PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
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2. STATISTICAL INFERENCE<br />
According to Rao-Blackwell, T ∗ = E(T |S) will be an improved (unbiased) estimator<br />
for µ.<br />
( T<br />
If x = (x 1 , . . . , x n ) T for X ∼ N n (µ1, σ 2 I), then = Ax, where A is a matrix<br />
S)<br />
⎛<br />
A = ⎝<br />
1 0 . . . 0<br />
1 1 . . . 1<br />
⎞<br />
⎠<br />
Hence,<br />
⎛⎛<br />
( T<br />
∼ N 2 (A(µ1), σ<br />
S)<br />
2 AA T ) = N 2<br />
⎝⎝<br />
µ<br />
nµ<br />
⎞ ⎛<br />
⎠ , ⎝<br />
⎞⎞<br />
σ 2 σ 2<br />
⎠⎠<br />
nσ 2<br />
σ 2<br />
=⇒ T |S = s ∼<br />
)<br />
N<br />
(µ + σ2<br />
nσ (s − nµ), 2 σ2 − σ4<br />
nσ 2<br />
=<br />
( ( 1<br />
N<br />
n s, 1 − 1 ) )<br />
σ 2<br />
n<br />
∴ E(T |S) = 1 n<br />
is a better (or equal) estimator for µ.<br />
∑<br />
xi = ¯x = T ∗<br />
Finally, observe Var( ¯X) = σ2<br />
n ≤ σ2 = Var(X 1 ) with < for n > 1, σ 2 > 0.<br />
Remarks<br />
(1) We have given only an incomplete outline <strong>of</strong> theory.<br />
(2) There are also concepts <strong>of</strong> minimal sufficient & complete statistics to be<br />
considered.<br />
2.3 Methods Of Estimation<br />
2.3.1 Method Of Moments<br />
Consider a random sample X 1 , X 2 , . . . , X n from F (θ) & let µ = µ(θ) = E(X).<br />
The method <strong>of</strong> moments estimator ˜θ is defined as the solution to the equation<br />
¯X = µ(˜θ).<br />
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