2015-7-3 xlpeng 1 Ratio Estimation and Regression Estimation (Chapter 4, Textbook, Barnett, V., 1991) 2.1 Estimation of a population ratio:

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Ratio Estimation and Regression Estimation(Chapter 4, Textbook, Barnett, V., 1991)

2.1 Estimation of a population ratio: The ratio estimator In some situations it is useful to estimate a (positive) ratio of two

population characteristics: the totals, or means, of two (positive) variables X and Y.

The sample average of ratio

unbiased for estimating the population mean

Two obvious estimators of R are

The ratio of the sample averages

is widely used.


1 1

1 1( / )

n n

i i ii i

r y x rn n

/ /T Tr y x y x

1 1

1 1( / )

N N

j j jj j

R R Y XN N

but biased for estimating R

The bias in estimating R by r

The bias in estimating R by r is the expectation of the following difference:

(2.3)


( ) /r R y Rx x 1

1y Rx x X

X X

2

1 .y Rx x X x X

X X X

2

[( )( )]( )

y Rx E y Rx x XE r R E

X X

Discussion about the bias


≈


(2.5)

2

21

2 2 22

( )1

1

12

Nj j

j

Y YX X

Y RXf

nX N

fS RS R S

nX

( ) ( )j j j j jZ Y RX Y Y RX RX


A (slightly) biased estimate of the true variance parameter ( )V r

2 2

2 21

( ) (1 )1ˆ( )1

ni i r

i

y rx f sfV r

nx n nx

For large n, an approximate 100 (1-α) % (symmetric) two-sided confidence interval for the population ratio R is:

)(ˆ)(ˆ rVzrRrVzr

And the required sample size is

2 2 2/V d x z

2 2 / 4d x

2.2 Ratio estimation of a population mean or total


( / )Ry rX X x y

( / )TR T Ry rX NX x y Ny

Variance of ratio estimator



Example: (Food additive)

A researcher was investigating a new food additive for cattle. Midway through the two-month study, he was interested in estimating the average weight for the entire herd of N = 500 steers. A simple random sample of n = 12 steers was selected from the herd and weighed. These data and prestudy weights are presented in the accompanying table for all cattle sampled. Assume the prestudy average = 880 pounds.

Estimate the ratio of present weight to prestudy weight of the herd, and provide an estimate of the standard error for your answer.

Which points have greatest influence on the estimate?




Solution:


The estimate of the ratio R of the present weight to prestudy weight for the herd is:

Solution:

000929.012

646.848,8)

500

121(

880

11)(

22

2

rSXn

frVar

030485.0000929.0)( rse

Sample size determination


Hence we should sample 94 steers to estimate R, the change in weight of herd after the study with error bound of 1%.

Example: (Sugar content)


In a study to estimate the total sugar content of a truckload of oranges, a SRS of n = 10 oranges was juiced and weighted. The total weight of all the oranges, obtained by first weighing the truck loaded and then unloaded, was found to be 1800 pounds. Estimate Y , the total sugar content for the oranges and provide the standard error of the estimate.


Solution:

The scatter plot shows a strong positive association between sugar content and weight, making the ratio estimator a reasonable choice.

An estimate of Y is


Solution:

Example: (Promotional campaign)


An advertising firm is concerned about the effect of a new regional promotional campaign on the total dollar sales for a particular product. A SRS of n = 20

stores is drawn from the N = 452 regional stores in which the product is sold. Quarterly sales data are obtained for the current three-month period and the three-month period prior to the new campaign. The pre-campaign sales for all stores X = 260, 256. Check the scatter plot to see if these stores are in two different size groups.


Example: (Promotional campaign)


Solution:

(a) Without using the auxiliary information, the estimate of the average current three-month sales using ordinary estimator is


Solution:

(b) When the total pre-campaign three-month sales is known to be X = 260256, the average pre-campaign three-month sales is

Then the estimate of the average current three-month sales using ratio estimator is

which represent an average increase of 7.1% of the current three-month sales from the pre-campaign three-month sales.

Solution:


The ratio estimator here is much better than ordinary estimator since the current three-month sales yi is closely and positively related to the pre-campaign three-month sales xi with correlation coefficient 0.9986.



This examines when the variance of (2.10) could be less or greater than that of (1.9)


2.3 Regression estimation

Condition (2.15.1) demands that X and Y be linearly related, but, if the linear relationship does not pass through the origin, then, it suggests considering an alternative estimator known as regression estimator.




A practicable simple linear regression model is (2.17)

.

An ideal (perfect) linear relationship is

(2.16)

)( jj XXbYY

(2.18)

jjj EXXbYY )(



Consider the average (mean) of either (2.16) or (2.17),

( )Ly y b X x (2.19)



2( ) [( ) ]L LVar y E y Y 2

2 2 2

2 2

{[( ) ( )] }

1( 2 )

1(1 )

L

Y YX X

Y YX

E y Y b x X

fS bS b S

nfS

n

21( )Y

fS Var y

n

(2.20)

y



(2.21) 2 2 21ˆ( ) ( 2 )L y yx x

fV y s bs b s

n



From (2.20),

2 2 21min { ( )} min ( 2 )b L b Y YX X

fVar y S bS b S

n

2 21(1 )Y YX

fS

n

The minimum is obtained with 2min / /YX X YX Y Xb b S S S S

Y

Thus the most efficient regression estimator of is

( / )( )L YX Y Xy y S S X x



The optimal value of b of (2.22) suggests the obvious estimate:

1min 2 2

1

( )( )( )

( )

n

i iyx in

x ii

y y x xsb b

s x x

(2.24)

( )Ly y b X x (2.25)

which enjoys the following asymptotic properties:

1( ) ( )LE y Y O n



Asymptotic properties:

( )LVar y 2 2 2 3/21( / ) ( )Y YX X

fS S S O n

n

21( ) ( )L y yx

fV y s bs

n

(2.27)

2.4 Comparison of ratio and regression estimators



2.4 Comparison of ratio and regression estimators

2 2 2 21( ) ( ) 2R L X YX Y X YX Y

fV y Var y R S R S S S

n

21X YX Y

fRS S

n

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2015-7-3 xlpeng 1 Ratio Estimation and Regression Estimation (Chapter 4, Textbook, Barnett, V., 1991) 2.1 Estimation of a population ratio: