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Chapter 6 (cont.)Relative Efficiency of Estimators
Compare the variances of this chapter’s 3 estimators of the population mean (ratio, regression, difference)Compare these variances to that of the sample mean from a SRS
y
But First, Need to Address Bias Generally, it’s inappropriate to compare
variances of biased estimators
The bias becomes negligible if the relationship between x and y falls along a straight line through the origin (next slide)
For from a SRS, ( ) yy E y
ˆRatio estimator of , , is biased
since is biased for
y y x x
y
x
y rx
yr Rx
Approx. of Relative Bias of r2
2
( ) ˆ yx xss sE r R N n
R Nn x y x
Recall that the ratio estimator
works well when the relationshipbetween and can be describedby a line through the origin andthe point ( , ), that is, the line
through the origin with slope .
yrx
x y
x yyrx
400 500 600 700 800 900 1000 1100 1200 1300 14000
2004006008001000120014001600
Previous Steer Weight Example
x, Prestudy Steer Weight
y, P
rese
nt S
teer
Wei
ght
2
2ˆ ˆ ˆIf , then 0y y yx x x
x
s s sy s s ss x y x y xx
Relative Efficiency
How do we tell which one is best for a particular sampling situation?
Cannot always answer definitively, but there are some guidelines.
One such guideline: relative efficiency.
1
regression estimatorˆ (
differenc
ˆratio estimato
e estimatorˆ
The sample mean ,
,
,
can all be used to estimat
)
(
r
e .
)y
yL
x
y
y x
x
D
y b
y
y
x
yx
x
Relative Efficiency - 21 2
1 2
Suppose and are estimators of a population parameter.
If and are unbiased (or nearly unbiased), wegenerally choose the estimator with the smaller variance.
Should compare the variances assuming
E E
E E
1 2
1 2
2 1
equal sample sizes.
Compare the variances by looking at the r
Relative efficiency of to i
a
s
( )( )
tio.
E E
E V EREE V E
Relative Efficiency - 3
1 2
1 2
2 1
1 2
2 1
In most situations have only estimated v
Relative efficiency of to is
( )( )
ˆ( )So define ˆ( )
ariances.
E E
E V EREE V E
E V EREE V E
2
11
2
Be careful. does not necessarily mean ( ) ( )
since dealing only with estimators of variances based on sample data.
But if samples are large and we have good estimators of the varianc
1EREE
V E V E
1
1
2
es
and is considerably greater than 1, we can be confident that
is the better estimator.
RE
E
EE
Relative Efficiency-4 ˆ yRE
y
ˆRelative efficiency of the ratio estimator to the sample mean y x xyr yx
2
2
2 2
2 2 2 2
ˆˆ ( )ˆ ˆ( )
ˆ2
y
y
ry
y y
r y x x y
sV y nRE
sy Vn
s ss s r s r s s
2
2
2 1
ˆ ˆ( ) 1
( )where
1
ry
n
i ii
r
snVN n
y rxs
n
2 2 2 2
2 2
2
Soˆ
1if
ˆ2
ˆor 2
ˆor 2 (assuming 0)
1 ( )ˆor >2 2 2 ( )
y
y y x x y
x y x
x y x
x x
y y
REy
s s r s r s s
r s s r s
ys s rs rx
rs s x cv xs s y cv y
Relative Efficiency-5 ˆ yREy
ˆ 1 ( )
From previous slide
ˆ1 if >2 ( )
y cv xREy cv y
• In ratio estimation, the y values are frequently updated x values (for example, 1st quarter earnings this year compared to 1st quarter earnings last year).
• In such situations cv(y) is frequently very close in value to cv(x)
1ˆˆIn these cases, the ratio estimator is more efficient than when .2y y
ˆIn general, the ratio estimator is more efficient than if the variation
among the values is small relative to the variation among the valuesand when the correlation between and is a high po
y y
x yx y
sitive value.If possible, try to choose values that are nearly constant.x
Relative Efficiency-6 ˆ yLREy
2 2 21
1 1
2 2 21
1 Recall:
1 1ˆ ˆ( ) 1 ( ) ( )2
slight change: replace ( 2) with
regression estima
( 1)
1ˆ ˆ
torˆ ( )
( ) 1 1
.n n
yL i ii
yL
i
x
x
yL y
nV y y b x xn N n
n n
n nV s
y b x
b sn N N
2
2
1
ˆ1
ˆsince
y
y
x
sn
sb
s
ˆThis approximation to ( ) is good as long as is reasonably large.yLV n
1ˆRelative efficiency of the regression estimator ( )
to the sample mean yL xy b x
y
Relative Efficiency-7 ˆ yLREy
22
2
2 2 2
From previous slide
ˆ ˆˆ( ) 1 1 ,so (ignoring fpc)
ˆ 1 ˆ1 if 0ˆ ˆ(1 ) 1
yyL
yL y
y
snVN ns
REy s
Thus, is always more efficient than as an estimator of . (However, can have serious bias problems unless the regression of y on x is truly linear.
ˆ yL yy ˆ yL
Relative Efficiency-8 ˆˆyL
y
RE
1ˆRelative efficiency of the regression estimator ( )ˆto the ratio estimator
yL x
y
y b x
2 2 2
2 2
2 2 2 2 2
2 2 2 2
2
21 1
21
ˆ ˆˆ ˆ( ) 2ˆ ˆˆ (1 )ˆ( )
ˆ1
ˆ
ˆ ˆ2 (1 )
ˆ ˆ2 0
ˆ 0
ˆ0, since ,
0
yL y y x x y
y yyL
yL
y
y x x y y
x x y y
y x
yx x
x
V s r s r s sRE
sV
RE
s r s r s s s
r s r s s s
s rs
sb s rs b
s
b r
Relative Efficiency-9 ˆˆyL
y
RE
21
From previous slide
ˆ1 0
ˆyL
y
RE b r
So the regression estimator is more efficient than the ratio estimator unless , in which case they are equivalent.
ˆyLˆ y
1yb r x
1
Sinc
.
e the least squares line always goes through ( , ),
when the least squares line also goes through (0,0)
x yybx
Relative Efficiency-10ˆRelative efficiency of the difference estimator ( )
to the sample mean . Note that both are unbiased for .yD x
y
y x
y
ˆ yDREy
2
1
2
1
2 2
Recall
( )ˆ ˆ( ) 1 , where ,
( 1)
11 ( ) ( )( 1)
1 ˆ1 2
n
ii
yD i i i
n
i ii
y x x y
d dnV d y xN n n
n y y x xN n n
n s s s sn N
Relative Efficiency-11 ˆ yDREy
2 2
2
2 2
2
From previous slide1ˆ ˆˆ( ) 1 2 ,so (ignoring fpc)
ˆ;
ˆ2
ˆˆ ˆ1 2
2
yD y x x y
yD y
y x x y
yD xx y x
y
nV s s s sn N
sRE
y s s s s
sRE s s s
y s
If the variation in x and y values is about the same, then the difference estimator is more efficient than when the correlation between x and y is greater than ½.
ˆ yD
y
Relative Efficiency-12 ˆˆyL
yD
RE
1ˆRelative efficiency of the regression estimator ( )ˆto the difference estimator ( ).
yL x
yD x
y b x
y x
2 2
2 2
2 2 2
2
21 1
ˆ ˆˆ ˆ( ) 2ˆ ˆˆ (1 )ˆ( )
ˆˆ ˆ1 2 0
ˆ
ˆ( ) 0
ˆ( ) 0 since
yL yD y x x y
yD yyL
yLx x y y
yD
x y
yx x
x
V s s s sRE
sV
RE s s s s
s s
ss b s b
s
The regression estimator will be equivalent to the difference estimator when b1 = 1. Otherwise, the regression estimator will be more efficient than the difference estimator.
ˆyLˆ yD
Relative Efficiency-13 Summary
2
2 2 2
ˆ 1 ( )ˆ1 if >ˆ2 2 ( )
y y
y x x y
s cv xREy s r s r s s cv y
2
ˆ 1 ˆ1 if 0ˆ1
yLREy
2 2 2
212 2
ˆˆ 21 if 0
ˆˆ (1 )yL y x x y
y y
s r s r s sRE b r
s
2
2 2
ˆˆ1 if
ˆ2 2yD y x
y x x y y
s sRE
y s s s s s
2 2
212 2
ˆˆ 21 if ( ) 0
ˆˆ (1 )yL y x x y
x xyD y
s s s sRE s b s
s