attained when we increase the number of replications. In this paper, we present the results for B = 200 and B = 500 for the purpose of comparing the distribution of the bootstrap estimates. The results for

)2.0,2.0,3.0(=α and )1.0,3.0,5.0(=α shows that the Yule-Walker estimate have smallest standard error followed by maximum likelihood estimate, unbiased Yule-Walker and lastly by least squares estimate. However for

)6.0,7.0,8.0( −=α especially for 10α , the maximum likelihood estimate has smallest standard error followed by least square estimate, Yule-Walker estimate and lastly by

unbiased Yule-Walker estimate. For 01α , the maximum

likelihood estimate has smallest standard error followed by Yule-Walker estimate, unbiased Yule-Walker estimate and lastly by least squares estimate. For 11α , the Yule-Walker estimate has smallest standard error, followed by maximum likelihood estimate, least square estimate and lastly the unbiased Yule-Walker estimate.


B = 200Grid size = 20x20

Bootstrap Yule-Walker method Bootstrap Unbiased Yule-Walker method α10 = 0.8 α01 = 0.7 α11 = -0.6 α10 = 0.8 α01 = 0.7 α11 = -0.6

0.60 0.65 0.70 0.75 0.80

010

20

30

40

alphatstar[i, ]

0.45 0.50 0.55 0.60 0.65

010

2030

40

alphatstar[i, ] -0.55 -0.50 -0.45 -0.40 -0.35 -0.30

010

2030

alphatstar [i, ] 0.70 0.75 0.80 0.85

01

02

03

04

0

alphatstar[i, ]

0.60 0.65 0.70 0.75

010

2030

40

alphatstar[i, ]

-0.7 -0.6 -0.5 -0.4

02

04

06

0

alph at star [i, ]

Bootstrap Least Square method Bootstrap Maximum Likelihood method α10 = 0.8 α01 = 0.7 α11 = -0.6 α10 = 0.8 α01 = 0.7 α11 = -0.6

0.70 0.75 0 .80 0.8 5

01

02

03

04

05

0

rh hatstar[i, ]

0.50 0.55 0.60 0.6 5 0. 70

01

02

03

04

0

rh hatstar[i, ]

-0.7 -0.6 -0.5 -0.4

02

04

06

0

rhha tstar[i, ] 0.70 0. 75 0.80 0. 85

01

02

03

04

05

0

rhhatstar[ i, ]

0 .55 0.60 0.65 0.70

01

02

03

04

0

rh hatstar[i, ]

-0.7 -0.6 - 0.5 -0.4

020

4060

80

rhhatstar[i, ]

Figure 4 : Histogram of B = 200 bootstrap replications of 10α , 01α

and 11α , respectively for α = (0.8, 0.7, -0.6) and 20×20

grid size.

B = 500Grid size = 20x20

Bootstrap Yule-Walker method Bootstrap Unbiased Yule-Walker method α10 = 0.8 α01 = 0.7 α11 = -0.6 α10 = 0.8 α01 = 0.7 α11 = -0.6

0.55 0.60 0.6 5 0.70 0.7 5 0.80

02

04

06

08

01

00

alpha t star [i, ]

0.4 5 0.50 0.55 0.60 0.65 0.70

02

04

06

08

01

00

alphat star [i, ]

-0.55 -0.5 0 -0.45 -0. 40 -0 .35 -0.30

02

04

06

08

0

alpha t star [i, ] 0.65 0.70 0.75 0.80 0.85

020

40

60

80

100

120

alphatstar[i, ]

0.50 0.5 5 0.60 0.6 5 0.70 0.7 5 0.80

02

04

06

08

01

00

alpha tst ar[i, ]

-0 .7 -0.6 -0.5 -0.4

02

04

06

08

0

alpha tst ar[i, ]

Bootstrap Least Square method Bootstrap Maximum Likelihood method α10 = 0.8 α01 = 0.7 α11 = -0.6 α10 = 0.8 α01 = 0.7 α11 = -0.6

0.65 0.70 0.75 0.80 0.85

020

4060

8010

0

rhhatstar[i, ]

0.45 0.5 0 0.55 0.60 0.65 0.70 0 .75

02

04

06

08

01

00

r hhatstar[i, ]

-0.7 -0.6 -0.5 -0.4 -0.3

050

100

150

rhhatstar[i, ] 0.65 0.70 0.75 0.80 0.85

020

4060

8010

0

rhhatstar[i, ]

0.50 0.55 0.60 0.65 0.70 0.75

020

4060

8010

0

rhhats tar[i, ]

-0.70 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40

020

4060

8010

0

rhhatstar[i, ]

Figure 5: Histogram of B = 500 bootstrap replications of 10α , 01α

and 11α , respectively for α = (0.8, 0.7, -0.6) and 20×20

grid size.

6.0 NUMERICAL EXAMPLE

The numerical example is presented here to illustrate the procedure of the bootstrap estimate of the standard error of the parameters by fitting the AR(1,1) models to real data sets. The data set is the yield of wheat on 20×25 regular grid. We obtain this data set from Cressie (1991) but the original version is presented in Mercer and Hall (1911). The analysis is started by estimating the parameters using the above methods, namely the Yule-Walker, the unbiased Yule-Walker, the least squares and the maximum likelihood methods. The comparison among the estimates from these methods will then be made based on the value of the standard error which is obtained by bootstrapping method with B = 200 and 500.

Table 1 displays the value of estimates of the Yule-Walker, unbiased Yule-Walker, Least squares and Maximum likelihood methods with its standard error (in bracket). It shows that the Yule-Walker estimator has smallest standard

error as compared to unbiased Yule-Walker, maximum likelihood and least squares estimators. Figures 6 and 7 display the histograms of the estimates. It is clear that the distribution of the standard errors for B = 500 is more symmetric than the distribution of the standard errors of B = 200. This also indicates that the distribution will be more symmetric if B increases.

Table 1 : Estimates of the parameters with it estimated standard error (in bracket) from bootstrapping for wheat data.

Bootstrapreplication

Estimationmethod 10α̂ 01α̂

11α̂

B = 200

YWUnbiased YWLSML

0.2127 (0.0411)0.2116 (0.0430)0.2284 (0.0476)0.2228 (0.0443)

0.4693 (0.0352)0.4923 (0.0448)0.5218 (0.0413)0.5060 (0.0398)

-0.0696 (0.0432)-0.0707 (0.0513)-0.1031 (0.0437)-0.0927 (0.0433)

B = 500

YWUnbiased YWLSML

0.2127 (0.0440)0.2116 (0.0456)0.2284 (0.0442)0.2228 (0.0451)

0.4693 (0.0400)0.4923 (0.0386)0.5218 (0.0410)0.5060 (0.0378)

-0.0696 (0.0416)-0.0707 (0.0478)-0.1031 (0.0471)-0.0927 (0.0458)

B = 200 Bootstrap Yule-Walker method Bootstrap Unbiased Yule-Walker method

0 .1 0 0.1 5 0.2 0 0.2 5 0.3 0

010

2030

est ar[ i, ]

0.35 0.40 0.45 0.50

010

20

30

40

estar[i, ]

-0.15 -0.10 -0.05 0.0 0.05

010

20

30

40

estar[i, ] 0.10 0.15 0.20 0.25 0.30

010

20

30

40

estar[i, ]

0.35 0.40 0.45 0.50 0.55 0.60

010

20

30

estar[i, ]

-0.2 -0.1 0.0 0.1

020

40

60

estar[i, ]

Bootstrap Least Square method Bootstrap Maximum Likelihood method

0.10 0.15 0.20 0.25 0.30 0.35

010

2030

estar[i, ]

0.40 0.45 0.50 0.55 0.60 0.65

010

2030

40

estar[i, ]

-0.25 -0.20 - 0.15 -0.10 -0.05 0.0

010

2030

estar[i, ] 0.15 0.20 0.25 0.30

010

2030

estar[i, ]

0.40 0.45 0.50 0.55 0.60

010

2030

40

estar[i, ]

-0.20 -0.15 - 0.10 -0.05 0.0 0.05

010

2030

4050

es tar[i, ]

Figure 6: Histogram of the distribution of the bootstrap replications

of 10α , 01α and 11α , respectively for wheat data (B =

200).

B = 500 Bootstrap Yule-Walker method Bootstrap Unbiased Yule-Walker method

0.05 0.10 0.15 0.20 0.25 0.30 0.35

020

4060

8010

0

estar[i, ] 0.35 0.40 0.45 0.50 0.55

020

4060

8010

0

estar[i, ]

-0.20 -0.15 -0.10 -0.05 0.0 0.05

020

4060

8010

0

es tar[i, ] 0.05 0.10 0.15 0.20 0.25 0.30 0.35

020

4060

80

estar[i, ]

0.40 0.45 0.50 0.55 0.60

020

4060

8010

0

estar [i, ]

- 0.20 -0.15 -0.10 -0.05 0.0 0.05

020

4060

80

estar[i, ]

Bootstrap Least Square method Bootstrap Maximum Likelihood method

0.10 0.15 0.20 0.25 0.30 0.35 0.40

020

4060

80

estar[i, ]

0.40 0.45 0.50 0.55 0.60 0.65

020

40

60

80

100

estar[i, ]

-0.2 -0.1 0.0

020

4060

80

estar[i , ] 0.10 0.15 0.20 0.25 0.30 0.35

020

40

60

80

100

estar[i, ]

0.40 0.45 0.50 0.55 0.60

020

40

60

80

100

estar[i, ]

- 0.2 -0.1 0.0

020

4060

80

estar[i, ]

Figure 7: Histogram of the distribution of the bootstrap replications

of 10α , 01α and 11α , respectively for wheat data (B =

500).

7.0 CONCLUSION

In this paper, we have applied the bootstrapping procedure to obtain the standard error of the estimates of the parameters of the spatial unilateral AR(1,1) model. We make comparison among the estimates obtained from the Yule-Walker method, the least squares and the maximum likelihood methods based on their standard error value. The results show that, generally the Yule-Walker estimate has smallest standard error and the least square estimate has largest standard error. For


illustration, we apply the procedure to wheat data and make comparison among the estimators.

REFERENCES

Awang, N and Shitan, M. (2008), Maximum likelihood estimation for the non-separable spatial unilateral autoregressive model, Proceeding of The 3rd

International Conference on Mathematics and Statistics (ICoMS-3), Institut Pertanian Bogor, Indonesia.

Basu, S., and Reinsel, G. C. (1992). A note on the properties of spatial Yule-walker estimators. Journal of Statistical Computing and Simulation, 41, 243-255.

Basu, S. and Reinsel, G. C. (1993), Properties of the spatial unilateral first order ARMA model, Advance in Applied Probability, 25, 631-648.

Cressie, N. A. C. (1991), Statistics for Spatial Data, New York : Wiley.

Davison, A. C. and Hinkley D. V. (1997), Bootstrap Methods and Their Application, Cambridge University Press, USA.

Efron, B. and Tibshirani, R. J. (1993), An Introduction to the Bootstrap. New York : Chapman & Hall.

Guyon, X. (1982), Parameter estimation for a stationary process on a d-dimensional lattice, Biometrika, 69, 95-105.

Ha, E. and Newton, H. J. (1993). The bias of estimators of causal spatial autoregressive processes, Biometrika, 80, 1, pp. 242-5.

Mercer, W.B. and Hall, A.D. 91911). The experimental error of field trials, Journal of Agriculture Science, 4, pp. 107-132.

Tjøstheim, D. (1978), Statistical spatial series modelling, Advances in Applied Probability, 10, 130-154.

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