243
Acta Math. Univ. Comenianae
Vol. LXI, 2(1992), pp. 243–250
UNIFORMLY BEST LINEAR–QUADRATIC ESTIMATOR IN A
SPECIAL STRUCTURE OF THE REGRESSION MODEL
G. WIMMER
Abstract. The paper shows the uniformly best linear-quadratic unbiased estimator of the covariance matrix element related to the repeated measurement in a
regression model where dispersions depend quadratically on mean value parameters.
Its consistency with respect to increasing number of repeated measurements is also
investigated.
Introduction
e X1 β, Σ),
e where the observation
Let us have the well known regression model (Y,
q
e with mean
e ∈ R is a realization of the normally distributed random vector Y
y
k
e
value E(Y) = X1 β (X1 is a known q × k design matrix, β ∈ R is a vector of
e A large class of measurement
unknown parameters) and covariance matrix Σ.
devices has its dispersion characteristic of the form σ2 (a + b|ϕ|)2 , where σ2 , a and
b are known positive constants and ϕ is the true measured value (see e.g. [1, p. 28],
e of the
[3, p. 456, 914]). If we assume independent measurements, we obtain Σ
form

(a + b|e01 X1 β|)2

0

e
e = σ2 Σ(β)
= σ2 
Σ
..

.
0
0
(a + b|e02 X1 β|)2
..
.
...
...
...

0
0

.

(a + b|e0q X1 β|)2
Further let us suppose that the rank of the matrix X1 is
R(X1 ) = q ≤ k ,
(there are less linearly independent observations as unknown parameters) but one
measurement (say s-th measurement) is repeated n − q times.
Received March 18, 1992; revised May 18, 1992.
1980 Mathematics Subject Classification (1991 Revision). Primary 62J05,62F10.
Key words and phrases. Regression model with repeated measurement and dispersions depending on the mean value parameters, uniformly best linear-quadratic unbiased estimator
(UBLQUE) and its consistency.
244
G. WIMMER
We obtain the model
(1)
(Y, Xβ, Σ),
where


Iq,q
 e0s 


=  .  X1 ,
 .. 
Xn,k
e0s
Σn,n = σ2 Σ(β)
 (a+b|e0 Xβ|)2 ...
1
..
..

.
.

0


..

.
= σ2 

0

0


..
.
0
(a+b|e0s Xβ|)2
...
..
.
0
0
...
0





,




(a+b|e0q Xβ|)2
(a+b|e0s Xβ|)2
..
0
.

(a+b|e0s Xβ|)2
s ∈ {1, 2, . . . , q}, n > q and e0i is the transpose of the i-th unit vector.
Continuated the considerations concerning the special case of the regression
model with dispersions depending on mean value parameters stated in [4], [5]
and [6] there arises the problem of finding the UBLQUE (uniformly best linear2
quadratic unbiased estimator) of the functional σ2 (a + b|e0s Xβ|) of parameter β
in model (1).
Based on analyses and methods in [5] and [6] it can be shown that the β◦ 2
LBLQUE (β◦ -locally best linear-quadratic unbiased estimator) of σ2 (a + b|e0s Xβ|)
exists iff the linear system (2) is soluble. In this case is the desired β◦ -LBLQUE
2
given by (3) and (4). Of course the UBLQUE of σ2 (a + b|e0s Xβ|) in model (1) is
related to such a solution of (2) that does not depend on β◦ .
In this paper is proved that the “natural” estimator (9) based only on the re2
peated measurements is the desired UBLQUE of σ2 (a + b|e0s Xβ|) (in Corollary 4).
An easy consequence is the consistency of (9) with respect to increasing number
n of the repeated s-th measurement. This forms a base for further investigations
with repetitions of more than one measurement.
UBLQUE
First let us introduce some denotations.
UNIFORMLY BEST ESTIMATOR IN A SPECIAL MODEL
0
is the (k 2 + q) × n2 matrix
X(s)















X0 ⊗ X0
e01 ⊗ e01
..
.
e0s−1 ⊗ e0s−1
e0s+1 ⊗ e0s+1
..
.
0
eq ⊗ e0q
Pn
0
0
es ⊗ es + j=q+1 e0j ⊗ e0j
245








,






where ⊗ means the Kronecker product (see e.g. [2, p. 11]) and I? is the nonsingular
matrix with next property:
I? vec A = vec A0
∀ {An,n }
(vec An,n = (a11 , a21 , . . . , an1 , a12 , a22 , . . . , an2 , . . . , a1n , a2n , . . . , ann )0 ).
Further
Cn = ((Iq,q , es , . . . , es ) ⊗ (Iq,q , es , . . . , es ))(I + I? )(Σ−1 (β◦ ) ⊗ Σ−1 (β◦ ))
 


Iq,q
Iq,q
 e0s   e0s 
 


· (I + I? )( .  ⊗  . )
 ..   .. 
e0s
e0s
A0k2 +q−1,q2
 X0 ⊗ X0 
1
1
 e01 ⊗ e01 


..




.
 0

0

e
⊗
e
=
s−1  .
 s−1
 e0 ⊗ e0 
s+1 
 s+1


..


.
e0q ⊗ e0q
and
(We only note that A0 is a matrix of full rank in columns.)
It holds
0
A ((Iq,q , es , . . . , es ) ⊗ (Iq,q , es , . . . , es ))
0
P
X(s)
=
.
n
e0s ⊗ e0s + j=q+1 e0j ⊗ e0j
As stated in Introduction it can be shown (based on analyses and methods
in [5] and [6]) that there exists in model (1) the βo -LBLQUE of the functional
σ2 (a + b|e0s Xβ|)2 of parameter β iff
Ok2 +q−1,1
0
?
−1
−1
?
(2)
X(s) (I + I )(Σ (β◦ ) ⊗ Σ (β◦ ))(I + I )X(s) δ =
8
246
G. WIMMER
is soluble. In this case a0 Y + Y0 AY is the desired β◦ -LBLQUE, where
a = −(A + A0 )Xβ◦
(3)
and
(4)
vecA =
1 −1
(Σ (β◦ ) ⊗ Σ−1 (β◦ ))X(s) δ.
2
It is easy to seen that equation (2) can be rewritten as
0
A Cn A g
O
δ=
,
(5)
g0
g
8
where
n
X
0
0
0
g1,k
2 +q−1 = (es ⊗ es +
e0j ⊗ e0j )(I + I? )(Σ−1 (β◦ ) ⊗ Σ−1 (β◦ ))
j=q+1

 

Iq,q
Iq,q
 e0s   e0s 

 

· (I + I? )( .  ⊗  . )A
 ..   .. 
e0s
e0s
−4
= 4(n − q + 1)(a + b|e0s Xβ◦ |)
(e0s X1 ⊗ e0s X1 , 0, . . . , 0)
and
g = (e0s ⊗ e0s +
n
X
e0j ⊗ e0j )(I + I? )(Σ−1 (β◦ ) ⊗ Σ−1 (β◦ ))
j=q+1
· (I + I? )(es ⊗ es +
n
X
ej ⊗ ej )
j=q+1
−4
= 4(n − q + 1)(a + b|e0s Xβ◦ |)
Let us denote
(6)
δk2 +q
.


X01 (X1 X01 )−1 es ⊗ X01 (X1 X01 )−1 es


0

2(a + b|e0s Xβ◦ |)4 
..


=

.
.

(n − q)(1 − n + q) 


0
1−n+q
Lemma 1. It holds
(A0 Cn A
g)δ = Ok2 +q−1,1 .
247
UNIFORMLY BEST ESTIMATOR IN A SPECIAL MODEL
Proof. As
 X0 (X X0 )−1 e ⊗ X0 (X X0 )−1 e 
1 1
s
1 1
s
1
1


0

A
.


..
0
·
k2 +q−1,1
0
b|es Xβ◦ |)4
2(a
2(a +
+ b|e0s Xβ◦ |)4
=
(es ⊗ es )q2 ,1
(n − q)(1 − n + q)
(n − q)(1 − n + q)
and
Cn (es ⊗ es )
2(a + b|e0s Xβ◦ |)4
(n − q)(1 − n + q)

0
?
= 2(I + I )q2 ,q2 ·



⊗







(a+b|e1 Xβ◦ |)−2
..
.
..
(n−q+1)(a+b|e0s Xβ◦ |)−2
0
...
..
.
..
.
0
..
.
(n−q+1)(a+b|e0s Xβ◦ |)−2
0
...
..
.
0
...
0

!

0


0
...
(a+b|e0q Xβ◦ |)−2
2(a + b|e0s Xβ◦ |)4
(n − q)(1 − n + q)
n−q+1
= −8(es ⊗ es )
,
n−q
· (es ⊗ es )
we obtain
 X0 (X X0 )−1 e ⊗ X0 (X X0 )−1 e 
1 1
s
1 1
s
1
1

 2(a + b|e0s Xβ◦ |)4
0

A0 Cn A 
.

 (n − q)(1 − n + q)
..

(7)
0
0
X ⊗X
 e01 ⊗ e01

= −
..

.
e0q ⊗ e0q

0

8(n − q + 1)

 (es ⊗ es )

n−q
 X0 e ⊗ X0 e 
1 s
1 s

0
n−q+1


= −8
..


n−q
.
0
k2 +q−1,1
.

...
(a+b|e0q Xβ◦ |)−2
...
0
..
.
.
0
(a+b|e01 Xβ◦ |)−2
..
.
0




248
G. WIMMER
Further
 X0 e ⊗ X0 e 
1 s
1 s

0
2(a + b|e0s Xβ◦ |)4
n−q+1 


g
=8
.

..
(n − q)
n−q 
(8)
0
.
k2 +q−1,1
After an easy calculation (by the help of (7) and (8)) we can finish the proof of
the lemma.
Lemma 2. It holds
(g0
g)δ = 8 .
Proof. As
 X0 (X X0 )−1 e ⊗ X0 (X X0 )−1 e 
1 1
s
1 1
s
1
1
 2(a + b|e0s Xβ◦ |)4

0
8

g0 
.
 (n − q)(1 − n + q) = − n − q

..
0
and
g
2(a + b|e0s Xβ◦ |)4
8(n − q + 1)
=
,
(n − q)
n−q
we easy obtain
(g0
g)δ = −
8
8(n − q + 1)
+
= 8.
n−q
n−q
The lemma is proved.
According to Lemma 1 and Lemma 2, δ (given in (6)) is a solution to (5), or,
equivalently to (2). Thus we obtain the next corollary:
Corollary 3. The random variable
(9)
1
n−q
where
Y =
n
X
(Yj − Y )2 ,
j=s,j=q+1
n
X
1
Yj
n − q + 1 j=s,j=q+1
2
is the β◦ -LBLQUE of σ2 (a + b|e0s Xβ|) .
Proof. As δ (given in (6)) is a solution to (2), a0 Y + Y0 AY is the β◦ -LBLQUE
2
of σ2 (a + b|e0s Xβ|) , where (3) and (4) hold.
UNIFORMLY BEST ESTIMATOR IN A SPECIAL MODEL
It is easy to see that
249
(e0i ⊗ e0j )vec A
is
1
n−q
and
0 for i ∈
/ {s, q + 1, . . . , n} or j ∈
/ {s, q + 1, . . . , n} ,
1
−
for i ∈ {s, q + 1, . . . , n}, j ∈ {s, q + 1, . . . , n}, i 6= j
n−q+1
1
n−q
1−
1
n−q+1
for i ∈ {s, q + 1, . . . , n}, j = i .
So
Y0 AY = (Y0 ⊗ Y0 )vec A
X
n
n
X
1
1
−
=
Yi Yj
n−q
n − q + 1 i=s,i=q+1 j=s,j=q+1
+
=
n
X
1
1
(1 −
)
Y2
n−q
n − q + 1 i=s,i=q+1 i
1
n−q
n
X
(Yj − Y )2 .
j=s,j=q+1
We see that for A given by (4) is A = A0 . After a short computation also
2
X A = O is obvious. So (9) is the β◦ -LBLQUE of σ2 (a + b|e0s Xβ|) .
0
As (9) does not depend on the β◦ , we finally have
Corollary 4. The random variable
1
n−q
with
Y =
n
X
(Yj − Y )2 ,
j=s,j=q+1
n
X
1
Yj
n − q + 1 j=s,j=q+1
is in model (1) the UBLQUE (uniformly best linear-quadratic unbiased estimator)
2
of σ2 (a + b|e0s Xβ|) .
We only note that evidently (9) is a consistent estimator with dispersion
4
Dβ =
2σ4 (a + b|e0s Xβ|)
.
n−q
Acknowledgement. The author thanks to the referee for helpfull comments
which substantialy improved the paper in the way of finding the UBLQUE in
model (1).
250
G. WIMMER
References
1. Fajt V., Elektrická měřenı́, SNTL/ALFA, Praha, 1978.
2. Rao C. R. and Mitra S. K., Generalized Inverse of Matrices and Its Applications, J. Wiley,
New York, 1971.
3. Rinner K. and Benz F., Jordan/Eggert/Kneissl Handbuch der Vermessungskunde, Band VI,
Stuttgart, 1966.
4. Wimmer G., Linear model with variances depending on the mean value, Mathematica Slovaca
42 (1992), 223–238.
5.
, Estimation in a special structure of the linear model Mathematica Slovaca, (to
appear).
, Linear-quadratic estimators in a special structure of the linear model, Submitted to
6.
Applications of Mathematics.
G. Wimmer, Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Czechoslovakia