MSE Performance of the Weighted Average
Estimators Consisting of Shrinkage Estimators

Akio Namba
Kazuhiro Ohtani
March 2015
Discussion Paper No.1513

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANMSE Performance of the Weighted Average
Estimators Consisting of Shrinkage Estimators
Akio Namba∗ and Kazuhiro Ohtani†
Graduate School of Economics
Kobe University
Rokko, Nada-ku, Kobe 657-8501
Japan

Abstract
In this paper we consider a regression model and propose estimators which are the weighted
averages of two estimators among three estimators; the Stein-rule (SR), minimum mean squared
error (MMSE) and the adjusted minimum mean squared error (AMMSE) estimators. We derive theformula for the mean squared error (MSE) of the proposed estimators. It is shown by numericalevaluations that one of the proposed estimators has smaller mean squared error (MSE) than thepositive-part Stein-rule (PSR) estimator over a moderate region of parameter space when the numberof the regression coeﬃcients is small (i.e., 3). Also, its MSE performance is comparable to the PSR
estimator even when the number of the regression coeﬃcient is not so small.

Key Words: Mean squared error, Stein-rule estimator, Minimum mean squared error estimator, Adjusted Minimum mean squared error estimator, weighted average estimator∗ Akio Namba

Graduate School of Economics, Kobe University
Rokko, Nada-ku, Kobe, 657-8501, Japane-mail: namba@econ.kobe-u.ac.jp† Kazuhiro Ohtani
Graduate School of Economics, Kobe University
Rokko, Nada-ku, Kobe, 657-8501, Japane-mail: kotani@econ.kobe-u.ac.jp

1

Introduction

In the context of a linear regression, the Stein-rule (SR) estimator proposed by Stein (1956) and James andStein (1961) dominates the ordinary least squares (OLS) estimator in terms of mean squared error (MSE)
if the number of the regression coeﬃcients is larger than or equal to three. Though the SR estimator
dominates the OLS estimator, Baranchik (1970) showed that the SR estimator is further dominated by
the positive-part Stein-rule (PSR) estimator.
Also, as one of improved estimators, Theil (1971) proposed the minimum mean squared error (MMSE)
estimator. However, Theil’s (1971) MMSE estimator includes unknown parameters, Farebrother (1975)
suggested an operational variant of the MMSE estimator which is obtained by replacing the unknown
parameters by the OLS estimators. As an extension of the MMSE estimator, Ohtani (1996b) considered
the adjusted minimum mean squared error (AMMSE) estimator which is obtained by adjusting thedegrees of freedom of the operational variant of the MMSE estimator. He showed by numerical evaluations
that the AMMSE estimator has smaller MSE than the SR and PSR estimators in a wide region of thenoncentrality parameter when k ≤ 5, where k is the number of regression coeﬃcient. In particular,when k = 3, the MSE of the AMMSE estimator can be much smaller than that of the PSR estimatorfor small values of the noncentrality parameter. Therefore, Ohtani (1999) considered a heterogeneous
pre-test estimator such that the AMMSE estimator is used if the null hypothesis that all the regression
coeﬃcients are zeros (in other words, the value of the noncentrality parameter is zero) is accepted,
and the SR estimator is used if the null hypothesis is rejected. Although the results were obtained bynumerical evaluations, he showed that a heterogeneous pre-test estimator dominates the PSR estimatorwhen k = 3 and the critical value of the pre-test is chosen appropriately. Extending the result of
Ohtani (1999), Namba (2000) proposed another heterogeneous pre-test estimator and numerically showedthat the proposed estimator has smaller MSE than the PSR estimator even when k ≥ 4. However, since
the estimators considered by Ohtani (1999) and Namba (2000) connect two diﬀerent estimators via a
pre-test based on the F -statistic, they are not smooth. This is a drawback of their estimators common
to the PSR estimator because the smoothness is required for the estimator to be admissible.
In this paper, we propose estimators which are weighted averages of diﬀerent kinds of estimators.
The proposed estimators are smooth since they do not incorporate the pre-test. In the next section, we
introduce the estimators, and derive the explicit formula for MSE of the estimators in section 3. Using
this formula, we examine the MSE performance of the estimators by numerical evaluations in section 4.
Our numerical results show that one of our estimators has smaller MSE than the PSR estimator over amoderate region of parameter space. Finally, some concluding remarks are given in section 5.

12Model and estimators

Consider a linear regression model,
y = Xβ + ,  ∼ N (0, σ 2 In ).

(1)

where y is an n × 1 vector of observations on the dependent variable, X is an n × k matrix of full column
rank of observations on nonstochastic independent variables, β is a k × 1 vector of coeﬃcients, and  isan n × 1 vector of normal error terms with E[] = 0 and E[0 ] = σ 2 In .
The ordinary least squares (OLS) estimator of β isb = S −1 X 0 y,(2)where S = X 0 X. In the context of linear regression, the Stein-rule (SR) estimator proposed by Stein(1956) is defined as
(
)
ae0 e
bSR = 1 − 0
b,
b Sb

(3)

where e = y − Xb, and a is a constant such that 0 ≤ a ≤ 2(k − 2)/(ν + 2), where ν = n − k. If we use
the loss functionL(β̄; β) = (β̄ − β)0 S(β̄ − β),(4)where β̄ is any estimator of β, the SR estimator dominates the OLS estimator in terms of mean squared
error (MSE) for k ≥ 3. As is shown in James and Stein (1961), the MSE of the SR estimator is minimized
when a = (k −2)/(ν +2). Thus we use this value of a hereafter. Although the SR estimator dominates the
OLS estimator, Baranchik (1970) showed that the SR estimator is further dominated by the positive-part
Stein-rule (PSR) estimator defined as
[
]
ae0 e
bPSR = max 0, 1 − 0
b.
b Sb

(5)

As one of improved estimators, Theil (1971) proposed the minimum mean squared error (MMSE)
estimator. However, since Theil’s (1971) MMSE estimator includes unknown parameters, Farebrother(1975) suggested the following operational variant of the MMSE estimator
(
)
b0 Sb
bM =
b.
b0 Sb + e0 e/ν(6)Hereafter, we call the operational variant of the MMSE estimator the MMSE estimator simply. There areseveral studies about the MMSE estimator and its variants. Some examples are Vinod (1976), Dwivedi
and Srivastava (1978), Tracy and Srivastava (1994) and Ohtani (1996a, 1996b, 1997). Ohtani (1996a)2derived the exact MSE of the MMSE estimator and a suﬃcient condition for the MMSE estimator todominate the OLS estimator.
Furthermore, as an extension of the MMSE estimator, Ohtani (1996b) considered the following estimator which is obtained by adjusting the degrees of freedom of b0 Sb (i.e., k),()
b0 Sb/kbAM =b.
b0 Sb/k + e0 e/ν

(7)

We call this estimator the adjusted MMSE (AMMSE) estimator. Ohtani (1996b) showed by numericalevaluations that if k ≤ 5 the AMMSE estimator has the smaller MSE than the SR, PSR and MMSE
estimators in a wide region of noncentrality parameter defined as λ = β 0 Sβ/σ 2 . Thus, Ohtani (1999)
considered the following heterogeneous pre-test estimator:
βbPT (τ ) = I(F ≤ τ )bAM + I(F > τ )bSR ,(8)

where F = (b0 Sb/k)/(e0 e/ν) is the test statistic for H0 : β = 0 against H1 : β 6= 0, τ is the critical valueof the pre-test, and I(A) is an indicator function such that I(A) = 1 if an event A occurs and I(A) = 0
otherwise. He showed by numerical evaluations that the heterogeneous pre-test estimator dominates the
PSR estimator for k = 3 if the critical values of the pre-test is chosen appropriately. Also, extending
the result of Ohtani (1999), Namba (2000) proposed another pre-test estimator obtained by replacing
the AMMSE estimator with the 2SHI estimator proposed by Tran Van Hoa and Chaturvedi (1990),
and showed the dominance of the proposed estimator over the PSR estimator by numerical evaluations.
Though the estimators proposed by Ohtani (1999) and Namba (2000) dominates the PSR estimator, they
are not smooth because of the pre-test. This is a drawback of their estimators, since the smoothness isa necessary condition for the admissibility of the estimator.
Thus, in this paper, we propose the following two estimators which are weighted averages of theestimators introduced above:
)
(
)
(
F
F
b
βWA1 =
bM + 1 −
bAM ,
1+F
1+F
(
)
(
)
F
F
βbWA2 =bSR + 1 −
bAM .
1+F
1+F

(9)
(10)

These estimators are smooth since they do not incorporate any pre-test. Similar to βbPT (τ ) given in (8),bAM plays an important role in both βbWA1 and βbWA2 when the value of F is small.
In the next section, we derive the explicit formula for the MSEs of βbWA1 and βbWA2 .

3

MSE of the estimators

In this section, we derive the formula for the MSEs of βbWA1 and βbWA2 .

3

Noting that b0 Sb/e0 e = kF/ν, substituting (3), (6) and (7) into (9) and (10), and conducting some
calculations, we have[]
F2
F
βbWA1 =b+
(F + 1)(F + 1/k) (1 + F )2

(11)

and
[
]
F − aν/kFb
βWA2 =+b.
F +1(1 + F )2(12)

Thus, the MSE of βbWA1 is given by
MSE[βbWA1 ] = E[(βbWA1 − β)0 S(βbWA1 − β)]
[{
}
]
F4
F3
F2
0
=E
+
2
+
b
Sb
(F + 1)2 (F + 1/k)2(F + 1)3 (F + 1/k) (F + 1)4
[{
}
]
F
F2
0
+
−2E
β
Sb
+ β 0 Sβ.
(F + 1)(F + 1/k) (F + 1)2(13)

Similarly,
[{

1
F
F2
+
2
+
(F + 1)2 (F − aν/k)−2(F + 1)3 (F − aν/k)−1(F + 1)4
[{
}
]
2
F
F
−2E
+
β 0 Sb + β 0 Sβ.
(F + 1)(F − aν/k)−1(F + 1)2MSE[βbWA2 ] = E

}

]
b Sb
0

Here, we define the functions H(p, q, r, m; α) and J(p, q, r, m; α) as
]
[
Fr
0
m
(b Sb)
H(p, q, r, m; α) = E(F + 1)p (F + α)q

(14)

(15)

and
[
J(p, q, r, m; α) = E

]
Fr
0
m 0
(b
Sb)
β
Sb
.
(F + 1)p (F + α)q(16)Then, we can express the MSEs of βbWA1 and βbWA2 as
MSE[βbWA1 ] = H(2, 2, 4, 1; 1/k) + 2H(3, 1, 3, 1; 1/k) + H(4, 0, 2, 1; 1/k)
−2J(1, 1, 2, 0; 1/k) − 2J(2, 0, 1, 0; 1/k) + β 0 Sβ

(17)

and
MSE[βbWA1 ] = H(2, −2, 0, 1; −aν/k) + 2H(3, −1, 1, 1; −aν/k) + H(4, 0, 2, 1; −aν/k)−2J(1, −1, 0, 0; −aν/k) − 2J(2, 0, 1, 0; −aν/k) + β 0 Sβ.
As is shown in appendix, explicit formulae for H(p, q, r, m; α) and J(p, q, r, m; α) are given by( )p+q−r ∑∞k2 mH(p, q, r, m; α) = (2σ )wi (λ)Gi (p, q, r, m; α),νi=0( )p+q−r ∑∞kwi (λ)Gi+1 (p, q, r, m; α),J(p, q, r, m; α) = β 0 Sβ(2σ 2 )m
ν
i=0

4

(18)

(19)
(20)

where wi (λ) = exp(−λ/2)(λ/2)/i! andGi (p, q, r, m; α) =Γ((ν + k)/2 + m + i)
Γ(ν/2)Γ(k/2 + i)

∫ 1
0

tk/2+m+r+i−1 (1 − t)ν/2+p+q−r−1
dt.
[t + k(1 − t)/ν]p [t + αk(1 − t)/ν]q(21)

Substituting (19) and (20) into (17) and (18), we obtain explicit formulae for the MSEs of βbWA1 andβbWA2 .
Since further theoretical analysis of the MSEs of βbWA1 and βbWA2 is diﬃcult, we execute numericalevaluation in the next section.

4

Numerical results

In this section, we compare the MSEs of estimators by numerical evaluations.
The parameter values used in the numerical evaluations were k = 3, 4, 5, 8, 10, n = 20, 30, 50,
and λ = various values. The numerical evaluations were executed on a personal computer, using the
FORTRAN code. In evaluating the integral in Gi (p, q, r, m; α) given in (21), we used Simpson’s rule with
500 equal subdivisions. The infinite series in H(p, q, r, m; α) and J(p, q, r, m; α) were judged to converge
when the increment of the series gets smaller than 10−12 . In order to compare the MSE performancesof the estimators, we evaluated the values of relative MSE defined as MSE(β̄)/MSE(b), where β̄ is any
estimator of β. Thus, the estimator β̄ has smaller MSE than the OLS estimator when the value of relative
MSE is smaller than unity. Since the results for k = 3, k = 5 and n = 30 are qualitatively typical, weshow the results only for these cases.
Table 1 shows the results for k = 3 and n = 30. We can see that all estimators considered here except
for bAM dominates the OLS estimator b in terms of MSE, and βbWA2 has smaller MSE than the otherestimators considered here except for bAM when λ is small (i.e., λ ≤ 2.0). Moreover, βbWA2 has smaller
MSE than bPSR when λ is small and moderate (i.e., λ ≤ 8.0). Though bAM has smallest MSE among all
the estimators when λ is small and moderate (i.e., λ ≤ 12.0), it has slightly larger MSE than the OLS
estimator for large values of λ (i.e., λ ≥ 22.0). Also, though βbWA2 has smaller MSE than βbWA1 around
λ = 0, the region where βbWA1 has smaller MSE than bPSR (i.e., λ ≤ 10.0) is slightly wider than that ofβbWA2 (i.e., λ ≤ 8.0). Comparing βbWA1 and βbWA2 , the MSE of βbWA2 around λ = 0 is smaller than that ofβbWA1 while the MSE of βbWA2 is slightly larger than that of βbWA1 for λ ≥ 3.0). Comparing the gain andloss of MSE, βbWA2 may be preferred to bPSR and βbWA1 .
Table 2 shows the results for k = 5 and n = 30. Comparing Tables 1 and 2, we see that whenk increases from 3 to 5, the MSE performance changes largely. Though all estimators considered here
dominate the OLS estimator, βbWA1 is dominated by bPSR , and both βbWA1 and βbWA2 no longer havesmaller MSE than bPSR around λ = 0. Also, βbWA2 dominates βbWA1 and bM , and βbWA2 has smaller MSE

5than bPSR for 0.4 ≤ λ ≤ 22.0. This indicates that the MSE performances of two estimators, βbWA2 andbPSR , are comparable.
When it is diﬃcult to derive an exact distribution of an estimator, the bootstrap methods proposedby Efron (1979) have often been applied. (Some examples are Chi and Judge (1985), Brownstone (1990)
and Yi (1991).) However, as is shown in Zaman (1996), the bootstrap methods are not so valid when an
estimator is not smooth. Since the MSE performances of βbWA2 and bPSR are comparable and βbWA2 issmooth while bPSR is not smooth, we can say that βbWA2 can be a good smooth alternative to the PSR
estimator, in particular when we apply the bootstrap methods.

5Concluding remarks

In this paper, we proposed estimators for regression coeﬃcients which are weighted averages of two
shrinkage estimators. Our numerical results show that the estimator βbWA2 which is a weighted averageof the SR and the AMMSE estimators has smaller MSE than the PSR estimator over a moderate regionof parameter space when k = 3. Even when k > 3, βbWA2 has comparable MSE performance to the
PSR estimator bPSR . Also, the proposed estimators βbWA1 and βbWA2 have smaller MSE than the OLS
estimator for all parameter values considered in this paper. Moreover, because of their structure, theproposed estimators are smooth. Compared with the PSR estimator, this is the virtue of proposed
estimators.
Considering this result, we may able to construct smooth estimators which have more preferable
performance by taking weighted averages of possibly some other estimators. However, seeking for such
estimators is beyond the scope of this research and a remaining problem for future research.

Appendix
First, we derive the formula for H(p, q, r, m; α). Let u1 = b0 Sb/σ 2 and u2 = e0 e/σ 2 . Then, u1 is distributed
as the noncentral chi-square distribution with k degrees of freedom and noncentrality parameter λ =β 0 Sβ/σ 2 , and u2 is distributed as the chi-square distribution with ν = n − k degrees of freedom.
Using u1 and u2 , H(p, q, r, α) is expressed as][( νk uu21 )r2
m(σ u1 )
H(p, q, r, m; α) = E ν u1
( k u2 + 1)p ( νk uu12 + α)q= ν r k p+q−r (σ 2 )m

∞
∑

Ki
i=0
∫ ∞∫ ∞ k/2+m+r+i−1 ν/2+p+q−r−1u1u2
×exp[−(u1 + u2 )/2]du1 du2 ,p (νu + αku )q
(νu
+
ku
)
1
2
1
2
0
0

6

(22)

Table 1: MSEs for k = 3 and n = 30.
λ

βbWA1

βbWA2

bSR

bPSR

bM

bAM

0.0

0.50673

0.482030.68966

0.56393

0.631630.359910.1

0.52033

0.49737

0.699800.578200.641150.374820.2

0.533540.512230.709540.591960.650380.389350.3

0.54636

0.52663

0.718900.605220.65932

0.403520.4

0.558800.540590.72791

0.61800

0.66799

0.41733

0.6

0.58260

0.567210.74487

0.642200.68455

0.443940.8

0.605030.592210.76056

0.664690.700120.469231.0

0.626170.61569

0.775070.685590.71478

0.493291.5

0.67386

0.668360.80679

0.731640.74775

0.548452.0

0.71505

0.713440.833010.77006

0.776140.597203.0

0.781450.785190.87286

0.82900

0.821840.67859

4.0

0.831250.838060.900690.870370.856190.74270

5.00.86875

0.877160.92046

0.899640.88228

0.79347

6.0

0.897120.906200.934760.92055

0.902290.833917.0

0.91867

0.92784

0.945310.935650.917810.866288.0

0.935120.94404

0.95324

0.946710.92997

0.89233

9.0

0.94774

0.956210.95933

0.954930.93960

0.91341

10.0

0.957470.965390.96409

0.961140.94732

0.93055

12.0

0.97087

0.977640.97098

0.969650.95868

0.95604

14.0

0.979100.98478

0.97565

0.975070.966440.97336

16.0

0.98426

0.98901

0.97902

0.97876

0.971960.9853118.0

0.98759

0.99157

0.98156

0.981450.97602

0.99369

20.0

0.98979

0.99315

0.98354

0.98349

0.979100.9996222.0

0.991280.994140.985130.985110.98151

1.00386

24.0

0.99233

0.99478

0.986440.98643

0.983431.00692

26.0

0.99309

0.995210.987540.98753

0.98500

1.0091228.0

0.99366

0.99550

0.98847

0.98847

0.986311.0107230.0

0.994100.99571

0.98927

0.98927

0.987411.0118640.0

0.99535

0.99626

0.99203

0.99203

0.99102

1.0139950.0

0.99601

0.99659

0.99366

0.99366

0.99304

1.01374

100.0

0.99756

0.99770

0.99686

0.99686

0.99672

1.00951150.0

0.99824

0.99830

0.99792

0.99792

0.99786

1.00694

7

Table 2: MSEs for k = 5 and n = 30.
λ

βbWA1

βbWA2

bSR

bPSR

bM

bAM

0.0

0.52260

0.33082

0.444440.32567

0.72274

0.327310.1

0.53091

0.34252

0.455400.338990.72730

0.336600.2

0.53904

0.35396

0.46605

0.35200

0.73175

0.34573

0.3

0.54700

0.365140.476400.364700.736100.354690.4

0.55478

0.37609

0.48646

0.377090.74035

0.36349

0.6

0.56984

0.39726

0.505760.40100

0.74856

0.380630.8

0.584270.417510.524020.42378

0.756390.39718

1.0

0.59808

0.436900.54130

0.44549

0.76387

0.413151.5

0.63013

0.481840.580620.495390.781150.45072

2.0

0.65896

0.522190.615070.53964

0.79660

0.485173.0

0.70835

0.59118

0.67204

0.61375

0.822890.54592

4.0

0.748650.64733

0.71666

0.672250.844220.59743

5.0

0.78172

0.69330

0.752100.71865

0.861700.64132

6.0

0.809020.731190.780610.75567

0.876170.678907.0

0.83169

0.76261

0.80386

0.785410.888250.711238.00.85063

0.788820.82306

0.80950

0.898430.739189.0

0.866540.810830.83908

0.829180.90708

0.76346

10.0

0.88000

0.829430.85262

0.845430.91450

0.7846312.0

0.901220.85875

0.874100.870360.926480.81949

14.0

0.916920.880440.890290.88837

0.93567

0.84667

16.0

0.92879

0.89686

0.90287

0.901900.942900.8681818.0

0.937960.90955

0.912910.912420.94871

0.88544

20.0

0.945180.91957

0.921090.920850.953480.8994822.0

0.95097

0.927630.92787

0.92775

0.957450.91103

24.0

0.95569

0.934220.93359

0.933530.960800.92063

26.0

0.95959

0.93969

0.938470.938440.96367

0.9287028.0

0.96287

0.94430

0.94269

0.94267

0.966150.9355430.0

0.96566

0.94823

0.94637

0.946360.96832

0.9413840.0

0.97498

0.96152

0.959400.959400.97601

0.9609450.0

0.98026

0.969190.96735

0.967350.98071

0.97164100.0

0.99026

0.98424

0.98350

0.98350

0.99025

0.98941150.0

0.993500.98934

0.98896

0.98896

0.99348

0.993888whereKi =wi (λ),
2(ν+k)/2+i Γ(ν/2)Γ(k/2 + i)

(23)

and wi (λ) = exp(−λ/2)(λ/2)i /i !.
Making use of the change of variables, v1 = u1 /u2 and v2 = u2 , the integral in (22) reduces to∫ ∞∫ ∞ k/2+m+r+i−1 (ν+k)/2+m+i−1v1v2
exp[−(v1 + 1)/2]dv1 dv2 .
p (νv + αk)q
(νv
+
k)
1
1
0
0
Again, making use of the change of variable, z = (1 + v1 )v2 /2, (24) reduces to∫ ∞k/2+m+r+i−1v1(ν+k)/2+m+12
Γ((ν + k)/2 + m + i)dv1 .
(νv1 + k)p (νv1 + αk)q (v1 + 1)(ν+k)/2+m+1
0

(24)

(25)

Further, making use of the change of variable, t = v1 /(1+v1 ), and substituting (25) in (22) and performingsome manipulations, (22) reduces to (19) in the text.
Next, we derive the formula for J(p, q, r, m; α). Diﬀerentiating H(p, q, r, α) given in (19) with respect
to β, we obtain( )p+q−r ∑]∞ [k∂wi (λ)Gi (p, q, r, m; α)ν∂βi=0( )p+q−r ∑
]
∞ [
Sβ
k
Sβ
2 m
− 2 wi (λ) + 2 wi−1 (λ) Gi (p, q, r, m; α)= (2σ )
νσ
σi=0( )p+q−r ∑
∞
Sβ
Sβ
k
= − 2 H(p, q, r, m; α) + 2 (2σ 2 )mwi (λ)Gi+1 (p, q, r, m; α),
σ
σ
ν
i=0

∂H(p, q, r, α)= (2σ 2 )m
∂β

(26)

where we define w−1 (λ) = 0.
Expressing H(p, q, r, m; α) by b0 Sb and e0 e, we have:][0r( νk be0Sb
0me )(b Sb)
H(p, q, r, m; α) = E ν b0 Sb0q( k e0 e + 1)p ( νk be0Sb
e + α)0∫ ∞∫ ∞r( νk be0Sbe )=(b0 Sb)m fN (b)fe (e0 e)db d(e0 e),
ν b0 Sbν b0 Sb
p
q
(
+
1)
(
+
α)
0
−∞ k e0 ek e0 ewhere fe (e0 e) is the density function of e0 e, and[]1(b − β)0 S(b − β).
fN (b) =exp−
2σ 2(2π)k/2 |σ 2 S −1 |1/2
Diﬀerentiating H(p, q, r, m; α) given in (27) with respect to β, we obtain:[]0r( νk be0Sb∂H(p, q, r, m; α)
1
Sβ
0
m
e )
= 2 E ν b0 Sb(b Sb) Sb − 2 H(p, q, r, m; α).
ν b0 Sbpq∂βσσ( k e0 e + 1) ( k e0 e + α)
Equating (26) and (27), and multiplying β 0 from left, we obtain (20) in the text.

Acknowledgment
This work was supported by JSPS KAKENHI Grant Number 23243038, 26780136.

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(27)

(28)

(29)

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