Stat 312: Lecture 03
Minimum Variance Unbiased Estimator
Moo K. Chung
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 .
1 ¯ 1 ¯
S2 = (Xi −X)2 = Xi2 −nX 2 . ¯
n−1 n−1 Then it can be shown that X is the MVUE for
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
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
is minimized when c1 = 1/2.
 0 0 1 0 0 1 0 1 1 0 0 ...
Review Problems. You are not required to do these
problems but these are problems you should be able to
answer after each lecture. What is an unbiased estima-
tor of population parameter µ2 ? Exercise 6.3.
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-
cient estimator θ will produce an estimate closer