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CPS1416 Jungtaek O. et al.
3. Suggested generalized ratio-type estimator

3.1 Ratio-type estimator with p auxiliary variables
Cochran (1977) studied in detail about the ratio estimator. Also Srivastava
(1967) proposed the ratiotype estimator following as

̅
̂
Y ̅ = ̅ ( ) . (5)

Now we suggest a generalized ratio-type estimator defined by

̅
̂

Y ̅ = ̅ ∏ ( ) . (6)
=1

Then using the first order Taylor approximation, we obtain

̅ ̅
̂ ≈ ̅ (1 + ∑ ( − 1)) = ̅ (1 − ∑ ) + ∑ . (7)
̅
Y ̅

=1 =1 =1
̂
̂
Therefore by comparing (2) and (7) we have = ̅ = . Taking ̂ =

̂ , the generalized ratio-type estimator becomes approximately equal to the
̂
̂
multiple regression estimator. Also we obtain the relationship of =
0
y (1 − ∑ ̂ ). For the simple regression case, ̂ = ̂ 1 is well known to
=1 ̂
̅
minimize the first order approximate MSE where = . It can be confirmed by
̂

the results of Srivastava (1967) and Tailor et al (2015).
Also as a generalized version of the ratio estimator, a multiple ratio
estimator with p auxiliary variables which is suitable for the multiple regression
model without intercept can be simply obtained by taking ∑ ̂ = 1. As
=1
results, we suggest two estimators, the multiple ratio estimator and the
̂
generalized ratio-type estimator, Y ̅ defined by
̅
̂

Y ̅ = ̅ ∏ ( ) ̂ , (8)
=1

̅
̂

Y ̅ = ̅ ∏ ( ) ̂ , (9)
=1

̂ ̅
̂
̂
where ̂ = ̂ , ̂ = , = and , = 1, … , are estimated

∑
=1 ̂ ̂
regression coefficients.

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