10.5.2 Consistent estimates An estimate a i is consistent if it converges stochastically to a as m becomes large. An explicit statement of consistency is that for every small number,.∈>0，the estimate concentrates on the parameter a as m becomes large limP[o-∈<am<x+∈]→l m→00 or imP[la-al<∈]→l UESTC 11
11 UESTC 10.5.2 Consistent estimates An estimate is consistent if it converges stochastically to as m becomes large. An explicit statement of consistency is that for every small number, , the estimate concentrates on the parameter as m becomes large or ˆ m 0 ˆ m lim 1 ˆ m m P → − + → lim 1 ˆ m m P → − →
10.5.2 Consistent estimates Mean square consistent estimate m-.P]-0 In practice,it is more useful. UESTC 12
12 UESTC 10.5.2 Consistent estimates Mean square consistent estimate 2 lim 0 ˆ m m E → − = In practice, it is more useful
10.5.4 Estimators based on sufficient statistics Whether the full information is used to estimate a=T()in the? T()is a sufficient statistic for a if the conditional pdf p(yT()),does not depend on a. A method used to determine whether or not an estimate is a sufficient estimate is to check if the Fisher factorization theorem is satisfied. p(h,ynma）=g(T(y,ym),a）h(,ym) Depends only on the Independent of UESTC estimate parameter a 13 parameter a
13 UESTC Independent of parameter Depends only on the estimate & parameter 10.5.4 Estimators based on sufficient statistics is a sufficient statistic for if the conditional pdf , does not depend on . A method used to determine whether or not an estimate is a sufficient estimate is to check if the Fisher factorization theorem is satisfied. T y( ) p y y y T y ( 1 2 , , , m ( )) p y y g T y y h y y ( 1 1 1 ,... ; ,... , ,... m m m ) = ( ( ) ) ( ) Whether the full information is used to estimate in the ? ˆ =T y y ( )
Consider the sample mean where the individual random variables are independent and each is normally distributed with mean u and variance o2.The joint pdf of measurements is given by o As 2(y-2=2[(,-)+(厅-] UESTC =∑(y-'+m(厅-02+2∑(y-)(-4 ）14
14 UESTC Consider the sample mean where the individual random variables are independent and each is normally distributed with mean and variance . The joint pdf of measurements is given by As 2 ( ) ( ) ( ) 2 2 1 2 2 1 2 1 ,... ; exp 2 2 m i m m i y p y y = − = − ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 1 1 2 2 1 1 2 m m i i i i m m i i i i y y y y y y m y y y y = = = = − = − + − = − + − + − −
Where y=a=2.s,2=62-2(0y-j ,i=1 The pdf can be expressed as py,ym4,G2）= h(y,2,…,ym)=1 exp - 2 1 -m- exp -m ×1 2o2 2o2 (2o2） In this case Depends only on the Independent of estimate parameter parameter UESTC Both are sufficient estimates! 15
15 UESTC Where The pdf can be expressed as In this case 1 1 ˆ = , m i i y = y m = ( ) 2 2 2 1 1 s ˆ m y i i y y m = = = − ( ) ( ) ( ) 2 1 2 2 2 2 / 2 2 ,... ; y 1 exp exp 1 2 2 2 m y m p y y s m m = − − − h y y y ( 1 2 , , , 1 m ) = Independent of parameter Depends only on the estimate & parameter Both are sufficient estimates!