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
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
accordingly, the specific conditional root square estimation of 
is
0 < k < 1.
similarly, the generalized conditional root square estimation of 
is
.
Lemma 6
, where 
.
Proof:
when
,
.
Because
,
so 
.
Because
So
Lemma 7
,
.
Lemma 8 When
, exist
, when
, then
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,
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
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
Assume
then
so
.
Theorem 4 In the model (1), for
, if 
, 
, then
, Where
.
Proof:
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
,
Then
As 
are not all equal
, therefore
, also
, then
, thus
.
That
.