电子科技大学：《信号检测与估计 Signal Detection and Estimation》课程教学资源（课件讲稿）Chapter 10 Fundamentals of Estimation Theory

Chapter 10 Fundamentals of Estimation Theory UESTC 1

1 UESTC Chapter 10 Fundamentals of Estimation Theory

10.1 Chapter highlights Parameter estimation problem formulation Properties of estimators ·Bayes estimation ·inimax estimation Maximum-likelihood estimation Comparison of estimators of parameters UESTC 2

2 UESTC 10.1 Chapter highlights • Parameter estimation problem formulation • Properties of estimators • Bayes estimation • Minimax estimation • Maximum-likelihood estimation • Comparison of estimators of parameters

10.2 Formulation of the general parameter estimation problem The parameter estimation problem is to find a parameter estimate vector that depends on the observables. Estimator (估计子) Signal-processing 寸={y1w…,yml algorithms computing estimate of parameters &(G=a,,a from measurements Figure 10.1.Parameter estimation from a set of measurements Construct a function &(to estimate parameters! UESTC 3

3 UESTC 10.2 Formulation of the general parameter estimation problem The parameter estimation problem is to find a parameter estimate vector that depends on the observables. Estimator (估计子) Construct a function to estimate parameters! α ˆ( y)

5 The parameter estimate (depends on the random observables and is itself a random variable. The parameter a may be random vector or a nonrandom parameter vector. Specific cases of parameter estimation occur depending on whether pdf is known or unknown. UESTC 4

4 UESTC • The parameter estimate depends on the random observables and is itself a random variable. • The parameter may be random vector or a nonrandom parameter vector. • Specific cases of parameter estimation occur depending on whether pdf is known or unknown. α ˆ( y) α

Example 10.1 Constant number plus zero mean noise are yi=u+n, i=1,.,m The estimate can be defined by 空小点m m i=1 A direct estimation way (moment based method) 62=2(0y- 1ni1 i=1 UESTC 5

5 UESTC Example 10.1 Constant number plus zero mean noise are , 1,...., i i y n i m = + =  The estimate can be defined by 1 1 ˆ m i i y m  = =  A direct estimation way (moment based method): 1 1 ˆ m i i y m  = =  ( ) 2 2 1 1 ˆ ˆ m i i y m   = = −  ( ) ( ) 2 2 1 ; , exp 2 2 i i y p y           − = −     

10.3 Types of estimation problems 36 Parameter estimation Known distribution observables Distribution-free estimation Bayes (optimal estimate) Minimum variance (MV) (cost function needed) Mean-square error cost Least-square function (minimum MSE) Maximum a posteriori(MAP) Durbin (a priori pdf of a needed) Levinson/Weiner Absolute error cost function Cholesky Maximum likelihood (ML) (conditional pdf p(la)needed) Burg Minimax (MMX)estimate Lattice Kalman Square-root Kalman Figure 10.3.Classification of parameter estimation problems

6 UESTC 10.3 Types of estimation problems

10.5 Properties of estimators For same observables,we can define different ( How can we comment each a()? ·Unbiased ·Consistent ·Sufficient Minimum-variance estimates (efficient) Asymptotically efficient Asymptotically normal UESTC 7

7 UESTC 10.5 Properties of estimators For same observables, we can define different . How can we comment each ? • Unbiased • Consistent • Sufficient • Minimum-variance estimates (efficient) • Asymptotically efficient • Asymptotically normal α ˆ( y) α ˆ( y)

10.5.1 Unbiased estimates An estimate a(y)is unbiased,if Ea-a is a constant vector E{a=E{a}-n。 is a random vector Asymptotically unbiased estimate lim=F)- UESTC 8

An estimate is unbiased, if 8 UESTC 10.5.1 Unbiased estimates ( ) ˆ  y       ˆ is a constant vector ˆ E = is a random vector         = =   Asymptotically unbiased estimate     ˆ lim = m    E  →  =

10.5.1 Unbiased estimates In the example 10.1 ·the sample mean Unbiased i)2y· m i= ·The sample variance 6=12(y-) m i=l UESTC 9

9 UESTC 10.5.1 Unbiased estimates In the example 10.1 • the sample mean • The sample variance ( ) 1 1 ˆ=ˆ m i m i=  α y y =  Unbiased ( ) 2 2 1 1 ˆ ˆ m i m i   = = −  y

However,From(10.9)-(10.14) Ea}=2c-} m 2 m-1o2 m m lim EG2)-1 lim m-lo2=g2 m→0 m The sample variance is biased but asymptotically unbiased estimate. UESTC 10

10 UESTC However, From(10.9)-(10.14) The sample variance is biased but asymptotically unbiased estimate.   ( )  2 2 2 1 2 2 2 1 ˆ 1 m i i E E m m m m m       = = − − − = − =  y   2 2 2 1 lim lim ˆ m m m E m    → → − = =