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文件名称:因样本联合密度函数为-最优状态估计 卡尔曼,h∞及非线性滤波
文件大小:6.53MB
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更新时间:2021-06-15 20:42:23
概率论 茆诗松 答案
解:(1)因样本联合密度函数为
∑∑
== ==
−−
−
=
−
=
∏∏
m
i
i
n
i
i
ii
yx
mny
m
i
x
n
i
mn yyxxp 1
2
1
1
21 eee),;,,,,,( 21
1
2
1
12111
λλ
λλ λλλλλλLL ,
则似然函数
∑∑
= ==
−−
m
i
i
n
i
i yx
mnL 1
2
1
1
e),( 2121
λλ
λλλλ , ∑∑
==
−−+=
m
i
j
n
i
i yxmnL
1
2
1
12121 lnln),(ln λλλλλλ ,
令
⎪
⎪
⎩
⎪⎪
⎨
⎧
=−=
∂
∂
=−=
∂
∂
∑
∑
=
=
.0
ln
;0
ln
122
111
m
i
i
n
i
i
y
nL
x
nL
λλ
λλ
得
∑
=
= n
i
ix
n
1
1λ ,
∑
=
= m
i
iy
m
1
2λ ,
则
mn
mm
i
i
nn
i
i
mn
mn
yx
mn
yyxxp −−
==
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
∑∑
e),;,,,,,(sup
11
2111
, 21
λλ
λλ
LL ,
当λ1 = λ2 时,似然函数
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−
+
∑∑
= ==
m
i
i
n
i
i yx
mnL 11
1
e)( 11
λ
λλ , ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−+= ∑∑
==
m
i
j
n
i
i yxmnL
11
111 ln)()(ln λλλ ,
令 0
)(ln
1111
1 =⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−
+
= ∑∑
==
m
i
j
n
i
i yx
mn
d
Ld
λλ
λ
,得
∑∑
==
+
+
= m
i
i
n
i
i yx
mn
11
1λ ,
则
mn
mnm
i
i
n
i
i
mn
mn
yx
mn
yyxxp −−+
==
+
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
+
=
∑∑
e
)(
),;,,,,,(sup
11
2111
21
λλ
λλ
LL ,
故似然比检验统计量为
mm
i
i
nn
i
i
mn
mnm
i
i
n
i
i
nn
mn
mn
mn
YXmn
YXmn
YYXXp
YYXXp
YYXX
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
==Λ
∑∑
∑∑
==
+
+
==
=
11
11
2111
2111
,
11
)(
),;,,,,,(sup
),;,,,,,(sup
),,,,,(
21
21
λλ
λλ
λλ
λλ
LL
LL
LL
m
m
i
i
n
i
i
n
n
i
i
m
i
i
mn
nn
m
m
i
i
m
i
i
n
i
i
n
n
i
i
m
i
i
n
i
i
mn
nn
Y
X
X
Y
mn
mn
Y
YX
X
YX
mn
mn
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
+⋅
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
+
+
=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
+
⋅
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
+
+
=
∑
∑
∑
∑
∑
∑∑
∑
∑∑
=
=
=
=
+
=
==
=
==
+
1
1
1
1
1
11
1
11 11
)()(
;