A New Method of Construction of Robust Second Order Slope Rotatable Designs Using Pairwise Balanced Designs

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A New Method of Construction of Robust Second Order

Slope Rotatable Designs Using Pairwise Balanced Designs

Bejjam Re. Victorbabu, Kottapalli Rajyalakshmi

Department of Statistics, Acharya Nagarjuna University, Guntur, India Email: [email protected], [email protected]

Received May 10,2012; revised June 12, 2012; accepted June 28,2012

ABSTRACT

In this paper, a new method of construction of robust second order slope rotatable designs (RSOSRD) using pairwise balanced designs (PBD) is suggested and also obtained the variance of the estimated derivatives for the factors 6 ≤ v ≤ 15. It is shown that the new method sometimes leads to designs with less number of design points compared to designs constructed with the help of balanced incomplete block designs (BIBD).

Keywords: Response Surface Designs; Slope Rotatability; Correlated Errors; Robustness; Second Order Slope

Rotatable Designs (SOSRD)

1. Introduction

In response surface methodology, rotatability is a natural and highly desirable property. This was introduced and developed by Box and Hunter [1] assuming the errors to be uncorrelated and homoscedastic. Das and Narasim-ham [2] constructed second order rotatable designs (SORD) through balanced incomplete block designs (BIBD). Tyagi [3] constructed SORD using pairwise balanced designs (PBD). Panda and Das [4] studied first order rotatable designs with correlated errors. In order to study the na-ture of robust rotatable designs, rotatability conditions for second order regression designs have been derived, assuming the errors to be correlated. These conditions have been further studied under different variance co-variance structures of errors. Das [5,6] introduced robust second order rotatable designs (RSORD). Rajyalakshmi and Victorbabu [7] constructed robust rotatable central composite designs (RRCCD) for factors 2 ≤ v ≤ 17. Vic-torbabu and Rajyalakshmi [8] constructed a new method of construction of robust second order rotatable designs using BIBD. Victorbabu and Rajyalakshmi [9] studied a new method of construction of robust second order ro-tatable designs using PBD.

In response surface methodology, good estimators of the derivatives of the response function may be as im-portant or perhaps more imim-portant than estimation of mean response.

Estimation of differences in responses at two different points in the factor space will often be of great impor-tance. If a difference in responses at two points close together is of interest then estimation of local slope (rate of change) of the response is required. Estimation of

slopes occurs frequently in practical situations. For in-stance, there are cases in which we want to estimate rate of reaction in chemical experiment, rate of change in the yield of a crop to various fertilizer doses, rate of disinte-gration of radioactive material in animal etc. [9].

Murty and Studden [10] suggested optimal designs for estimating the slope of a polynomial regression. Hader and Park [11] introduced slope rotatable central compos-ite designs assuming errors are uncorrelated and homo-scedastic. Park [9] studied a class of multifactor designs for estimating the slope of response surfaces. Victorbabu and Narasimham [12] constructed second order slope rotatable designs (SOSRD) using BIBD assuming errors are uncorrelated and homoscedastic. Victorbabu and Na-rasimham [13] constructed SOSRD using PBD. Several authors have studied slope rotatable designs assuming errors to be uncorrelated and homoscedastic. However it is not uncommon to come across some practical situa-tions when the errors are correlated, violating the usual assumptions. Specifically Das [14] introduced the con-cept of slope rotatability with correlated errors, which requires that the variance of the estimated derivative to be constant, independent of correlation parameter in-volved in the variance-covariance structure of errors. They have studied slope rotatability conditions for a second order design with correlated errors. Victorbabu and Rajyalakshmi [15] studied robust slope rotatable central composite designs (RSRCCD) for the factors 2 ≤

v ≤ 8. Further, Victorbabu and Rajyalakshmi [16] studied robust slope rotatable designs (RSOSRD) using BIBD for the factors 3 ≤ v ≤ 8.

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SRD using PBD and also obtained the variance of the estimated derivatives for factors 6 ≤ v ≤ 15.

2. Second Order Response Surface Designs

with Correlated Errors

Assuming that the response surface is of second order, we adopt the model:

2 0

1 1

v v

u i iu ii iu

i i

Y  x  x

 

 









_{ }

1

v

ij iu ju u i j

x x e



 



, , ,

(2.1)

where xiu denotes the level of the ith_{factor (}_i_{= 1,2, ···,}_v₎ in the uth_{run (}_u_{= 1,2, ···,}_N_{) of the experiment, e}

u’s are

correlated errors. Here    0 i ii ij are the parameters

of the model and is the observed response at the uth design point. u

Y

Second Order Slope Rotatable Designs with Correlated Errors

Following Hader and Park [11], Victorbabu and Nara-simham [12], Das [14], the necessary and sufficient con-ditions for slope rotatability in second order model with correlated errors are as follows:

Conditions for Second Order Slope Rotatable Designs with Correlated Errors.

The estimated response at x is given by 2

0

1 1

ˆ ˆ ˆ

ˆ v v

1

ˆ v

x i i i i

i i i j ij i j

x x  x x

 



 

y   

 

 







(2.2)

For the second order model as in (2.2), we have

1; 1

ˆ v

i j ij j

j j

ˆ _ˆ _ˆ

2 x

i ii

i

x  x

  



x y x    _ _

 (2.3)

The variance of ˆx i

x y



 is given by,

 



 









The variance of estimated first order derivative with respect to each independent variable xi as in (2.4) to be a







2

1; 1 1;

1;

. 2 . .

2 . 1;

4 4

2 ,

4

ˆ _ˆ _ˆ _ˆ

ˆ ˆ ˆ ˆ ˆ ˆ , 4 4 x

i i ii i i

i v v , , ˆ ˆ ˆ ii

j ij j s

j j i j s j s i

v

j i ij

j j i v

i j ii ij j j i

i i ii ii i ii x i i i ij ij j j j

V V x V x Co

x

x V x x

x Cov

x x Cov

V v x v x v

x

v y

x

y _ _

ij is v Cov   

_

  is j 1 2 v 2

iu i                      _         _ _ _ _    



. 1 1 . . 1; 1; 2 4 v v ij j s

i j s j s i

v v

i ij ii ij

j i

j j i j j i

x x v

x v x x v

          



 



(2.4) d  

function of



x if and only if,

. _0;1 _, . _0;

i ii i ij

v i v v

1)    

1i j v i,  ,  j

. _0;1 _,

ij ij

v  i j j v

2)     

. _0;1 _, _,

ii ij

v  i j v i  j

. _constant;1 _,

i i

v i v

3)   

. _constant;1 _,

ii ii

v i v

4)   

. _{constant 1} _{, and}

ij ij

v    i j v

5)

6) . 1 . _;1 _.

4

ii ii ij ij

v  v   i j v

0.j 0.jl 0;1 j

v v v

(2.5) The following are the equivalent conditions of (1) through (5) in (2.5) for slope rotatability in secondorre-lated errors model (2.1).

1)*: a)    

. 0

i j v

;

 ; 1≤ i, j ≤ v, i ≠ j; b)

. 0

ii j

v  ; 1 ≤ i, j ≤ v; c): i)

. 0

i jl

v  ; 1 ≤ i, j < l ≤ v; ii)

  

.. 0

ii jl

v  ; 1 ≤ i, j < l ≤ v, j l,  i i,



.. 0 ii jl v ; iii)

  

 ;

iv) 1i l,  j t v i j,  , ,  l t,



0.

; 2)*: a) v _jj a₀

.

i i v

= constant = , say; 1 ≤ j ≤ v; b) = constant = 1

e

.

ii ii v

, say; 1 ≤ i ≤ v;

c) = constant = 2 f g _  _   . ii jj v . ij ij

, say; 1 ≤ i ≤ v; 3)*: a) = constant = f; 1 ≤ i ≠ j ≤ v;

b) v = constant = 1; 1 ≤ i < j ≤ v. (2.6)

g

Following (2.6), the necessary and sufficient condi-tions for second order slope rotatability under the intra- class structure after some simplifications turn out to be

3

1 2 4

1 2 3 4

1

0;

N

i u i u i u i u u

x x x x   



1)*:



 4

1 i

i for any αi odd and 





≤ 4. 2 1 constant,1 N iu u

x i v



2)*: a)



  

4 1 constant,1 N iu u ;

x i v



b)



  

2 2 1

constant,1 , , ;

N

iu ju u

;

x x i j v i j



3)*



   

2 2 1 , 1 N iu u (2.7)

Using x N i v



  



2 2

4 1 , N iu ju u

x x N

and 





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1i j v i,  ,  j









, the second order slope rotatable design

parameters under the intra-class structure are as follows:









1) 2

1 1 N N 0 2 a      









;

2)













2 2 4 2 1 N N 2 1 1 1 1 N N

f    

          ; 3)



2



2 1 1 N e      ; 4)



4



2 1 1 N g     









;

5) ₀₀ ₂

1 1 N v N      ;

6)





















2 2 4 2 2

1 1 3

1 1 1

N N N

g N

2

f     

  

  

 

  

3

1 2 4

1 2 3 4

1

0;

N

i u i u i u i u u

x x x x   

 



4 1 i i _ _

  . (2.8)

Note that if ρ = 0, the conditions (2.7) and (2.8) re-duces to

1): for any αi odd and 





≤ 4.

2): a) 2

2 1

constant , 1

iu u

;

N

x N i v



  

4, 1



b) 4 1

constant

N

iu u

x cN  i v

4,1 , ,



 



;

3) 2 2 1

constant

N

iu ju u

x x N



 



 i j v i  j (2.9) where c = 3η, and 4, 2, η are constants.

The variances and covariances of the estimated pa-rameters of the model (2.1) under the slope rotatability are as follows:

1)

 





0 _T



2

ˆ g f v 1 f _;1

V i v

_  _ 

 

  

 

ˆi ; 1 ;

V  e  i v

 

ˆ ; 1 ;

V  g   i j v

 

;

2)

3) ij

4)

2

00 0

2 ₂ ₁

, 2

ˆ ii

v f v f v a

g V

T f f

g                                   

1 i v;





5) 0

0, ; ;

ˆ ˆ ₁

ii

a

Cov i v

T      









6) 2 0 00 , ; 2

ˆ ˆ ₁

ii jj

a fv

Cov i j v

T f f

g           _ _          (2.10) where





2

00 0

2

1

T v f v f va

g                          ˆ

and other covariance’s are zero.

An inspection of the variance of 0 shows that a

necessary condition for the existence of a non-singular second order design is T > 0 i.e.,

4)*:





2

00 0

2

1 0

v f v f va

g     _ _{ } _ _              

  (2.11)

Using (2.8), the expression 





2

00 0

2 ₁

v f v f va

g       _ _{ } _                









simplified to













4 2 2

2 2 2

2 2

( 1) 1 ( 1)

1 1

( 1)

N

c v N N

N

v N vN

        _                _

Hence the non-singularity condition is





















4

2 2 2

2 2

1 1 1

( 1) 0

c v N N

v N vN

 

   

    



    _ (2.12) For any general second order slope rotatable designs (c.f. Hader and Park [11]), we have

 

_ˆ 1

 

_ˆ _{. .,} . 1 .

4 4

ii ii ij ij

ii ij

V   V  i e v  v (2.13)

On simplification of (2.13), using (2.10), we get

5)*:









 





2

00 00 00 0 00 00

2 _f ₄ 2 ₁ ₄ ₂ ₁









2

0 4 1 0.

v ev f gv f v gva v fg v f v v fg

g g

_  __  _  _    _   

    

a v vfg

   

(2.14)

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





































































 































2 2 4 2

2 2 2

4 2 2

4 4

2 2 2 2

4 2 4 2

4 2 2

2

1 1 3

1

1 1 3 1 1 1

4 2

1 1 1 1

4 2 1

1 1 1

1 1

4 1

N N N

N N N N N

N v v

N N

N N N N N N N

v v

N N

N v v

    



2 2 4 2 4 1 1

N N N                                  _                                _  _      _   _        









           2 2 4 2 4 0 1 1 N N N                  (2.15)

For ρ = 0, Equation (2.15) reduces to (see Victorbabu and Narasimham [12])





2 2

4 v 5 c (c 3)

 _    _₂_v c



 5



4 =0_ (2.16) On simplification of Equation (2.4) using Equation (2.6) and (2.13), we get

2 2

1;

2 2

v

i j

j j i

i

x g

x e gd

          



 

2t k 2k

  t k

 

2t k

 

2t k 2

b v n

1 4 4 ˆ_x i v i g

V e x

x e g y      _     



(2.17)

3. New Method of Construction of RSOSRD

Using Pairwise Balanced Designs

Let there be an equi-replicated PBD with parameters (v,

b, r, k1, k2, ···, kp, λ) and k = sup (k1, k2, ···, kp),

de-note a fractional replicate of in ± levels, in which no interaction with less than five factors is confounded. [1-(v, b, r, k1, k2, ···, kp, λ)] denote the design points

gen-erated from the transpose of incidence matrix of PBD. [1-(v, b, r, k1, k2, ···, kp, λ)] are the b design points

generated from PBD (c.f. Raghavarao [17], pp. 298-300)),

n0 denote the number of central points and (±α, 0, 0, ···, 0) 21_{denote the design points generated from (±}_α_{, 0, 0, ···, 0)} point set.

2

Here we start with usual SOSRD using PBD having “n” (where n = b + 2v) non-central design points involving v-factors. For this n-non-central design points we add (n + 1) (n + n0 = m say) central points in the fol-lowing way.

One central point is placed in between each pair of non-central design points in the sequence, resulting thereby in (n − 1) such central points. The other two cen-tral points are placed one at the beginning and one at the end.

If the number of central points of the usual SOSRD with which we started is greater than (n + 1), the remain-ing central points are placed in any manner, if the num-ber is less, we need to include the requisite numnum-ber of additional central points. Here we examine the non-sin-

gularity condition for the newly constructed design. Let N (N =   ₀

 

2t k

 

2t k _₂t k   

2t k 2

N b  v m

2

) be the number of design points of an SOSRD using PBD with which we started. Out of N, let n be the number of non-central design points and n0 be the number of central points. i.e., N = n

+ n0. Let N1 be the number of design points of the newly constructed RSOSRD using PBD, where N1 = n + m > N. For the SOSRD using PBD with which we started, the following are the moment relations:

Theorem (3.1):

If (v, b, r, k1, k2, ···, kp, λ) is an equi-replicate PB

de-sign and k = sup (k1, k2, ···, kp) denotes a resolution

V fractional replicate of of 2k_{in ± levels and n0 is the}

number of central points, then the design points, [1-(v, b,

r, k1, k2, ···, kp, λ)] U (α, 0, 0, ···, 0) U(m)

give a v-dimensional RSOSRD in ₁

(where m = n + n0) design points, where  is a posi-tive real root (if it exists) of the biquadratic equation,





   

















 





 









 





  2 8 2 1 4 1 2 2 2 2 2 2 1 3 2

8 4 8 2 2 2

12 2 4 16 20 4 2

4 16 20 2

5 9 6 2

4 5 2 0.

t k t k

t k

v N vr vr

v r N v vr

vr v v r

v v r r N

vr v r

                      _   _   _     _         2 (3.1) If at least one positive real root of  exists in (3.1) then the design exists.

c can be obtained from

    4 2 2 2 t k t v r

c  

 

1

2 2

1 2 1

2 2 constant ,

N

t k iu u

x r N

. (3.2)

Proof: For the design points generated from the PBD,

simple symmetry conditions 1), 2), 2 of (2.7) are true. Condition 1) of (2.7) is true obviously. Conditions 2) and 3) of (2.7) are true as follows.

2) a)  



   



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1 i v;

1 4 tant cN , b) 1 4   4

1

2 2 cons

N

t k iu u

x r   

1 i v;

1 4 onstant N , 

  



3) 1 2 2   1

2 c

N

t k iu ju u

x x   

1  i j v

, 

 



(3.3) From 2) b) and 3) of (3.3), we get c given in (3.2). Substituting for  2 4

2

and c in (2.16), and on simplifi-cation we get the fourth degree equation in  given in (3.1)

Corollary: If k1 = k2 = ··· = kp = k, then Theorem 3.1

reduces to the method of construction of RSOSRD using BIBD.

The RSOSRD using pairwise balanced designs values “α” and the variances of estimated slopes for the factors 6 ≤ v ≤ 15 are given in Appendix.

Example: We illustrate the use of Theorem 3.1 by

constructing a RSOSRD for 6-factors with the help of the PB design (v = 6, b = 7, r = 3, k1 = 3, k2 = 2, λ = 1).

[1-(6, 7, 3, 3, 2, 1)] × 23_{U (}_α_{, 0, 0, ···, 0) × 2}1_U₍_m₌ 69) will give a RSOSRD in N1 = 137 design points for six factors. From (3.3), we have

2) a) 1 2 2 1

24 2

N iu u

1 2,1 ;

x  N  i v

1 4,1 ; 

  



b) 1 4 4

1

24 2

N iu u

x cN



  



   i v

1 ;

3) 1 2 2

1 4 1

8 ,

N iu ju u

x x N



 



  i j v

,

(3.4)

Substituting for  ₂ ₄

2 _{68352 0}

  

and c in (2.16), we get the fol-lowing biquadratic equation.

8 6 4

500 1152 6496 6144 . (3.5) (3.5) has only one positive real root α2_{= 2.7093.} It may be pointed out here that this RSOSRD using PBD has only 137 design points for 6-factors, where as the corresponding RSOSRD obtained using a BIB design (v = 6, b = 10, r = 5, k = 3, λ = 2) needs 185 design points. Thus the new method sometimes leads to RSO-SRD us-ing PBD with lesser number of design points than the RSOSRD obtained through BIB designs.

The Appendix gives the appropriate robust slope rota-tability values of the parameter “α” for designs using a PBD, star points and for different number of central points and also variances and covariances of the factors for 6 ≤ v ≤ 15.

4. Acknowledgements

The authors are thankful to the referee and the editor for

the valuable suggestions which helped in improving the quality of the paper.

REFERENCES

[1] G. E. P. Box and J. S. Hunter, “Multifactor Experimental Designs for Exploring Response Surfaces,” The Annals of Mathematical Statistics, Vol. 28, No. 1, 1957, pp. 195- 241. doi:10.1214/aoms/1177707047

[2] M. N. Das and V. L. Narasimham, “Construction of Ro- tatable Designs through Balanced Incomplete Block De- signs,” Annals of Mathematical Statistics, Vol. 33, No. 4, 1962, pp. 1421-1439.

[3] B. N. Tyagi, “On the Construction of Second Order and Third Order Rotatable Designs through Pairwise Bal- anced Designs and Doubly Balanced Designs,” Calcutta Statistical Association Bulletin, Vol. 13, 1964, pp. 150- 162.

[4] R. N. Panda and R. N. Das, “First Order Rotatable De- signs with Correlated Errors,” Calcutta Statistical Asso-ciation Bulletin, Vol. 44, 1994, pp. 83-101.

[5] R. N. Das, “Robust Second Order Rotatable Designs. Part- I RSORD,” Calcutta Statistical Association Bulletin, Vol. 47, 1997, pp. 199-214.

[6] R. N. Das, “Robust Second Order Rotatable Designs. Part- II RSORD,” Calcutta Statistical Association Bulletin, Vol. 49, 1999, pp. 65-76.

[7] K. Rajyalakshmi and B. Re. Victorbabu, “Robust Second Order Rotatable Central Composite Designs,” JP Journal of Fundamental and Applied Statistics, Vol. 1, No. 2, 2011, pp. 85-102.

[8] B. Re. Victorbabu and K. Rajyalakshmi, “A New Method of Construction of Robust Second Order Rotatable De- signs Using Balanced Incomplete Block Designs,” Open Journal of Statistics, Vol. 2, No. 1, 2012, pp. 39-47. [9] S. H. Park, “A Class of Multifactor Designs for Estimat-

ing the Slope of Response Surfaces,” Technometrics, Vol. 29, No. 4, 1987, pp. 449-453.

[10] V. N. Murty and W. J. Studden, “Optimal Designs for Estimating the Slope of a Polynomial Regression,” Jour-nal of the American Statistical Association, Vol. 67, No. 340, 1972, pp. 869-873.

[11] R. J. Hader and S. H. Park, “Slope-Rotatable Central Com- posite Designs,” Technometrics, Vol. 20, No. 4, 1978, pp. 413-417.

[12] B. Re. Victorbabu and V. L. Narasimham, “Construction of Second Order Slope Rotatable Designs through Bal- anced Incomplete Block Designs,” Communications in Statistics—Theory and Methods, Vol. 20, No. 8, 1991, pp. 2467-2478. doi:10.1080/03610929108830644

[13] B. Re. Victorbabu and V. L. Narasimham, “Construction of Second Order Slope Rotatable Designs Using Pairwise Balanced Designs,” Journal of the Indian Society of Ag- ricultural Statistics, Vol. 45, No. 2, 1993, pp. 200-205. [14] R. N. Das, “Slope Rotatability with Correlated Errors,”

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[15] B. Re. Victorbabu and K. Rajyalakshmi, “Robust Slope Rotatable Central Composite Designs,” Paper Submitted for the Possible Publication, 2012.

[16] B. Re. Victorbabu and K. Rajyalakshmi, “Robust Second Order Slope Rotatable Designs Using Balanced

Incom-plete Block Designs,” Paper Submitted for the Possible Publication, 2012.

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Appendix

The Variance of Estimated Derivatives Slopes) for the Factors 6 ≤v≤ 15

v = 6, b = 7, r = 3, k1 = 3, k2 = 2, λ = 1

N1 = 137 λ4 = 0.0584 λ2 =0.2147 c = 4.8351 η = 1.6117 α = 1.6460 ρ

 

ˆ0

V 

 

ˆ

i

V 

 

ˆ

ij

V 

 

ˆ

ii

V  Cov



 ˆ ˆ0, ii





ˆij



ˆ_x

i

ˆ ,ii

Cov x

y V_ _



 

 

0 0.0141σ2_0.0340_σ2_0.1250_σ2_0.0310_σ2 ₋_0.0053_σ2 ₋_0.0013_σ2_0.0340_σ2_{+ 0.1250}_σ2_d2

0.1 0.1127σ2_0.0306_σ2_0.1125_σ2_0.0281_σ2 ₋_0.0047_σ2 ₋_0.0012_σ2_0.0306_σ2_{+ 0.1125}_σ2_d2

0.2 0.2113σ2_0.0272_σ2_0.1000_σ2_0.0250_σ2 ₋_0.0042_σ2 ₋_0.0011_σ2_0.0272_σ2_{+ 0.1000}_σ2_d2

0.3 0.3099σ2_0.0238_σ2_0.0875_σ2_0.0219_σ2 ₋_0.0037_σ2 ₋_0.0009_σ2_0.0238_σ2_{+ 0.0875}_σ2_d2

0.4 0.4084σ2_0.0204_σ2_0.0750_σ2_0.0187_σ2 ₋_0.0032_σ2 ₋_0.0008_σ2_0.0204_σ2_{+ 0.0750}_σ2_d 2

0.5 0.5070σ2_0.0170_σ2_0.0625_σ2_0.0156_σ2 ₋_0.0026_σ2 ₋_0.0007_σ2_0.0170_σ2_{+ 0.0625}_σ2_d 2

0.6 0.6056σ2_0.0136_σ2_0.0500_σ2_0.0125_σ2 ₋_0.0021_σ2 ₋_0.0005_σ2_0.0136_σ2_{+ 0.0500}_σ2_d2

0.7 0.7042σ2_0.0102_σ2_0.0375_σ2_0.0094_σ2 ₋_0.0016_σ2 ₋_0.0004_σ2_0.0102_σ2_{+ 0.0375}_σ2_d2

0.8 0.8028σ2_0.0068_σ2_0.0250_σ2_0.0062_σ2 ₋_0.0011_σ2 ₋_0.0003_σ2_0.0068_σ2_{+ 0.0250}_σ2_d2

0.9 0.9014σ2_0.0034_σ2_0.0125_σ2_0.0031_σ2 ₋_0.0005_σ2 ₋_0.0001_σ2_0.0034_σ2_{+ 0.0125}_σ2_d2 v = 8, b = 15, r = 6, k1= 4, k2 = 3, k3 = 2, λ = 2

N1 =513 λ4 = 0.0624 λ2 =0.2081 c = 4.8044 η = 1.6015 α = 2.3180 ρ

 

ˆ0

V 

 

ˆ

i

V 

 

ˆ

ij

V 

 

ˆ

ii

V  Cov



 ˆ ˆ0, ii





ˆij



ˆ_x

i

ˆ ,ii

Cov x

y V_ 

 

 

0 0.0037σ2_0.0094_σ2_0.0312_σ2_0.0078_σ2 ₋_0.0010_σ2 ₋_0.0004_σ2_0.0094_σ2_{+ 0.0312}_σ2_d2

0.1 0.1033σ2_0.0084_σ2_0.0281_σ2_0.0070_σ2 ₋_0.0009_σ2 ₋_0.0004_σ2_0.0084_σ2_{+ 0.0281}_σ2_d2

0.2 0.2029σ2_0.0075_σ2_0.0250_σ2_0.0062_σ2 ₋_0.0008_σ2 ₋_0.0003_σ2_0.0075_σ2_{+ 0.0250}_σ2_d2

0.3 0.3026σ2_0.0066_σ2_0.0219_σ2_0.0055_σ2 ₋_0.0007_σ2 ₋_0.0003_σ2_0.0066_σ2_{+ 0.0219}_σ2_d2

0.4 0.4022σ2_0.0056_σ2_0.0187_σ2_0.0047_σ2 ₋_0.0006_σ2 ₋_0.0002_σ2_0.0056_σ2_{+ 0.0187}_σ2_d2

0.5 0.5018σ2_0.0047_σ2_0.0156_σ2_0.0039_σ2 ₋_0.0005_σ2 ₋_0.0002_σ2_0.0047_σ2_{+ 0.0156}_σ2_d2

0.6 0.6015σ2_0.0037_σ2_0.0125_σ2_0.0031_σ2 ₋_0.0004_σ2 ₋_0.0002_σ2_0.0037_σ2_{+ 0.0125}_σ2_d2

0.7 0.7011σ2_0.0028_σ2_0.0094_σ2_0.0023_σ2 ₋_0.0003_σ2 ₋_0.0001_σ2_0.0028_σ2_{+ 0.0094}_σ2_d2

0.8 0.8007σ2_0.0019_σ2_0.0062_σ2_0.0016_σ2 ₋_0.0002_σ2 ₋_0.0001_σ2_0.0019_σ2_{+ 0.0062}_σ2_d2

0.9 0.9004σ2_0.0009_σ2_0.0031_σ2_0.0008_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0009_σ2_{+ 0.0031}_σ2_d2 v = 9, b = 11, r = 5, k1 = 5, k2 = 4, k3 = 3, λ = 2

N1 = 389 λ4 = 0.0823 λ2 =0.2369 c = 4.8094 η = 1.6031 α = 2.4655 ρ

 

ˆ0

V 

 

ˆ

i

V 

 

ˆ

ij

V 

 

ˆ

ii

V  Cov



 ˆ ˆ0, ii





ˆij



ˆ_x

i

ˆ ,ii

Cov x

y V_ _



 

 

0 0.0049σ2_0.0109_σ2_0.0312_σ2_0.0078_σ2 ₋_0.0011_σ2 ₋_0.0004_σ2_0.0109_σ2_{+ 0.0312}_σ2_d2

0.1 0.1044σ2_0.0098_σ2_0.0281_σ2_0.0070_σ2 ₋_0.0010_σ2 ₋_0.0004_σ2_0.0098_σ2_{+ 0.0281}_σ2_d2

0.2 0.2039σ2_0.0087_σ2_0.0250_σ2_0.0062_σ2 ₋_0.0009_σ2 ₋_0.0003_σ2_0.0087_σ2_{+ 0.0250}_σ2_d2

0.3 0.3035σ2_0.0076_σ2_0.0219_σ2_0.0055_σ2 ₋_0.0008_σ2 ₋_0.0003_σ2_0.0076_σ2_{+ 0.0219}_σ2_d2

0.4 0.4030σ2_0.0065_σ2_0.0187_σ2_0.0047_σ2 ₋_0.0007_σ2 ₋_0.0002_σ2_0.0065_σ2_{+ 0.0187}_σ2_d2

0.5 0.5025σ2_0.0054_σ2_0.0156_σ2_0.0039_σ2 ₋_0.0006_σ2 ₋_0.0002_σ2_0.0054_σ2_{+ 0.0156}_σ2_d2

0.6 0.6020σ2_0.0043_σ2_0.0125_σ2_0.0031_σ2 ₋_0.0004_σ2 ₋_0.0002_σ2_0.0043_σ2_{+ 0.0125}_σ2_d2

0.7 0.7015σ2_0.0033_σ2_0.0094_σ2_0.0023_σ2 ₋_0.0003_σ2 ₋_0.0001_σ2_0.0033_σ2_{+ 0.0094}_σ2_d2

0.8 0.8010σ2_0.0022_σ2_0.0062_σ2_0.0016_σ2 ₋_0.0002_σ2 ₋_0.0001_σ2_0.0022_σ2_{+ 0.0062}_σ2_d2

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v = 10, b = 11, r = 5, k1 = 5, k2 = 4, λ = 2

N1 = 393 λ4 = 0.0814 λ2 = 0.2345 c = 4.8162 η = 1.6054 α = 2.4673 ρ

 

ˆ0

V 

 

ˆ

i

V 

 

ˆ

ij

V 

 

ˆ

ii

V  Cov



 ˆ ˆ0, ii





ˆij



ˆ_x

i

ˆ ,ii

Cov x

y V_ 

 

 

0 0.0050σ2_0.0109_σ2_0.0313_σ2_0.0078_σ2 ₋_0.0010_σ2 ₋_0.0004_σ2_0.0109_σ2_{+ 0.0313}_σ2_d2

0.1 0.1045σ2_0.0098_σ2_0.0281_σ2_0.0070_σ2 ₋_0.0009_σ2 ₋_0.0003_σ2_0.0098_σ2_{+ 0.0281}_σ2_d2

0.2 0.2040σ2_0.0087_σ2_0.0250_σ2_0.0063_σ2 ₋_0.0008_σ2 ₋_0.0003_σ2_0.0087_σ2_{+ 0.0250}_σ2_d2

0.3 0.3035σ2_0.0076_σ2_0.0219_σ2_0.0055_σ2 ₋_0.0007_σ2 ₋_0.0003_σ2_0.0076_σ2_{+ 0.0219}_σ2_d2

0.4 0.4030σ2_0.0065_σ2_0.0188_σ2_0.0047_σ2 ₋_0.0006_σ2 ₋_0.0002_σ2_0.0065_σ2_{+ 0.0188}_σ2_d2

0.5 0.5025σ2_0.0054_σ2_0.0156_σ2_0.0039_σ2 ₋_0.0005_σ2 ₋_0.0002_σ2_0.0054_σ2_{+ 0.0156}_σ2_d2

0.6 0.6020σ2_0.0043_σ2_0.0125_σ2_0.0031_σ2 ₋_0.0004_σ2 ₋_0.0002_σ2_0.0043_σ2_{+ 0.0125}_σ2_d2

0.7 0.7015σ2_0.0033_σ2_0.0094_σ2_0.0023_σ2 ₋_0.0003_σ2 ₋_0.0001_σ2_0.0033_σ2_{+ 0.0094}_σ2_d2

0.8 0.8010σ2_0.0022_σ2_0.0063_σ2_0.0016_σ2 ₋_0.0002_σ2 ₋_0.0001_σ2_0.0022_σ2_{+ 0.0063}_σ2_d2

0.9 0.9005σ2_0.0011_σ2_0.0031_σ2_0.0008_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0011_σ2_{+ 0.0031}_σ2_d2 v = 12, b = 16, r = 6, k1 = 6, k2 = 5, k3 = 4, k4 = 3, λ = 2

N1 = 1073 λ4 = 0.0596 λ2 = 0.1932 c = 4.8199 η = 1.6066 α = 2.7625 ρ

 

ˆ0

V 

 

ˆ

i

V 

 

ˆ

ij

V 

 

ˆ

ii

V  Cov



 ˆ ˆ0, ii





ˆij



ˆ_x

i

ˆ ,ii

Cov x

y V_ _



 

 

0 0.0018σ2_0.0048_σ2_0.0156_σ2_0.0039_σ2 ₋_0.0004_σ2 ₋_0.0002_σ2_0.0048_σ2_{+ 0.0156}_σ2_d2

0.1 0.1016σ2_0.0043_σ2_0.0141_σ2_0.0035_σ2 ₋_0.0003_σ2 ₋_0.0002_σ2_0.0043_σ2_{+ 0.0141}_σ2_d2

0.2 0.2014σ2_0.0039_σ2_0.0125_σ2_0.0031_σ2 ₋_0.0003_σ2 ₋_0.0001_σ2_0.0039_σ2_{+ 0.0125}_σ2_d2

0.3 0.3012σ2_0.0034_σ2_0.0109_σ2_0.0027_σ2 ₋_0.0003_σ2 ₋_0.0001_σ2_0.0039_σ2_{+ 0.0109}_σ2_d2

0.4 0.4011σ2_0.0029_σ2_0.0094_σ2_0.0023_σ2 ₋_0.0002_σ2 ₋_0.0001_σ2_0.0029_σ2_{+ 0.0094}_σ2_d2

0.5 0.5009σ2_0.0024_σ2_0.0078_σ2_0.0020_σ2 ₋_0.0002_σ2 ₋_0.0001_σ2_0.0024_σ2_{+ 0.0078}_σ2_d2

0.6 0.6007σ2_0.0019_σ2_0.0063_σ2_0.0016_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0019_σ2_{+ 0.0063}_σ2_d2

0.7 0.7005σ2_0.0014_σ2_0.0047_σ2_0.0012_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0014_σ2_{+ 0.0047}_σ2_d2

0.8 0.8004σ2_0.0010_σ2_0.0031_σ2_0.0008_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0010_σ2_{+ 0.0031}_σ2_d2

0.9 0.9002σ2_0.0005_σ2_0.0016_σ2_0.0004_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0005_σ2_{+ 0.0016}_σ2_d2 v = 13, b = 16, r = 6, k1 = 6, k2 = 5, k3 = 4, k4 = 3, λ = 2)

N1 = 1077 λ4 = 0.0594 λ2 = 0.1925 c = 4.8273 η = 1.6091 α = 2.7653 ρ

 

ˆ0

V 

 

ˆ

i

V 

 

ˆ

ij

V 

 

ˆ

ii

V  Cov



 ˆ ˆ0, ii





ˆij



ˆ_x

i

ˆ ,ii

Cov x

y V_ _



 

 

0 0.0018σ2_0.0048_σ2_0.0156_σ2_0.0039_σ2 ₋_0.0003_σ2 ₋_0.0002_σ2_0.0048_σ2_{+ 0.0156}_σ2_d2

0.1 0.1016σ2_0.0043_σ2_0.0141_σ2_0.0035_σ2 ₋_0.0003_σ2 ₋_0.0002_σ2_0.0043_σ2_{+ 0.0141}_σ2_d2

0.2 0.2014σ2_0.0039_σ2_0.0125_σ2_0.0031_σ2 ₋_0.0003_σ2 ₋_0.0001_σ2_0.0039_σ2_{+ 0.0125}_σ2_d2

0.3 0.3013σ2_0.0034_σ2_0.0109_σ2_0.0027_σ2 ₋_0.0002_σ2 ₋_0.0001_σ2_0.0034_σ2_{+ 0.0109}_σ2_d2

0.4 0.4011σ2_0.0029_σ2_0.0094_σ2_0.0023_σ2 ₋_0.0002_σ2 ₋_0.0001_σ2_0.0029_σ2_{+ 0.0094}_σ2_d2

0.5 0.5009σ2_0.0024_σ2_0.0078_σ2_0.0020_σ2 ₋_0.0002_σ2 ₋_0.0001_σ2_0.0024_σ2_{+ 0.0078}_σ2_d2

0.6 0.6007σ2_0.0019_σ2_0.0063_σ2_0.0016_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0019_σ2_{+ 0.0063}_σ2_d2

0.7 0.7005σ2_0.0014_σ2_0.0047_σ2_0.0012_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0014_σ2_{+ 0.0047}_σ2_d2

0.8 0.8004σ2_0.0010_σ2_0.0031_σ2_0.0008_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0010_σ2_{+ 0.0031}_σ2_d2

0.9 0.9002σ2_0.0005_σ2_0.0016_σ2_0.0004_σ2 ₋_0.0001_σ2 ₋_0.0001_σ2_0.0005_σ2_{+ 0.0016}_σ2_d2

(9)

v = 14, b = 16, r = 6, k1 = 6, k2 = 5, k3 = 4, λ = 2

N1 = 1081 λ4 = 0.0592 λ2 =0.1918 c = 4.8342 η = 1.6114 α = 2.7679 ρ

 

ˆ0

V  V

 

ˆi V

 

ˆij V

 

ˆii Cov