Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 11

An estimator ʸ

is said to be unbiased for ʸ if

E ʸ ¼ ʸ;

ð 1 :

17 Þ

where E ʸ ¼ X
s2Ω

ʸ

ðÞpðÞ is the expected value of the estimator ʸ . This means

that the average value of the estimates over all possible samples must be equal to the
value of the (unknown) population parameter to be estimated. It will never system-
atically under- or overestimate the population value.
The precision of an estimator is represented by the mean square error, defined as

¼ X
s2Ω

2 2 pðÞ:

MSE ¼ E ʸ ʸ

ʸ

ðÞʸ ð 1 : 18 Þ

ʸ 1 and ʸ 2 , for the parameter ʸ 1 is said to be more

Consider two estimators, ʸ .

efficient than ʸ 2 if

MSE ʸ 1 MSE ʸ 2 : ð 1 : 19 Þ

This property means that the variation over all possible outcomes must be small.
Note that if ʸ

is unbiased for ʸ , Eq. ( 1.19 ) reduces to
Var ʸ 1 Var ʸ 2

; ð 1 : 20 Þ

ʸ ¼ X
s2S

ʸ þ B ʸ

2 , where Var

ðÞE ʸ

2 pðÞ is

ʸ

because MSE ¼ Var

¼ E ʸ ʸ

the variance of the estimator ʸ , and B ʸ is the bias.

Consider a sequence of samples of sizes (2,3, ... , n ), and a corresponding sequence

.Anestimator, ʸ n ,for

ʸ 2 ; ʸ 3 ; ...; ʸ n

of estimators ʸ is said to be consistent if

< ˅

¼ 1

ʸ n ʸ

lim
n!1

Pr ; ð 1 : 21 Þ

is constant for increasing n .
Linear estimators are usually preferred because of their simplicity. This consid-
eration led Horvitz and Thompson (HT 1952 ) to define a well-known estimator that
is widely used in practical applications. It is the main benchmark for the remainder
of this topic. The HT estimator for the population total t ¼ X U y k

˅ ʸ

where is small, and

is

t HT ¼ X s

y k
π k :

ð 1 : 22 Þ

The estimator t HT is unbiased for t ¼ X U y k (S ¨ rndal et al. 1992 ). The higher the
value of

π k (the inclusion probability of unit k ), the less weight given to y k (the