Download 1 Important Formulas in Ratio Estimation

STAT 152, Fall 2003
Summary for Chapter 3
The theme for this chapter is using known information to improve the precision of estimates of means and totals. Ratio and Regression estimation improve
precision when the variability of the residuals (yi − yi ), where yi is the predicted
value from the ratio or regression model, is smaller than the variability of the
(yi − y).
1
Important Formulas in Ratio Estimation
For ratio estimation to apply, two quantities yi and xi must be measured on
each sample unit. If an SRS is taken, natural estimators for ratio B, population
total ty , and population mean y U are:
=
• B
y
x
ty
tx
=
x
• tyr = Bt
U
• y r = Bx
Bias and mean squared error of ratio estimators:
= −Cov(B,
x)/xU
• |Bias(B)|
•
|Bias(B)|
1/2
[V (B)]
≤ CV (x)
− B] ≈ (1 −
• E[B
− B)2 ] ≈
• E[(B
= (1 −
• V [B]
n
1
N ) nx2U
1
x2U
(BSx2 − RSx Sy ) =
1
x2U
[BV (x) − Cov(x, y)]
V (d), where d = y − Bx
s2e
n
N ) nx2U
= (1 −
= N 2 (1 −
• V [
tyr ] = V [tx B]
= (1 −
• V [
y r ] = V [xU B]
n
1
N ) nx2U
2
n se
N) n
2
n se
N)n
1
i∈S
i )2
(yi −Bx
n−1
i
, where ei = yi − Bx
2
Regression Estimation
Ratio estimation works best if the data are well fit by a straight line through
the origin. Sometimes, data appear to be evenly scattered about a straight line
that does not go through the origin-that is, the data look as though the usual
straight line regression model: y = B0 + B1 x
Important formulas in regression estimation:
0 + B
1 xU , estimator of y U .
y reg = B
• 0 and B
• B
1 are the ordinary least squares regression coefficients:
(xi −x)(yi −y)
rs
i∈S
1 = 0 = y − B
1 x.
B
= sxy , B
(x −x)2
i∈S
i
1 , x)
• Bias: E[
yreg − y U ] = −Cov(B
2
(yi −[y U +B1 (xi −xU )])2
n Sd
• Mean Squared Error: MSE(
y reg ) ≈ (1− N
) n , where Sd2 = N
i=1
N −1
n s2e
0 + B
1 xi ).
• Standard Error: SE(
y reg ) = (1 − N
) n , where ei = yi − (B
3
Estimation in Domains
Suppose there are D domains. Let Ud be the index set of the units in the population that are in domain d and let Sd be the index set of the units in the sample
that are in domain d, for d = 1, 2, ..., D. Let Nd be the number of population
units in Ud , and nd be the number of sample units in Sd .
Important formulas in domain estimation:
• y Ud = i∈Ud Nyid .
• y d = i∈Sd nydi , a natural estimator of y Ud
2
n syd
• Standard Error of y d : SE(y d ) ≈ (1 − N
) nd
yi if i ∈ Ud
• tyd = N u, where ui =
0 if i ∈
/ Ud
n s2u
• Standard Error of tyd : SE(tyd ) = N (1 − N
)n
2
4
Comparison
SRS
Ratio
Regression
Population Total
Estimator
Population Mean
Estimator
ei
ty
ty ( tx )
tx
1 (xU − x)]
N [y + B
y
y( tx )
tx
1 (xU − x)]
y+B
yi − y
i
yi − Bx
0 − B
1 xi
yi − B
The population total estimator variance is N 2 (1 −
The population mean estimator variance is (1 −
3
2
n se
N)n.
2
n se
N)n .