Point Estimation: A Statistical Approach to Unknown Parameters, Study notes of Statistics

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Chapter

(^7) Point Estimation

0 Basis^ setting :

Assume a^ random^ sample {^ X.^ Hi,

• • • , Xnl (^) from a population F- (x:O^ ) make (^) inference ( estimation or testing ) (^) for the^ unknown (^) parameters 0 Inference based^ on^ ①^ A^ set^ of^ data^ FX.^.

" Xnl ③

Assumption of^ Flx

:O) (^) for the (^) Joint distribution (^) of IX. (^) ,^ -^ -^ - Xn (^) } f is^ a point estimator^ of^ O^ such^ that^ value^ of f^ is^ taken^ as^ an^ estimate^ of^0 A particular value (^) of f (^) leg. A -^ t^ ) is a

point

estimate (^) of O %cEshmatimCriten 0 Bias

Bias (8) =^ Elf ) - O

Unbiased (^) if Bias (^) (f) =^ Elf (^) ) - 0= positively biased (^) if Elf) -^ O^ - o negatively biased^ tf^ Elf^ ) - O so E. g

. For X as a^ point estimator of

population mean^

N E (^) ( X) -^ n^ → ECI)^ =n Sampling

error is^ different^ from estimation^ error^.

On

average

.^ under^ repeated sampling ,^ an^ unbiased^ estimator^ correctly

estimates O^. 0 Variance

For the^ sample mean^ Varix)

= As (^) sample size^ n increases^ ,^ the^ estimator^ 's (^) variance and the standard error decreases 0 Mean (^) squared error and^ mean^ absolute (^) deviation MSE (^) (f) =^ E^ ( tf-^ OT)

T.tmethodefmomenLEs-hma.hn#nh--NacO)--ElXk

) is^ the^ Kth^ population moment

Mr =^ th^ FE,Xik = (^) th

( Xie^ -1^ "^ t^ Xnk^ )^ is the^ Kth sample moment

MME (^) of O^ B the^ solution to^ the^ p equations Cp (^) components of^

O )

Unf (^) ) =^ Mk (^) for Kel (^) , (^2).

• (^) - - ,

P

{ Xi^ , "- , Xn^ )^ is^ a^ random^ sample from a^ population with^ mean^ it^ and^ variance 62 - o '

find the^ NME^ of C M^ ,^621

ME ECX ) - re ha - - ML

M (^) , E. Xi -^ - X (^) ECX4= (^) TEXT m (^) n ,

A = MFF 6- = FfEH⑤

Ma - MR

LEE Xi^

• X ' = ht E. Hi^
• F) 2 Eroh (^) Elting (Xi - XY (^) ) =

f- El EE

.^ Hi

- XT

= (^) 's EFE, Kixx - 2X - Xi) ) =L (^) EEL Exit

• Etxy ]^ Eixy = (^) ECX ' ) - Eazy

= Vartxlt LEEDY

= ( 64M )

• tri) = to

ECT- ) - 62 = - Eco →

negatively biased

Hence (^) , (^) sample variance 5k (^) # E.

( Xi-^ X^ )

'

NICE.^ Xi^ '

• NE) is unbiased^ estimator^ of 6-

Mr (^) avenges to^ Uk^ as^ n→^ a^ due^ -6^ law^ of (^) large numbers

No information^ about^ distribution^ is^ used^ beyond the^ moments D (^) Least Square Estimation

Lse .tn.^.

ri =mianE* Proof Given S=^ ¥ +^ zrxaiscxi.TN , ( Xi - al '

= Hi-

XT 't (^) n Tx- as ' 70 30 =D

The value^ of a that^ minimises S^ is when^ F-a -

A-- I ② 5-^ E. Hi- al ' Set (^) data = -273 Hi - a) =o II. Hi^

• a) (^) =o E. Xi^
• na - HEE Xi^ -^ a^ a^ I^ -_ a MSE (^) Chl =^ Ella
• ng 't = It

Estimated standard Error

E. see^. CX)

In

¥, .EE (^ Xi^

• FI (^) 'T 't

LSE B^ a^ geometrical solution It makes^ no use^ of any information^ about (^) distribution ( No^

Assumption )

D (^) Maximum likelihood^ Estimation

let think^ ,^ -^

" ixn! (^) O ) be the^ joint probability density function^ for^ random

variables ( X, ,^ -^ -^ - ,^ Xn^ )

The MLE of 0 based on^ IX. .^.^ -^ - Xn )^ is

f

mqxf IX. (^) ,^ -^ -^ - , Xn^ :O (^) )

• (^) When n B large ,^ Fr N^ ( O^ , (^) ¥ ) approx. For Mn^ ,^67

fix in)^

= (^) 122625£ exp [^

• ¥ ( x- us ' ] log flxin)^

= - d-

log @^64 - ¥ (^ x

• up d (^) log fixing x- n (^) d'

logfcxioy

In = T^ - day

= - F

Fisher (^) Information Itu)^ = - f!^

• f. fix;a,^ d×^ = pl x f. Ifcxinldx = (^) Text = ft i (^). f= In^ N^ (^ n^ ,