Allocation Problem in Bayesian Stratified
Sampling with Nonlinear Cost Function
Let the population
of ,21 units
is divided
into
subpopulations
of
Y,, X,? .‘) NI
units,
The
are nonoverlapping
and 3, + X, + ._.+ .I I = h’
respectiv el>‘. These subpopulations
1. - number of strata;
subpopulations
are catled strata, Notation for stratified sampling:
W, = Xh / N - stratum weight: Yhi- be the variable of interest for the /th unit in the kth stratum
Also let YLand S’; be the mean and variance ofthe
K =
_! iii;,
Ni
s; = $(,;,
x
izl
Ii{ s in the kth stratum, i.e ,
- ,;)2
1-1
and let i’ be the overall finite population mean givsen b>
Now. suppose that a priori within each stratum the Ykt‘s are exchangeable
for I # ,l
varianccr a,
and uncorrelated,
information
is such that Iii’s in different strata are uncorrelated,
i .e. covf 1 i, , I;, ) = 0 ,
n, - number of sampliny units selected from stratum k:
sample, f1 = il, + II, + . . + n, :
_Yk,-
j7 --
with mean q
and
Finall‘ , assume that one’s prior
i.e. cov( Ii,, Y\,,) = 0, for k + ,\.
total number of sampling units in the
value obtained for the ith unit in the kth stratum.
Let -Vibe the mean ofthe sample observations
in the kth stratum.
The liner Bayes estimator of r is given by (Ericson ( 1988)).
1’
~
and the Bayes risk is satistied the inequality
l’heorem 1. In stratified random sampling with a cost function
the Bayes risk is a minimum
vyhen
for a specified cost (” and the cost is a minimum
for a specitied
I’,
If{ ’ is f’ixed then
if I ’ is fised then
rI
c
=
f
@;
L’h
1’ (ii
11
h-1
j=!
1-1
Theorem 2. If the cost function IS of the form C”= 2~4~ in fzL1 C’ is a minimum
fi)r a specified
i=I
cost ( ‘, and the cost is a minimum for a specified variance, when
%
_=-
ka 2;)
If ( ’ is fked then
i
Fl =
,v c(q vf ).exp
h:i
!
.
1
If k’ is fixed then
1.
n=
c
(CA,
h =l
,=l
1=1
In case of the liner cost functmn optimal allocation IS giwn by Ericson ( 1988 1.
REFERENCES
Ericson W.A. ( 1988). Bayesian Inference in Finite Populations. In Krishnalah P.R., Rao C.R., eds.
Handbook of Statisks, ~01.6, Sampling. Hsevier Science Publishers B V. knsterdam,
2 13-246.
k