# Stat 312 Lecture 03 Minimum Variance Unbiased Estimator

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```					                                 Stat 312: Lecture 03
Minimum Variance Unbiased Estimator
Moo K. Chung
mchung@stat.wisc.edu

September 9, 2004

1. Population of interest is a collection of measur-                                          ¯
to the true parameter θ. X is MVUE for µ (we
able objects we are studying. Let X1 , · · · , Xn be            will not prove this statement).
a random sample from the population. Then sam-
¯
ple mean X and sample variance S 2 are unbiased              4. Given a random sample X1 , · · · , Xn , a linear es-
estimators of population mean µ and population                  timator of parameter θ is an estimator of form
variance σ 2 respectively.                                                              n
Proof. Note that                                                                  ˆ
θ=         ci Xi .
n                                             i=1
1                ¯        1                 ¯
S2 =                (Xi −X)2 =                Xi2 −nX 2 .                                    ¯
n−1                       n−1                            Then it can be shown that X is the MVUE for
i=1                       i=1
population mean µ among all possible linear esti-
¯         ¯
Then using the fact E(X)2 = VX + (EX)2 =    ¯                   mators.
σ 2 /n + µ2 , it can be shown that ES 2 = σ 2 .
Proof. Case n = 2 will be proved. The general
statement follows inductively. Consider linear es-
2. There may be many unbiased estimators of θ.
ˆ       ˆ                      timators
Given two unbiased estimators θ1 and θ2 of θ. We
ˆ                                ˆ
µ = c1 X1 + c2 X2 .
choose one that gives less variance. If V(θ1 ) ≤
ˆ ˆ                                   ˆ
V(θ2 ), θ1 is called more efﬁcient than θ2 . An efﬁ-            To be unbiased, c1 + c2 = 1. To be most efﬁcient
cient estimator has less variability so we are more             among all unbiased linear estimators, the variance
likely to make an estimate close to the true param-             has to be minimized. The variance is
eter value. The following coin ﬂipping example
clearly demonstrate this.                                         Vˆ = c2 VX1 + c2 VX2 = c2 + (1 − c1 )2 σ 2
µ    1        2        1

The quadratic term in the bracket 2c2 − 2c1 + 1
1
> a<-rbinom(1000,1,0.5)
is minimized when c1 = 1/2.
> a
[1] 0 0 1 0 0 1 0 1 1 0 0 ...
Review Problems. You are not required to do these
> mean(a)
problems but these are problems you should be able to
[1] 0.517
answer after each lecture. What is an unbiased estima-
> mean(a[1:3])
tor of population parameter µ2 ? Exercise 6.3.
[1] 0.3333333
> mean(a[1:9])
[1] 0.4444444
> mean(a[1:11])
[1] 0.3636364

3. Among all unbiased estimators, we choose the
most efﬁcient estimator called the minimum vari-
ance unbiased estimator (MVUE). The MVUE is
an unbiased estimator with the smallest variance.
MVUE is the most efﬁcient estimator. An efﬁ-
ˆ
cient estimator θ will produce an estimate closer

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