Introduction to unequal-probability sampling; PPS sampling with
replacement
Unequal probability sampling with replacement: As we saw in
the jobs example, there are situations in which it is desirable to have un-
equal probabilities of selecting elements into the sample. In section 3.3 of
the text, it is stated that for a population with elements {u1, u2, ..., uN}, we
might choose to sample elements with replacement with respective selec-
tion probabilities of {δ1, δ2, ..., δN}.In that situation the estimator
bτ=1
n
n
X
i=1
yi
δi
is an unbiased estimator for τ, and an unbiased estimator of V(bτ) is given
by:
b
V(bτ) = 1
n"n
X
i=1 µyi
δi
−bτ¶2
/(n−1)#=1
n"n
X
i=1
(bτi−bτ)2/(n−1)#.
Note that this variance estimator is of the form s2/n, and is unbiased
for V(bτ) because the with replacement sampling scheme yields a sample of
nseparate independent estimates of τ, namely bτi=yi/δi. This estima-
tor and its’ variance estimator are due to Hansen and Hurwitz (1943), and
thus the estimator is called the Hansen-Hurwitz estimator. Sampling
with replacement yields estimators whose theoretical properties are easy to
understand, but sampling with replacement is often inefficient and can be
impractical in some situations. Later we will discuss general approaches to
unequal-probability sampling that have sampling without replacement, using
estimators developed by Horvitz and Thompson (1952).
PPS sampling with replacement: In cluster sampling, it is often use-
ful to use unequal-probability sampling with replacement with probabilities
proportional to size (PPS). In this case, δi=mi/M, and our estimator of
the total is:
bτpps =1
n
n
X
i=1
yi
δi
=1
n
n
X
i=1
yi
(mi/M)=M
n
n
X
i=1
yi
mi
=M
n
n
X
i=1
yi,
where yi=yi/miis the average of the observations in cluster i. To
obtain an estimator of the mean we can just divide by M, giving
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