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