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The_Horvitz_Thompson_Estimator.pdf
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详细说明：霍维茨汤姆森估计量。Horvitz-Thompson Estimator. 用于不等概不放回抽样的总体量估计。Example: The H-T Estimator for SRS without replacement
Consider taking a simple random sample(SRS. without replacement, of size n from a
population of size N. The inclusion and joint inclusion probabilities are
丌;
丌一
Note that the Horvitz-Thompson estimator for Srs without replacement becomes
∑"-∑
泛=1
√
the usual estimator of the population total T for an Srs derived earlier
Do you think the Horvitz-Thompson variance will be Var(T)=VN(N-n)o2/m as it
was before for srs?
Example: The H-T Estimator for sampling with replacement
Reconsider the scenario for the Hansen-Hurwitz estimator, where we sample with replace
ment from a population such that the probabilities of selection on any given draw are unequal
These probabilities were denoted p1,.,PN for a population of size N. The inclusion and
joint inclusion probabilities are
With these then, the Horvitz-Thompson estimator is: Tn=>
yi
(1-m n, and the H-T
variance is a, horrendous mess
For sampling with replacement, it is generally easier to use the Hansen-Hurwitz esti-
mator
Although the Horvitz-Thompson estimator can be used for any probability sampling
plan, there is often a simpler way to derive the estimator and its variance than through
inclusion probabilities
Comparison of H-H and H-T Estimators for the Farm Example
Consider taking a random sample of size n= 5(with replacement) from the N= 625
pixels in the map of the farms given in class, and estimating the total number of workers on
all the farms. This was done earlier for the Hansen-Hurwitz estimator, and will be repeated
here for the Horvitz-Thompson estimator
Ten samples of size n= 5 will be taken where the individual farms will be selected accordin
to the probability proportional to size"(PPS)sampling described earlier. Specifically, a
pair of integers between 1 and 25 will be chosen at random and the farm with the
sponding coordinates on the map will be selected
First, consider the Horvitz-Thompson estimator for the single sample of size 5 given in class
estinate the total number of workers using the Hansen-Hurwitz estinator. The sanple
is repeated in the table below, along with the relevant components for the H-T estimator
Coordinates Data
2=1-(1-p)
D25/625=.0080
0394
19,25
C828/625=.0448
2048
21,21
B412/625=.0192
0924
15.4
A814/625-.0224
.1071
7.20
A313/625=.0208
0998
As the samples here were distinct, the horvitz -Thompson estimator of the total number of
2
4
N-.0394.204+.0924,1071.098
237.94 worker
e Recall that the estimated number of workers for the hansen -hurwitz estimator com-
puted earlier was 227.66 workers. Since the true total number of workers was T= 247
does this make the Horvitz-Thompson estimator better
To compute the estimated variance of this estimated total number of workers, we need first
to compute the joint inclusion probabilities for each pair of units in the sample. Using the
formula derived in class, given as: Ti;=Ti+T-(1-(1-p; m), the table below gives
the ten Tii values corresponding to the ten pairs of units
Unit nuinber
Unit
4
0066.0029.0034.0032
0156.01810169
0081.0075
0087
The estimated variance is then computed as
(x)+()m
≠
1-.0394
1-.0998
03942
+2
006-(0394)(2048))(2)(8)
0087-(1071)(099(8)(3)
(.0394)(2048
0066
4071)(.099.008
11191.15-2(4922037)=1347077
giving a standard error of SE(TT)=v1347077=36 70 workers. This is essentially the same
as that found (36.75) with the Hansen-Hurwitz estimator
This task of sampling 5 pixels from the farm area was repeated 10 times, producing the
following H-H and H-t estimators. R code to compute these estimates is given below the
table
Hansen-Hurwitz Horvitz-Thompson
Sample
SE(GD)
SE(TT)
2276636.75237.9436.70
221446.7622
47.01
3
160.9020.24171.3520.66
4
2065544.93212.70
4110134.56417473462
6
2648847752760548.04
了
1178719.611257419.48
236.552444242642414
2482169832568369.82
10
2479257942519557.80
A
e233.27
241.33
Anything interesting about this table
R Code
Sets the sample size
y<-c(2,8,4,83)
Sets the vector of y-values
Compute H-H estimate and se
p <-(1/625)*C(5, 28, 12, 14, 13)# Computes the vector of selection probs
tau.p <-(1/n)*sum(y/p)
t Computes the H-H estimate
print(tau. p)
i prints the h-h estimate
var tau. p
var(y/p)/n
t Computes the variance of the H-H estimate
print(sqrt(var tau. p))
Prints se of h-h estimate
f Compute h-t estimate and se
1-(1-p)^n
#f Computes the vector of inclusion probs
tau.pi <-sum(y/pi)
Computes the H-T estimate
print (tau. pi)
Prints the h-t estimate
Compute the estimated variance of the H-T estimate by computing the
two terms (the single sum and the double sum) separately
var1 < sum(y 2*(1-p1/pi 2)# First term of variance
Second term of variance: the multiplier 2 below is because the pair i,j
is the same as ]
1
var2 <-o
for(i in 1:(n-1))t
for(j in (1+1): n)t
piij <-piliJ pi[j]-(1-(1-p[i]-p[])n)#joint inclusion probability
var2 <-var2+ 2*(y [i]*y[j1piij)*(pi. 1j- pi[i]*pi [j])/(pili]*pi [j])
var tau. pi < var1 var2
f Computes the h-t estimated var
sd. tau. pi < sqrt(var tau. pi)# Computes the standard dev'n
print(sd. tau. pi)
Prints the h-T sD
f Note: the above program is inefficient in that it uses loops and
it can be done more efficiently, but less transparently, with
clever use of matrix commands. However, the above program still
runs in just a few seconds for n=500
29
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