【8】p-Normal Distribution's MLE
考虑(p)元正态总体(Xsim N_p(mu,Sigma)),设(X_{(i)}=(x_{i1},dots,x_{ip})',)((i=1,dots,n))为(p)元正态总体(X)的简单随机样本,则有观测数据库:
[X=
left(
egin{array}
{cccc}
x_{11} & x_{12} & dots & x_{1p}\
x_{21} & x_{22} & dots & x_{2p}\
vdots & vdots & & vdots \
x_{n1} & x_{n2} & dots & x_{np}\
end{array}
ight)
]
是一个随机阵。
引理(1)
对于矩阵(A_{m imes n}),(A_{n imes m})有:(tr(AB)=tr(BA))。
[egin{align}
tr(AB)=&sum_{i=1}^m(AB)_{ii}\
=&sum_{i=1}^m(sum_{j=1}^na_{ij}b_{ji})\
同理:tr(BA)=&sum_{i=1}^n(sum_{j=1}^mb_{ij}a_{ji})
end{align}
]
由于(Sigma)的可交换性,因此:
[tr(AB)=sum_{i=1}^m(sum_{j=1}^na_{ij}b_{ji})=sum_{i=1}^n(sum_{j=1}^mb_{ij}a_{ji})=tr(BA)
]
于是可以推广到多个矩阵相乘:
[tr(prod_{i=1}^nA_i)=tr(A_nprod_{i=1}^{n-1}A_i)
]
似然函数(L(mu,Sigma))
对于样本(X_{(i)}=(x_{i1},dots,x_{ip})',)((i=1,dots,n)),其联合密度函数为:
[f(x_{(i)})=frac1{(2pi)^{p/2}|Sigma|^{1/2}}expleft{-frac12(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)
ight}
]
而由似然函数定义:
[egin{align}
L(mu,Sigma)=&prod_{i=1}^nf(x_{(i)})\
=&prod_{i=1}^nfrac1{(2pi)^{p/2}|Sigma|^{1/2}}expleft{-frac12(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)
ight}\
=&frac1{(2pi)^{np/2}|Sigma|^{n/2}}expleft{-frac12sum_{i=1}^n(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)
ight}
end{align}
]
由于:
[(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)=C_0(是一个数)
]
所以:
[(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)=tr{(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)}=tr(C_0)
]
于是:
[egin{align}
L(mu,Sigma)
=&frac1{(2pi)^{np/2}|Sigma|^{n/2}}expleft{-frac12sum_{i=1}^n(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)
ight}\
=&frac1{(2pi)^{np/2}|Sigma|^{n/2}}expleft{-frac12sum_{i=1}^ntr[(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)]
ight}\
(由引理)=&frac1{(2pi)^{np/2}|Sigma|^{n/2}}expleft{-frac12sum_{i=1}^ntr[Sigma^{-1}(x_{(i)}-mu)(x_{(i)}-mu)']
ight}\
=&frac1{(2pi)^{np/2}|Sigma|^{n/2}}expleft{tr(-frac12Sigma^{-1}sum_{i=1}^n(x_{(i)}-mu)(x_{(i)}-mu)')
ight}\
end{align}
]
其中:
[egin{align}
&sum_{i=1}^n(x_{(i)}-mu)(x_{(i)}-mu)'\
=&sum_{i=1}^n(x_{(i)}-overline{X}+overline{X}-mu)(x_{(i)}-overline{X}+overline{X}-mu)'\
=&sum_{i=1}^n(x_{(i)}-overline{X})(x_{(i)}-overline{X})'+n(overline{X}-mu)(overline{X}-mu)'\
=&A+n(overline{X}-mu)(overline{X}-mu)'
end{align}
]
带回似然函数可得:
[egin{align}
L(mu,Sigma)=&frac1{(2pi)^{np/2}|Sigma|^{n/2}}etrleft{-frac12Sigma^{-1}sum_{i=1}^n(x_{(i)}-mu)(x_{(i)}-mu)'
ight}\
=&frac1{(2pi)^{np/2}|Sigma|^{n/2}}etrleft{-frac12Sigma^{-1}(A+n(overline{X}-mu)(overline{X}-mu)')
ight}\
(两边求对数)ln{L(mu,Sigma)}=&
-frac{np}2ln{(2pi)}-frac{n}2ln{|Sigma|}-frac12trleft[Sigma^{-1}(A+n(overline{X}-mu)(overline{X}-mu)')
ight]\
=&
-frac{np}2ln{(2pi)}-frac{n}2ln{|Sigma|}-frac12trleft[Sigma^{-1}A+Sigma^{-1}n(overline{X}-mu)(overline{X}-mu)'
ight]\
=&
-frac{np}2ln{(2pi)}-frac{n}2ln{|Sigma|}-frac12tr[Sigma^{-1}A]-frac12tr[Sigma^{-1}n(overline{X}-mu)(overline{X}-mu)']\
(仅当mu=overline{X}时取等号)leq&-frac{np}2ln{(2pi)}-frac{n}2ln{|Sigma|}-frac12tr[Sigma^{-1}A]\
end{align}
]
求似然函数极大值:
方法一
- (引理二)
设(B)为(p)阶正定矩阵,则:(tr(B)-ln{|B|}geq p).
[ln{|B|}=sum_{i=1}^pln{lambda_i}=sum_{i=1}^pln{(1+lambda_i-1)}leqsum_{i=1}^p(lambda_i-1)=tr(B)-p
]
则有:
[tr{\,(B)}-ln{|B|}ge p ag{引理,证毕}
]
于是观察(ln{L(mu,Sigma)})得:
[egin{align}
ln{L(mu,Sigma)}=&
-frac{np}2ln{(2pi)}-frac{n}2ln{|Sigma|}-frac12tr[Sigma^{-1}A]\
=&-frac{np}2ln{(2pi)}-frac{n}2left[ln{|Sigma|}+tr[Sigma^{-1}frac{A}n]
ight]\
=&-frac{np}2ln{(2pi)}-frac{n}2left[tr[Sigma^{-1}frac{A}n]-ln{|Sigma^{-1}frac An|}+ln{frac{A}{n}}
ight]\
leq&-frac{np}2ln{(2pi)}-frac n2(p+ln{|frac An|})
end{align}
]
以上不等式的等号当且仅当(Sigma^{-1}frac{A}n=I_p)时成立,于是(Sigma=frac{A}n),则有:
[ln{L(overline{X},frac1nA)}=max_{overline{X},Sigma>0}ln{L(overline{X},Sigma)}=-frac{np}2(1+ln(2pi))-frac n2ln{|frac{A}n|}
]
其中
[overline{X}=frac1nsum_{i=1}^nX_{(i)}\
Sigma=frac1nsum_{i=1}^n(x_{(i)}-overline{X})(x_{(i)}-overline{X})'
]