The Relative Efficiency of the Conditional Root Square Estimation of Parameter in Inhomogeneous Equality Restricted Linear Model ()
1. Definition and Lemma
Definition 1 [1] In the model (1), defined as
is the specific conditional root square estimation of
:
where 0 < k < 1,
, W, V defined as above paper, Q is p-orthogonal matrix, make
,
is Non-zero characteristic values of W, and
.
Definition 2 [2] In the model (1), defined as
is the generalized conditional root square estimation of
:
.
where
,
.
said
is
-root square parameter, W, Q, V defined as above paper.
Definition 3 [3] Two estimation
and
of the model (1), defined as
is elative efficiency of estimation
for elative efficiency estimation of
. If
is the best linear unbiased estimation of
, then note
.
For the above definition 3, if
, then shows that
is better than
under mean squares error and if the bigger of
(that efficiency highter),
improve the degree of
bigger.
Lemma 1 [1]
,
.
Lemma 2 [1]
is positive semidefinite matrix, and rank of W is
.
Lemma 3 [1] Exist Q is p-order orthogonal matrix,
make
,
is Non-zero characteristic values of W, and
.
Lemma 4 [1] Mean squares error of
is
,
is Non-zero characteristic values of W, and
.
Lemma 5 [1] Assume
then
![](https://www.scirp.org/html/10-80340\131cff0d-315a-4413-9365-32338b3d9d9a.jpg)
where
.
2. Main Results
We can prove the following exist theorem
and bound of
and
. Now, we have the following lemma.
Assume
, then
.
And the RLSE of
is
![](https://www.scirp.org/html/10-80340\dd6c44a4-37e5-4854-9b98-1b9a612c5a19.jpg)
accordingly, the specific conditional root square estimation of
is
![](https://www.scirp.org/html/10-80340\60e45795-50f9-4562-9de0-346bb409a47d.jpg)
0 < k < 1.
similarly, the generalized conditional root square estimation of
is
![](https://www.scirp.org/html/10-80340\b5d027ff-cc5c-487b-b8fd-cce1ddf3080b.jpg)
.
Lemma 6
, where
.
Proof:
when
,
.
Because
,
so
.
Because
![](https://www.scirp.org/html/10-80340\db1a1c3a-2f48-4a40-a59c-40c91ecc3ecf.jpg)
So
![](https://www.scirp.org/html/10-80340\582d4b3a-b65a-4f5c-b37f-7540f9946185.jpg)
Lemma 7
,
.
Lemma 8 When
, exist
, when
, then
![](https://www.scirp.org/html/10-80340\c9a254cc-1fba-47bf-8feb-191245e57001.jpg)
has minimum value.
Proof: Note
,
, then
.
For
, we have
. When
, if
, then
,
; if
, then
,
. When
,
. so
is a monotonically decreasing function in
.
For
, when
, we have
. this means
is a monotonically increasing function in
. so, there always exist
, when
we have
, so
is a monotonically decreasing function in
,
<
,
has minimum value.
Lemma 9 In the model (1), for
, when
, then
has minimum value.
Proof: according to lemma 8,
![](https://www.scirp.org/html/10-80340\0582bcf4-4d47-4249-adb7-48c06688b028.jpg)
Let
, we get
when
, the solution of this equation is
; when
, the solution of this equation is
, that
.
so
. Therefore when
, then
has minimum value.
Lemma 10 In the model (1), exist root square parameter 0 < k < 1, then mean squares error of
is
.
Lemma 11 In the model (1),
, always exist
, then
.
Proof: Based on the lemma 9 and lemma 10.
Theorem 1 In the model (1),
, always exist
, then
.
Proof: Based lemma 11 and definition 3, we get the conclusion.
Theorem 2 In the model (1), for
, exist
, then
.
Proof: For
, if
,
, choose
,
we have
.
Assume at least exist i, that
, assume
, where
, based on lemma 6 and, we have
![](https://www.scirp.org/html/10-80340\ce942695-6e04-457e-a265-41af0d3335e7.jpg)
based on lemma 9, we have
, then
, so
.
For above theorem, then
.
Using theorem 2, we get the following the conclusion.
Inference 1 In the model (1), for
, exist
,
then
.
Inference 2 In the model (1), if
Are not all equal,
then
.
Proof: Because
, Q is orthogonal matrix, so when
, then
, based on theorem 2, we get the conclusion.
Theorem 3 In the model (1), when
then
, where
,
is
the largest component of module.
Proof: Assume
, then
![](https://www.scirp.org/html/10-80340\7858aae7-8a84-4a12-b151-8b6893028021.jpg)
Assume
then
so
.
Theorem 4 In the model (1), for
, if
,
, then
, Where
.
Proof:
![](https://www.scirp.org/html/10-80340\b4211b2f-f698-48fd-887f-0844d22ee9c3.jpg)
Therefore
, let
,
then
.
Theorem 5 In the model (1), assume the non-zero characteristic root
of W are not all equal
, for the efficiency lower bound
of
and the efficiency lower bound
of
, the relationship of them is
.
Proof: By theorems 3 and 4, we get
,
note
then
,
![](https://www.scirp.org/html/10-80340\a7d59a89-15be-43eb-b096-27d67f8f5856.jpg)
Then
![](https://www.scirp.org/html/10-80340\b5bdeb3b-ef2a-458a-844a-a6e90465bd4a.jpg)
As
are not all equal
, therefore
, also
, then
, thus
.
That
.