Singh, G. N.; Khe an, M.; Mau ya, S.
A icle
SOME EFFECTIVE ESTIMATION PROCEDURES UNDER NON-
RESPONSE IN TWO-PHASE SUCCESSIVE SAMPLING
S a is ics in T ansi ion New Se ies
P o ided in Coope a ion wi h:
Polish S a is ical Associa ion
Sugges ed Ci a ion: Singh, G. N.; Khe an, M.; Mau ya, S. (2016) : SOME EFFECTIVE ESTIMATION
PROCEDURES UNDER NON-RESPONSE IN TWO-PHASE SUCCESSIVE SAMPLING, S a is ics in
T ansi ion New Se ies, ISSN 2450-0291, Exeley, New Yo k, NY, Vol. 17, Iss. 2, pp. 163-182,
h ps://doi.o g/10.21307/s a ans-2016-012
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STATISTICS IN TRANSITION new se ies, June 2016 163
STATISTICS IN TRANSITION new se ies, June 2016
Vol. 17, No. 2, pp. 163–182
SOME EFFECTIVE ESTIMATION PROCEDURES
UNDER NON-RESPONSE IN TWO-PHASE
SUCCESSIVE SAMPLING
G. N. Singh
1
, M. Khe an
2
, S. Mau ya
3
ABSTRACT
This wo k is designed o assess he e ec o non- esponse in es ima ion o he
cu en popula ion mean in wo-phase successi e sampling on wo occasions.
Sub-sampling echnique o non- esponden s has been used and exponen ial
me hods o es ima ion unde wo-phase successi e sampling a angemen ha e
been p oposed. P ope ies o he p oposed es ima ion p ocedu es ha e been
examined. Empi ical s udies a e ca ied ou o jus i y he sugges ed es ima ion
p ocedu es and sui able ecommenda ions ha e been made o he su ey
p ac i ione s.
Key wo ds: non- esponse, successi e sampling, wo-phase sampling, mean
squa e e o , op imum eplacemen s a egy.
1. In oduc ion
In collec ing in o ma ion h ough sample su eys, he e may a ise nume ous
p oblems; one o hem is non- esponse. I equen ly occu s in mail su eys, whe e
some o he selec ed uni s may e use o e u n back he illed in ques ionnai es.
An es ima e ob ained om such an incomple e su ey may be misleading,
especially when he esponden s di e signi ican ly om he non- esponden s,
because he es ima e may be a biased one. Hansen and Hu wi z (1946) sugges ed
a echnique o sub-sampling o non- esponden s o handle he p oblem o non-
esponse. Coch an (1977) and Fabian and Hyunshik (2000) ex ended he Hansen
and Hu wi z (1946) echnique o he si ua ion when besides he in o ma ion on
he cha ac e unde s udy, in o ma ion on auxilia y cha ac e is also a ailable.
Recen ly, Choudha y e al. (2004) Singh and P iyanka (2007), Singh and Kuma
1
Depa men o Applied Ma hema ics, Indian School o Mines, Dhanbad 826004.
E-mail: [email protected].
2
Depa men o Applied Ma hema ics, Indian School o Mines, Dhanbad 826004.
E-mail: muk i.khe [email protected].
3
Depa men o Applied Ma hema ics, Indian School o Mines, Dhanbad 826004.
164 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion ....
(2009, 2010), Singh e al. (2011) and Ga cia Luengo (2013) used he Hansen and
Hu wi z (1946) echnique o he es ima ion o popula ion mean on he cu en
occasion in wo-occasion successi e sampling.
I he s udy cha ac e o a ini e popula ion is subjec o change o e ime, a
single occasion su ey is insu icien . Fo such a si ua ion successi e sampling
p o ides a s ong ool o gene a ing eliable es ima es o e di e en occasions.
Sampling on successi e occasions was i s conside ed by Jessen (1942) in he
analysis o a m da a. The heo y o successi e ( o a ion) sampling was u he
ex ended by Pa e son (1950), Eckle (1955), Rao and G aham (1964), Sen (1971,
1972, 1973), Gup a (1979), Das (1982) and Singh and Singh (2001) among o he s.
In sample su eys, he use o auxilia y in o ma ion has shown i s signi icance
in imp o ing he p ecision o es ima es o unknown popula ion pa ame e s. When
he popula ion pa ame e s o auxilia y a iable a e unknown be o e s a o he
su ey we go o wo-phase (double) sampling s uc u e o p o ide he eliable
es ima es o he unknown popula ion pa ame e s. Singh and Singh (1965) used
wo-phase (double) sampling o s a i ica ion on successi e occasions. Recen ly,
Singh and P asad (2011) and Singh and Homa (2014) applied wo-phase sampling
scheme wi h success in he es ima ion o he cu en popula ion mean in wo-
occasion successi e sampling.
The aim o he p esen wo k is o s udy he e ec o non- esponse when i
occu s on a ious occasions in wo-occasion successi e ( o a ion) sampling.
Recen ly, Bahl and Tu eja (1991), Singh and Vishwaka ma (2007) and Singh and
Homa (2013) sugges ed exponen ial ype es ima o s o popula ion mean unde
di e en ealis ic si ua ions. Mo i a ed wi h he domina ing na u e o hese
es ima o s and u ilizing he in o ma ion on a s able auxilia y a iable wi h
unknown popula ion mean o e bo h occasions, some new exponen ial me hods
o es ima ion ha e been p oposed o es ima e he cu en popula ion mean in wo-
phase successi e ( o a ion) sampling a angemen .. The Hansen and Hu wi z
(1946) echnique o sub-sampling o non- esponden s has been used o educe he
nega i e e ec s o non- esponse. P ope ies o he p oposed es ima o s a e
examined and hei empi ical compa isons a e made wi h he simila es ima o and
wi h he na u al successi e sampling es ima o when comple e esponse is
obse ed on bo h occasions. Resul s a e in e p e ed and ollowed by sui able
ecommenda ions.
2. Sample s uc u es and symbols
Le U = (U1, U2, -, -, -, UN) be he ini e popula ion o N uni s, which has been
sampled o e wo occasions. The cha ac e unde s udy is deno ed by x(y) on he
i s (second) occasion espec i ely. I is assumed ha he non- esponse occu s
only in s udy a iable x(y) and in o ma ion on an auxilia y a iable z (s able o e
occasion), whose popula ion mean is unknown on bo h occasions, is a ailable and
posi i ely co ela ed wi h s udy a iable. Since we ha e assumed ha non-
STATISTICS IN TRANSITION new se ies, June 2016 165
esponse occu s on bo h occasions, he popula ion can be di ided in o wo classes
– hose who will espond a he i s a emp and hose who will no on bo h
occasions. Le he sizes o hese wo classes be
*
1
N
and
*
2
N
espec i ely on he
i s occasion and he co esponding sizes on he cu en (second) occasion be N1
and N2, espec i ely. To u nish a good es ima e o he popula ion mean o he
auxilia y a iable z on he i s occasion, a p elimina y sample o size
n'
is d awn
om he popula ion by he simple andom sampling wi hou eplacemen
(SRSWOR) me hod, and in o ma ion on z is collec ed. Fu he , a second-phase
sample o size n (
n'
> n) is d awn om he i s -phase (p elimina y) sample by he
SRSWOR me hod and hence o h he in o ma ion on he s udy cha ac e x is
ga he ed. We assume ha ou o selec ed n uni s, n1 uni s espond and n2 uni do
no espond. Le n2h deno e he size o sub-sample d awn om he non- esponding
uni s in he sample on i s occasion. A andom sub-sample sm o m = n  uni s is
e ained (ma ched) om he esponding uni s on he i s occasion o i s use on
he second occasion unde he assump ion ha hese uni s will gi e comple e
esponse on he second occasion as well. Once again, o u nish a esh es ima e
o he popula ion mean o he auxilia y a iable z on he second occasion, a
p elimina y ( i s -phase) sample o size
u'
is d awn om he non-sampled uni s
o he popula ion by he SRSWOR me hod and in o ma ion on z is collec ed.
A second-phase sample o size u = (n-m) = nμ (
u'
> u) is d awn om he i s -
phase (p elimina y) sample by he SRSWOR me hod and he in o ma ion on s udy
a iable y is ga he ed. I is ob ious ha he sample size on he second occasion is
also n. He e λ and μ (λ+μ =1) a e he ac ions o he ma ched and esh samples,
espec i ely, on he second (cu en ) occasion. We assume ha in he unma ched
po ion o he sample on he cu en (second) occasion u1 uni s espond and u2
uni s do no espond. Le u2h deno e he size o he sub-sample d awn om he
non- esponding uni s in he esh sample (su) on he cu en (second) occasion.
Hence, onwa ds, we use he ollowing no a ions:
X, Y, Z
: The popula ion means o he a iables x, y and z espec i ely.
1 2 h 1 2h
m u u u n n n m m u
y , y , y , y , x , x , x , x , z ,z
: The sample means o he
espec i e a iables based on he sample sizes shown in su ices.
''
nu
z , z
: The sample means o he auxilia y a iable z and based on he i s -
phase samples o sizes u' and n' espec i ely.
yx xz yz
ρ , ρ , ρ
: The popula ion co ela ion coe icien s be ween he a iables
shown in su ices.
2 2 2
x y z
S , S , S
: The popula ion a iances o he a iables x, y and z espec i ely.
22
2x 2y
S ,S
: The popula ion a iances o he a iables x and y espec i ely in he
non- esponding uni s o he popula ion.
x y z
C , C , C
: The coe icien s o a ia ion o he a iables x, y and
z espec i ely.
166 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion ....
2x 2y
C , C
: The coe icien s o a ia ion o he a iables x and y in he non-
esponding uni s o he popula ion.
*
*2
N
W=N
: The p opo ion o non- esponding uni s in he popula ion a i s
occasion.
2
N
W= N
: The p opo ion o non- esponding uni s in he popula ion on
he cu en (second) occasion.
22
12
2h 2h
nu
= and =
nu
.
3. Fo mula ion o es ima ion s a egy
To es ima e he popula ion mean
Y
on he cu en (second) occasion, wo
di e en es ima o s a e conside ed – one es ima o
u
T
based on sample su o size
u d awn a esh on he second occasion and he second es ima o
m
T
based on he
sample sm o size m, which is common o bo h occasions. Since he non- esponse
occu s in he samples sn and su, we ha e used he Hansen and Hu wi z (1946)
echnique o p opose he es ima o s Tu and Tm. Hence, he es ima o s Tu and Tm
o es ima ing he cu en popula ion mean
Y
a e o mula ed as
'*'
*uu n m n
u u m m
'*
u u n m m
z -z x -x z
T = y exp and T = y exp
z +z x +x z
   
   
  

whe e
1 2h 1 2h
1 n 2 n 1 u 2 u
**
nu
n x +n x u y +u y
x = and y = .
nu
Combining he es ima o s Tu and Tm, inally we ha e he ollowing es ima o
o popula ion mean
Y
on he cu en (second) occasion
 
u m
T = φ T + 1-φ T
(3.1)
whe e
φ (0 φ 1)
is he unknown cons an (scala ) o be de e mined unde
ce ain c i e ions.
4. P ope ies o he es ima o T
Since he es ima o s
u m
T and T
a e exponen ial ype es ima o s, he
popula ion mean
Y
a e biased, he e o e he esul ing es ima o T de ined in

STATISTICS IN TRANSITION new se ies, June 2016 167
equa ion (3.1) is also a biased es ima o o
Y
. The bias B (.) and he mean squa e
e o M (.) o he es ima o T a e de i ed up o he i s o de o app oxima ions
using he ollowing ans o ma ions:
*
m 1 u 2 u 3 m 4 n 5
' * '
n 6 n 7 m 8 u 9 u 10
y (1 e )Y, y (1 e )Y, y (1 e )Y, x (1 e )X, x (1 e )X,
x (1 e )X, x (1 e )X, z (1 e )Z, z (1 e )Z, z (1 e )Z,
         
         
'
n 11
z (1 e )Z,
such ha
i
E(e ) = 0
,
i
e <1 i = 1, 2, 3, - - -, 11.
Unde he
abo e ans o ma ions, he es ima o s
u m
T and T
ake he ollowing o ms:
-1
u 3 10 9 10 9
11
T =Y(1+e )exp (e -e ) 1+ (e +e )
22







(4.1)
and
-1
-1
m 1 11 8 7 4 7 4
11
T =Y(1+e )(1+e )(1+e ) exp (e -e ) 1+ (e +e )
22







(4.2)
Thus, we ha e he ollowing heo ems:
Theo em 4.1.
Bias o he es ima o T o he i s o de o app oxima ions is ob ained as
u m
B(T) = φB(T )+(1-φ)B(T )
(4.3)
whe e
2
u z yz y z
'
1 1 3 1
B(T ) = Y - C - ρ C C
u u 8 2
  
  
  
and
 
 
2
x xz x z xy y x
m
* 2 2
12x z yz y z
'
1 1 3 1 1
- C + ρ C C - ρ C C
m n 8 2 2
B T =Y ( -1)
1 1 1
- W C + - C -ρ C C
8 n m n

  
  

   







P oo
The bias o he es ima o T is gi en by
um
B(T) = E T-Y = φE(T -Y) + (1-φ) E(T -Y)


 
um
= φB(T )+ 1-φ B(T )
(4.4)
whe e
u u m m
B(T ) = E T -Y and B(T ) = E T -Y .
   
   
168 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion ....
Subs i u ing he exp essions o Tu, and Tm om equa ions (4.1) and (4.2) in
equa ion (4.4), expanding he e ms binomially and exponen ially, aking
expec a ions and e aining he e ms up o he i s o de o sample sizes, we ha e
he exp essions o he bias o he es ima o T as desc ibed in equa ion (4.3).
Theo em 4.2.
The mean squa e e o o he es ima o T o he i s o de o app oxima ions
is ob ained as
       
2
2um
M( T) = φ M T + 1-φ M T +2φ 1-φ C
(4.5)
whe e
 
22
2
u u yz y
'
W( -1)
1 1 1 1 1
M(T ) = E T -Y = - -ρ + - + S
u u 4 u N u

    
    

    

(4.6)
 
 
xz yx
22
m m y
*
1
yz
'
1 1 1 1 1
- +ρ -ρ + -
m n 4 m N
M(T ) = E T -Y = S
( -1)
1 1 1
+ - 1 -2ρ + W
m n 4 n


    


    
    












(4.7)
and
  
2
y
um
S
C = E T -Y T -Y = - N


. (4.8)
P oo
I is ob ious ha he mean squa e e o o he es ima o T is gi en by
 
 
 
2
2
um
M(T) = E T-Y = E φ T -Y + 1-φ T -Y


 
 
 
 
 
  
22
2
2u m u m
= φ E T -Y + 1-φ E T -Y +2φ 1-φ E T -Y T -Y


     
2
2um
= φ M(T )+ 1-φ M T +2φ 1-φ C
(4.9)
Subs i u ing he exp essions o Tu, and Tm om equa ions (4.1)-(4.2) in
equa ion (4.9), expanding he e ms binomially and exponen ially, aking
expec a ions and e aining he e ms up o he i s o de o sample sizes, we ha e
he exp ession o he mean squa e e o o he es ima o T as i is gi en in equa ion
(4.5).
STATISTICS IN TRANSITION new se ies, June 2016 169
Rema k 4.1.
The exp ession o he mean squa e e o in he equa ion (4.5) is de i ed unde
he assump ions (i) ha he coe icien s o a ia ion o non- esponse class a e
simila o ha o he popula ion, i.e.
2x x 2y y C = C and C = C
, and (ii) since x and
y a e he same s udy a iable o e wo occasions and z is he auxilia y a iable
co ela ed o x and y, looking a he s abili y na u e o he coe icien s o a ia ion,
iz. Reddy (1978), he coe icien s o a ia ion o he a iables x, y and z in he
popula ion a e conside ed equal, i.e.
x y z
C = C = C .
5. Minimum mean squa e e o o he es ima o T
Since he mean squa e e o o he es ima o T in equa ion (4.5) is he unc ion
o unknown cons an φ, i is minimized wi h espec o φ, and subsequen ly he
op imum alue o φ is ob ained as
m
um
op
M(T )-C
φ = M(T )+M(T )-2C
. (5.1)
Now, subs i u ing he alue o
op
φ
in equa ion (4.5), we ge he op imum
mean squa e e o o T as
2
um
op
um
M(T ).M(T )-C
M(T) = .
M(T )+M(T )-2C
(5.2)
Fu he , subs i u ing he alues om equa ions (4.6)-(4.8) in equa ion (5.2),
we ge he simpli ied alue o M (T)op which is gi en below:
2
2y
3 2 1
op 2
6 5 4
S
a+μa +μ a
M(T) = a+μa +μ a n
. (5.3)
whe e
2 2 2 2
1 2 3 4 5 6
*
1 0 0 2 2 1 1 1
*
2 1 1 0 yz 1 xz xy
a = ac+k , a = ad+cb-k , a = bd, a = c-a+2k , a = a-b+d-2k , a = b,
1
a = -( + a ), b= a +1+( -1)W, c = d +c + - ( -1)W ,
4
1 1 1
d = 1- +(1- )d + ( -1)W , k =-1, a = -ρ , c = +ρ -ρ ,
4 4 4 1 yz
22
1 2 1 2
''
2h 2h
d = 1-2ρ,
nu
n n n
= , = , = , = and = .
N n u u n
170 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion ....
6. Op imum eplacemen s a egy
Since he mean squa e e o o he es ima o T gi en in equa ion (5.3) is he
unc ion o μ ( ac ions o he sample o be d awn a esh a he second occasion),
he op imum alue o μ is de e mined o es ima e he popula ion mean
Y
wi h
maximum p ecision and lowes cos . To de e mine he op imum alue o μ, we
minimized he mean squa e e o o he es ima o T gi en in equa ion (5.3) wi h
espec o μ, which esul s in quad a ic equa ion in μ and he espec i e solu ions
o μ, say
ˆ
μ
, a e gi en below:
2
1 2 3
pμ +2p μ+p =0
(6.1)
2
2 2 1 3
1
-p ± p -p p
ˆ
μ = p
(6.2)
whe e
1 1 5 2 4 2 1 6 3 4 3 2 6 3 5
p = a a -a a , p = a a -a a and p = a a -a a .
F om equa ion (6.2) i is ob ious ha eal alues o
ˆ
μ
exis i he quan i ies
unde squa e oo a e g ea e han o equal o ze o. Fo any combina ions o
co ela ions ρyx, ρxz and ρyz, which sa is y he condi ions o eal solu ions, wo eal
alues o
ˆ
μ
a e possible. Hence, while choosing he alues o
ˆ
μ
, i should be
emembe ed ha
ˆ
0μ1
. I bo h he alues o
ˆ
μ
sa is y he s a ed condi ion, we
chose he smalle alue o
ˆ
μ
as i will help in educing he cos o he su ey.
All o he alues o μ a e inadmissible. Subs i u ing he admissible alue o
ˆ
μ
, say
(0)
μ
, om equa ion (6.2) in o equa ion (5.3), we ha e he op imum alue o he
mean squa e e o o T, which is shown below:
2
(0) (0)2 y
03 2 1
op (0) (0)2
6 5 4
S
a+μ a +μ a
M(T ) = a+μ a +μ a n
. (6.3)
7. Some special cases
Case 1: When non- esponse occu s only a i s occasion
When non- esponse occu s only a i s occasion, he es ima o o he mean
Y
on he cu en occasion may be ob ained as
 
* * *
1u m
T = φ ξ + 1-φ T
(7.1)
STATISTICS IN TRANSITION new se ies, June 2016 177
Table 3. Pe cen age ela i e loss
*
1
L
in he p ecision o he es ima o T* wi h
espec o
1
ξ
No e: ‘*’ indica es
*(0)
μ
does no exis .
Table 4. Pe cen age ela i e loss
*
2
L
in he p ecision o he es ima o T* wi h
espec o
2
ξ
No e: ‘*’ indica es
*(0)
μ
does no exis .
W
0.05
0.10
0.15
0.20
ρyx
2
yz
*(0)
μ
*
1
L
*(0)
μ
*
1
L
*(0)
μ
*
1
L
*(0)
μ
*
1
L
0.6
1.5
0.6
0.7
0.8
0.9
*
0.5489
0.3748
0.2828
-
0.1481
0.3426
0.5457
*
0.5667
0.3911
0.2961
-
0.2827
0.6617
1.0599
*
0.5832
0.4065
0.3089
-
0.4057
0.9596
1.5452
*
0.5984
0.4212
0.3212
-
0.5185
1.2385
2.0041
2.0
0.6
0.7
0.8
0.9
*
0.5667
0.3911
0.2961
-
0.2827
0.6617
1.0599
*
0.5984
0.4212
0.3212
-
0.5185
1.2385
2.0041
*
0.6258
0.4484
0.3445
-
0.7180
1.7458
2.8508
*
0.6497
0.4732
0.3663
-
0.8890
2.1954
3.6142
0.8
1.5
0.6
0.7
0.8
0.9
0.6356
0.5141
0.4277
0.3613
0.1025
0.2075
0.3345
0.4977
0.6443
0.5234
0.4368
0.3698
0.1997
0.4057
0.6552
0.9754
0.6525
0.5324
0.4456
0.3780
0.2920
0.5951
0.9628
1.4345
0.6604
0.5411
0.4542
0.3861
0.3798
0.7764
1.2582
1.8759
2.0
0.6
0.7
0.8
0.9
0.6443
0.5234
0.4368
0.3698
0.1997
0.4057
0.6552
0.9754
0.6604
0.5411
0.4542
0.3861
0.3798
0.7764
1.2582
1.8759
0.6752
0.5574
0.4705
0.4016
0.5430
1.1163
1.8150
2.7098
0.6887
0.5727
0.4859
0.4163
0.6916
1.4292
2.3308
3.4842
W
0.05
0.10
0.15
0.20
yx
2
yz
*(0)
μ
*
2
L
*(0)
μ
*
2
L
*(0)
μ
*
2
L
*(0)
μ
*
2
L
0.6
1.5
0.6
0.7
0.8
0.9
*
0.5489
0.3748
0.2828
-
-22.1500
-41.1948
-69.5345
*
0.5667
0.3911
0.2961
-
-21.9853
-40.7427
-68.6579
*
0.5832
0.4065
0.3089
-
-21.8349
-40.3205
-67.8305
*
0.5984
0.4212
0.3212
-
-21.6969
-39.9255
-67.0482
2.0
0.6
0.7
0.8
0.9
*
0.5667
0.3911
0.2961
-
-21.9853
-40.7427
-68.6579
*
0.5984
0.4212
0.3212
-
-21.6969
-39.9255
-67.0482
*
0.6258
0.4484
0.3445
-
-21.4529
-39.2068
-65.6050
*
0.6497
0.4732
0.3663
-
-21.2436
-38.5697
-64.3036
0.8
1.5
0.6
0.7
0.8
0.9
0.6356
0.5141
0.4277
0.3613
-0.3271
-13.7064
-32.5814
-60.2711
0.6443
0.5234
0.4368
0.3698
-0.2294
-13.4806
-32.1549
-59.5015
0.6525
0.5324
0.4456
0.3780
-0.1367
-13.2648
-31.7456
-58.7621
0.6604
0.5411
0.4542
0.3861
-0.0486
-13.0583
-31.3527
-58.0511
2.0
0.6
0.7
0.8
0.9
0.6443
0.5234
0.4368
0.3698
-0.2294
-13.4806
-32.1549
-59.5015
0.6604
0.5411
0.4542
0.3861
-0.0486
-13.0583
-31.3527
-58.0511
0.6752
0.5574
0.4705
0.4016
0.1153
-12.6709
-30.6119
-56.7079
0.6887
0.5727
0.4859
0.4163
0.2646
-12.3144
-29.9258
-55.4606

178 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion ....
Table 5. Pe cen age ela i e loss
**
1
L
in he p ecision o he es ima o T** wi h
espec o
1
ξ
No e: ‘*’ indica es
**(0)
μ
does no exis .
Table 6. Pe cen age ela i e loss
**
2
L
in he p ecision o he es ima o T** wi h
espec o
2
ξ
No e: ‘*’ indica es
**(0)
μ
does no exis .
W
0.05
0.10
0.15
0.20
ρyx
2
yz
**(0)
μ
**
1
L
**(0)
μ
**
1
L
**(0)
μ
**
1
L
**(0)
μ
**
1
L
0.6
1.5
0.6
0.7
0.8
0.9
0.8515
0.4376
0.2964
0.2250
3.2070
2.3271
2.0922
2.1709
0.6634
0.3382
0.2305
0.1772
5.8247
4.1649
3.7242
3.8495
0.4538
0.2312
0.1600
0.1257
7.7922
5.5017
4.9027
5.0602
0.2206
0.1163
0.0847
0.0707
9.0181
6.3162
5.6273
5.8180
2.0
0.6
0.7
0.8
0.9
0.6634
0.3382
0.2305
0.1772
5.8247
4.1649
3.7242
3.8495
0.2206
0.1163
0.0847
0.0707
9.0181
6.3162
5.6273
5.8180
*
*
*
*
-
-
-
-
*
*
*
*
-
-
-
-
0.8
1.5
0.6
0.7
0.8
0.9
0.5785
0.4668
0.3883
0.3288
2.3221
2.2289
2.2786
2.4916
0.5297
0.4284
0.3573
0.3038
4.3575
4.1749
4.2545
4.6280
0.4801
0.3891
0.3254
0.2776
6.1286
5.8623
5.9574
6.4507
0.4298
0.3490
0.2925
0.2504
7.6552
7.3121
7.4125
7.9947
2.0
0.6
0.7
0.8
0.9
0.5297
0.4284
0.3573
0.3038
4.3575
4.1749
4.2545
4.6280
0.4298
0.3490
0.2925
0.2504
7.6552
7.3121
7.4125
7.9947
0.3272
0.2666
0.2245
0.1934
10.0446
9.5722
9.6656
10.3593
0.2222
0.1818
0.1538
0.1333
11.6475
11.0819
11.1591
11.9082
W
0.05
0.10
0.15
0.20
yx
2
yz
**(0)
μ
**
2
L
**(0)
μ
**
2
L
**(0)
μ
**
2
L
**(0)
μ
**
2
L
0.6
1.5
0.6
0.7
0.8
0.9
0.8515
0.4376
0.2964
0.2250
-6.8086
-19.4844
-38.7159
-66.7641
0.6634
0.3382
0.2305
0.1772
-3.9201
-17.2361
-36.4038
-63.9026
0.4538
0.2312
0.1600
0.1257
-1.7490
-15.6008
-34.7341
-61.8388
0.2206
0.1163
0.0847
0.0707
-0.3963
-14.6045
-33.7074
-60.5469
2.0
0.6
0.7
0.8
0.9
0.6634
0.3382
0.2305
0.1772
-3.9201
-17.2361
-36.4038
-63.9026
0.2206
0.1163
0.0847
0.0707
-0.3963
-14.6045
-33.7074
-60.5469
*
*
*
*
-
-
-
-
*
*
*
*
-
-
-
-
0.8
1.5
0.6
0.7
0.8
0.9
0.5785
0.4668
0.3883
0.3288
1.9021
-11.4032
-29.9953
-57.0593
0.5297
0.4284
0.3573
0.3038
3.9462
-9.1858
-27.3668
-53.6183
0.4801
0.3891
0.3254
0.2776
5.7249
-7.2633
-25.1016
-50.6823
0.4298
0.3490
0.2925
0.2504
7.2581
-5.6113
-23.1658
-48.1954
2.0
0.6
0.7
0.8
0.9
0.5297
0.4284
0.3573
0.3038
3.9462
-9.1858
-27.3668
-53.6183
0.4298
0.3490
0.2925
0.2504
7.2581
-5.6113
-23.1658
-48.1954
0.3272
0.2666
0.2245
0.1934
9.6578
-3.0360
-20.1687
-44.3867
0.2222
0.1818
0.1538
0.1333
11.2676
-1.3158
-18.1818
-41.8919
STATISTICS IN TRANSITION new se ies, June 2016 179
9. In e p e a ions o esul s
The ollowing conclusions may be d awn om Tables 1-6:
(1) F om Tables 1 and 5 i is clea ha
(a) Fo he ixed alues o W, ρyx and 2, he alues o
(0) **(0)
μμ,
dec ease wi h
he inc easing alues o ρyz. This implies ha he highe he alue o ρyz, he lowe
he ac ion o a esh sample equi ed on he cu en occasion.
(b) Fo he ixed alues o W, ρyx and ρyz, he alues o
(0) **(0)
μμ,
dec ease and
L1,
**
1
L
inc ease wi h he inc easing alues o 2.
(c) Fo he ixed alues o W, ρyz and 2, no pa e n is obse ed wi h he
inc easing alues o ρyx.
(d) Fo he ixed alues o 2, ρyz and ρyx, he alues o
(0) **(0)
μμ,
dec ease and
L1,
**
1
L
inc ease wi h he inc easing alues o W. This beha iou shows ha wi h
he highe non- esponse a e one may equi e o d aw he smalle sample on he
cu en occasion, which educes he cos o a su ey.
(2) F om Tables 2 and 6 i may be seen ha
(a) Fo he ixed alues o W, ρyx and 2, he alues o
(0) **(0)
μμ,
and L2,
**
2
L
dec ease wi h he inc easing alues o ρyz.. This implies ha i one uses he
in o ma ion on highly co ela ed auxilia y a iable, he e is a signi ican gain in
he p ecision o es ima es.
(b) Fo he ixed alues o W, ρyx and ρyz, he alues o
(0) **(0)
μμ,
dec ease
and L2,
**
2
L
inc ease wi h he inc easing alues o 2.
(c) Fo he ixed alues o W, ρyz and 2, he alues o L2,
**
2
L
inc ease wi h he
inc easing alues o ρyx.
(d) Fo he ixed alues o 2, ρyz and ρyx, he alues o
(0) **(0)
μμ,
dec ease and
L2,
**
2
L
inc ease wi h he inc easing alues o W. This pa e n shows ha he
highe he non- esponse a e, he g ea e he loss. This beha iou is p ac ically
jus i ied.
(3) F om Table 3 i is clea ha
(a) Fo he ixed alues o W, ρyx and 2, he alues o
*(0)
μ
dec ease and
*
1
L
inc ease wi h he inc easing alues o ρyz. This beha iou indica es ha i he
in o ma ion on highly co ela ed auxilia y a iable is a ailable, i plays an
impo an ole in imp o ing he p ecision o es ima es.
(b) Fo he ixed alues o W, ρyx and ρyz, he alues o
*(0)
μ
and
*
1
L
inc ease
wi h he inc easing alues o 2.
180 G. N. Singh, M. Khe an, S. Mau ya: Some e ec i e es ima ion ....
(c) Fo he ixed alues o W, ρyz and 2, no pa e n is isible wi h he inc easing
alues o ρyx.
(d) Fo he ixed alues o 2, ρyz and ρyx, he alues o
*(0)
μ
and
*
1
L
inc ease
wi h he inc easing alues o W.
(4) F om Table 4 i may be seen ha
(a) Fo he ixed alues o W, ρyx and 2, he alues o
*(0)
μ
and
*
2
L
dec ease
wi h he inc easing alues o ρyz. This implies ha nega i e loss is obse ed due o
he p esence o high co ela ion be ween he auxilia y a iables. This beha iou
is highly desi able.
(b) Fo he ixed alues o W, ρyx and ρyz, he alues o
*(0)
μ
and
*
2
L
inc ease
wi h he inc easing alues o 2. This indica es ha i a smalle size o sub-sample
is chosen, he loss in p ecision inc eases, as i was expec ed.
(c) Fo he ixed alues o W, ρyz and 2 no pa e n is seen wi h he inc easing
alues o ρyx.
(d) Fo he ixed alues o 2, ρyz and ρyx, he alues o
*(0)
μ
and
*
2
L
inc ease
wi h he inc easing alues o W.
10. Conclusions and ecommenda ions
I may be seen om he abo e ables ha o all cases he pe cen age ela i e
loss in p ecisions is obse ed whe e e he op imum alue o μ exis s, when non-
esponse occu s on bo h occasions. F om Tables 1, 3 and 5, i is seen ha he loss
is p esen due o he p esence o non- esponse on each occasion, bu he nega i e
impac o non- esponse is e y low, which jus i ies he use o Hansen and Hu wi z
(1946) echnique in he p oposed es ima ion p ocedu es. F om Tables 2, 4 and 6,
when he p oposed es ima o s a e compa ed wi h he na u al successi e sampling
es ima o , subs an ial p o i is isible, which jus i ies he in elligible use o
auxilia y in o ma ion in he o m o exponen ial me hods o es ima ion. Finally,
looking a good beha iou s o he p oposed es ima o s one may ecommend hem
o su ey s a is icians and p ac i ione s o hei p ac ical applica ions.
Acknowledgemen s
Au ho s a e hank ul o he e iewe s o hei aluable sugges ions, which
enhanced he quali y o his pape . Au ho s a e also hank ul o he Indian School
o Mines, Dhanbad o p o iding inancial assis ance and necessa y in as uc u e
o ca y ou he p esen esea ch wo k.
STATISTICS IN TRANSITION new se ies, June 2016 181
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