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STATISTICA, anno LXIX, n. 4, 2009

RATIO-CUM-PRODUCT TYPE EXPONENTIAL ESTIMATOR H.P. Singh, L.N. Upadhyay, R. Tailor

1. INTRODUCTION Consider a finite population U  (U1 , U 2 ,..., U N ) of N units. Suppose two auxiliary variables X 1 and X 2 are observed on U i (i=1,2,3,...,N), where X 1 is positively and X 2 is negatively correlated with the study variable Y. A simple random sample of size n is drawn without replacement from population U to estimate the population mean Y of the study variable Y assuming the knowledge of the population means X 1 and X 2 of the auxiliary variables X 1 and X 2 respectively. Singh (1967) proposed a ratio-cum-product estimator for population mean Y as t RP  y ( X 1/x 1 )( x 2 /X 2 ) ,

(1.1)

where n

n

n

i 1

i 1

i 1

y   y i / n , x 1   x 1i / n and x 2   x 2 i / n . Bahl and Tuteja (1991) envisaged ratio and product type exponential estimators for Y respectively as  X  x1  t Re  y exp  1 ,  X1  x1 

(1.2)

 x  X2  t Pe  y exp  2 .  x2  X2 

(1.3)

It is well known under simple random sampling without replacement (SRSWOR) that the variance/mean squared error (MSE) of usual unbiased estimator y is

V ( y )   S0 2  MSE( y ) ,

(1.4)

300

H.P. Singh, L.N. Upadhyay, R. Tailor

where N

  (1  f )/ n , f  n / N and S02   ( y j  Y )2 /( N  1) . j 1

To the first degree of approximation, mean squared errors of t RP , t Re and t Pe are respectively given by 2

MSE( t RP )   Y [C 02  C 12 (1  2 K 01 )  C 22{1  2( K 02  K 12 )}]

(1.5)

2

MSE( t Re )   Y [C 02  {C 12 (1  4 K 01 )/4}]

(1.6)

2

MSE( t Pe )   Y [C 02  {C 22 (1  4 K 02 )/4}] ,

Co 

where

So , Y

Ci 

Si , Xi

K oi  oi

N

N

j 1

j 1

(1.7)

Co Ci

(i=1,2),

K 12  12

C1 , C2

Si2   ( x ij  X i )2 /( N  1), Soi2   ( y i  Y )( x ij  X i )/( N  1), ( i  1, 2) ;  0 i  ( S0 i / S0 Si ) : is the correlation coefficient between Y and X i

( i  1, 2) , 12  S12 /( S1S2 ) : is the correlation coefficient between X 1 and X 2 , N

and S12   ( x 1 j  X 1 )( x 2 j  X 2 )/( N  1) . j 1

In this paper we have suggested a ratio-cum-product type exponential estimator for the population mean Y of the study variate Y using auxiliary information on X 1 and X 2 . The mean squared error of the suggested estimator has been derived under large sample approximation. Numerical illustration is given in support of the present study. 2. PROPOSED RATIO-CUM-PRODUCT TYPE EXPONENTIAL ESTIMATOR Motivated by Singh (1967) we propose a ratio-cum-product type exponential estimator for population mean Y as

 X  x1   x2  X2  t RPe  y exp  1  . exp    X1  x1   x2  X2   2( X 1x 2  x 1 X 2 )   y exp    ( X 1  x 1 )( x 2  X 2 ) 

(2.1)

301

Ratio-cum-product type exponential estimator

It is to be mentioned that when no auxiliary information is used, the estimator t RPe reduces to the usual unbiased estimator y . If the information on auxiliary variate X 1 is used, then the estimator t RPe tends to the ratio-type exponential estimator t Re given by (1.2). On the other hand, if the information on only auxiliary variate X 2 is available, the estimator t RPe reduces to the product-type exponential estimator t Pe given by (1.3). To the first degree of approximation, the mean squared error of t RPe is given by

MSE( t RPe )   Y 2 [C 02  {C 12 (1  4 K 01 )/4}  {C 22 (1  4 K 02  2K 12 )/4}] (2.2) 3. EFFICIENCY COMPARISONS In this section we have made comparison of the suggested estimator t RPe with the existing estimators such as usual unbiased estimator y , ratio and product estimators t R and t P , Singh’s (1967) estimator t RP and Bahl and Tuteja’s (1991) estimators t Re and t Pe . It follows from (1.4), (1.5), (1.6), (1.7) and (2.2) that the proposed ratio-cumproduct exponential estimator t RPe is more efficient than: (i) the usual unbiased estimator y if K 01  (1/4)

and

K 02  {(2 K 12  1)/4} ,

(3.1)

(ii) the Singh’s (1967) estimator t RP if K 01  (3/4)

and

K 02  {3(2 K 12  1)/4} ,

(3.2)

(iii) the ratio-type exponential estimator t Re if

K 02  {(2K 12  1)/4} ,

(3.3)

(iv) the product-type exponential estimator t Pe if K 01  {(1  2 K 21 )/4} .

(3.4)

Further to compare the ratio-cum-product type exponential estimator t RPe with usual ratio estimator t R  y( X 1 / x 1 ) and product estimator t P  y ( x 2 / X 2 ) we write the mean squared errors of t R and t P to the first degree of approximation, respectively as

302


MSE( t R )   Y 2 [C 02  C 12 (1  2 K 01 )]

(3.5)

MSE( t P )   Y 2 [C 02  C 22 (1  2 K 02 )]

(3.6)

From (2.2), (3.5) and (3.6) we note that the proposed estimator t RPe is more efficient than: (i) the usual ratio estimator t R if

K 01  (3/4)

and

K 02  {(2 K 12  1)/4} ,

(3.7)

(ii) the usual product estimator t P if

K 01  (1/4)

and

K 02  (2K 12  3)/4 .

4. CLASS OF ALMOST UNBIASED ESTIMATORS OF POPULATION MEAN

(3.8)

Y

It is observed that the suggested ratio-cum-product type exponential estimator is biased. In some applications biasedness is disadvantageous. This led authors to investigate unbiased estimators of the population mean Y . 4.1 Bias Subtraction Method The bias of t RPe to the first degree of approximation is given by  3 S12 S01 S02 S12 S 22      B( t RPe )  ( / 2)Y  2 2 4 X1 Y X 1 Y X 2 X1 X 2 4X 2 

(4.1)

Replacing Y , S12 , S 22 , S01 and S12 by their unbiased estimators y , n

n

j 1

j 1

s 12   ( x 1 j  x 1 )2 /( n  1), s 22   ( x 2 j  x 2 )2 /( n  1), n

s 01   ( y j  y )( x 1 j  x 1 )/( n  1) , j 1 N

N

s 02   ( y j  y )( x 2 j  x 2 )/( n  1)

and

j 1

s 12   ( x 1 j  x 1 )( x 2 j  x 2 )/( n  1) respectively in (4.1) we get a consistent esj 1

timate of the bias B( t RPe ) as


3 s2 s s s s2  Bˆ ( t RPe )  ( / 2) y  1 2  01  02  12  2 2  y X1 yX 2 X 1 X 2 4 X 2   4 X1

303

(4.2)

Thus an almost unbiased estimator of the population mean Y is given by

  2( X1x 2  x1X2  (1  f )  3 s12 s 01 s 02 s12 s 22  (u ) t RPe  y exp         2 2   ( X1  x1 )( x 2  X2 )  2n  4 X1 yX1 yX2 X1 X2 4 X2  (4.3) It can be easily shown to the first degree of approximation that the variance of is

(u ) t RPe

(u ) Var ( t RPe )  MSE( t RPe )

(4.4)

(u ) Thus if the bias is of considerable importance then the estimator t RPe is to be preferred over t RPe .

4.2 Jack knife Method Let a simple random sample of size n=g m drawn without replacement and split at random into g sub samples, each of size m. Then we consider the jackknife ratio-cum-product type exponential estimator of the population mean Y as t RPeJ 

 2( X 1x 2' j  x 1' j X 2  1 g ' y exp  j  ( X  x ' )( x '  X )  g j 1 1j 2j 2   1 

(4.5)

where y 'j  ( ny  my j )/( n  m ) and x ij'  ( nx i  mx ij )/( n  m ) , i=1,2; are the sample means based on a sample of (n-m) units obtained by omitting the jth group and y and x ij (i=1,2; j=1,2,...,g) are the sample means based on the jth sub samples of size m=n/g. To the first degree of approximation, the bias of t RPeJ is given by B( t RPeJ ) 

(N  n  m)  2  3 K 1    Y C 1   K 01   C 22  K 02 12    2N (n  m )   4 2 4   

From (4.1) and (4.6) we have B( t RPe ) ( N  n )( n  m )    ( Say ) B( t RPeJ ) n( N  n  m )

(4.6)

304


or B( t RPe )   B( t RPeJ )  0 or  B( t RPe )   B( t RPeJ )  0 , where  is a scalar. For any scalar  , we have  E( t RPe  Y )   E( t RPeJ  Y )  0 or  E( t RPe  y )   E( t RPeJ  y )  0 or

E[ t RPe   t RPeJ  y{ (1   )  1}]  Y .

Hence the general class of almost unbiased ratio-cum-product type exponential estimator of Y as   2( X 1x 2  x 1 X 2  t RPe ( u ) =  y{1   (1   )}   y exp    ( X 1  x 1 )( X 2  x 2 )   

 g

g



j 1

 2( X 1x 2' j  x 1' j X 2   ' y j exp   ' '  ( X 1  x 1 j )( x 2 j  X 2 )  

(4.7)

See Singh (1987 a) and Singh and Tailor (2005). Remark 4.1. For  =0, the estimator t RPe ( u ) reduces to the conventional unbiased estimator y while for   (1   )1 , t RPe ( u ) yields an almost unbiased estimator for Y as ( N  n  m )  2( X 1x 2  x 1 X 2  1 t RPe g y exp   (u )   N  ( X 1  x 1 )( X 2  x 2 )    2( X 1x 2 j  x 1 j X 2   ( N  n )( g  1) g  y j exp    Ng j 1  ( X 1  x 1 j )( x 2 j  X 2 )  

(4.8)

which is jackknifed version of the suggested estimator t RPe . A large number of almost unbiased ratio-cum-product type exponential estimators from (4.8) can be generated by substituting the suitable values of the scalar  . 5.

SEARCH OF ASYMPTOTICALLY OPTIMUM ALMOST UNBIASED RATIO-CUM-PRODUCT TYPE EXPONENTIAL ESTIMATOR IN THE CLASS OF ESTIMATORS t RPe ( u ) AT (4.7)

We write the estimator t RPe ( u ) at (4.8) as

305


t RPe ( u )  [ y {1   (1   )}   t RPe   t RPe J ]

(5.1)

The variance of t RPe ( u ) is given by

V ( t RPe ( u ) )    2{(1   )2 Var ( y )  Var ( t RPe )   2Var ( t RPeJ )  2 Cov ( t RPe , t RPeJ )  2(1   )Cov ( y , t RPe )  2 (1   )Cov ( y , t RPej )} 2{(1   )Var ( y )  Cov ( y , t RPe )   Cov ( y , t RPeJ )}  Var ( y ) (5.2) To the first degree of approximation, it can be easily shown that Var ( t RPeJ )  Var ( t RPe )  Cov ( t RPe , t RPeJ )  MSE( t RPe )

(5.3)

and Cov ( y , t RPe ) = Cov ( y , t RPeJ ) = ( / 2) [2C 02  K 02C 22  K 01C 12 )] ,

(5.4)

where MSE( t RPe ) is given by (2.2). (u ) Substitution of (1.4), (5.3) and (5.4) in (5.2), we get the variance of t RPe to the first degree of approximation as

Var ( t RPe ( u ) )  ( 2 ) Y 2 [C 02  (  2 /4)(1   )2 (C 12  C 22  2 K 12C 22 )   (1   )( K 02C 22  K 01C 12 )]

(5.5)

which is minimized for 

2( K 01C 12  K 02C 22  0 (say) (1   )(C 12  C 22  2K 12C 22 )

(5.6)

Thus the resulting minimum variance of t RPe ( u ) is given by

 ( K C 2  K C 2 )2  Var min( t RPe ( u ) )   Y 2 C 02  2 01 1 2 02 2 2  (C 1  C 2  2K 12C 2 )  

(5.7)

From (1.4), (2.2) and (5.7) we have  ( K C 2  K C 2 )2  Var ( y ) Var min( t RPe ( u ) )   Y 2  2 01 1 2 02 2 2   0  (C 1  C 2  2 K 12C 2 ) 

(5.8)

306


Var ( t RPe )  Var min( t RPe ( u )

  Y 2{C 2 (1  2 K )  C 2 (1  2 K  2 K )}2  1 01 2 02 12  0 2 2 2 4(C 1  C 2  2 K 12C 2 )   (5.9)

It follows from (5.8) and (5.9) that the proposed class of estimators t RPe ( u ) is more efficient than usual unbiased estimator y and the ratio-cum-product type exponential estimator t RPe at its optimum condition (i. e. when  coincides exactly with its optimum value 0 given by (5.6)). The optimum value 0 of  can be obtained quite accurately either through past data or experience gathered in due course of time. 6.

A GENERALIZED VERSION OF THE SUGGESTED RATIO-CUM-PRODUCT TYPE EXPONENTIAL ESTIMATOR

t RPe

We define the following class of ratio-cum-product type exponential estimator for the population mean Y as  a ( X  x1 )   a2 ( x 2  X 2 )   a1 ( X 1  x 1 ) a 2 ( x 2  X 2 )  (a1 ,a 2 ) t RPe  y exp  1 1   exp    y exp  ,   ( X x ) ( x X ) (x2  X2 )    2   ( X1  x1 ) 1 1  2

(6.1) where a1 and a 2 are suitably chosen constants. For ( a1 , a 2 )=(0,0), (1,1), (1,0), ( a1 , a 2 ) and (0,1), t RPe respectively reduce to y , t RPe , t Re and t Pe . To the first degree ( a1 , a 2 ) of approximation the bias and mean squared error of t RPe are respectively given by

a C 2  (2  a 2 )  a1C 12  (2  a1 )  a1a 2 12C 1C 2  ( a1 , a 2 ) B( t RPe )   Y  2 2  K 02  ,   K 01   4  2  4  4  2   (6.2)  a C2 a C2 aa  CC  ( a1 , a 2 ) )   Y 2 C 02  1 1 ( a1  4 K 01 )  2 2 ( a 2  4 K 02 )  1 2 12 1 2  . MSE( t RPe 4 4 2   (6.3) We mention that to the first degree of approximation the biases and MSEs of the estimators y , t RPe , t Re and t Pe can be easily obtained from (6.2) and (6.3) just be putting ( a1 , a 2 )=(0,0), (1,1), (1,0), and (0,1) respectively. ( a1 , a 2 ) The MSE ( t RPe ) at (6.3) is minimized for


a1 

2( K 01  K 02 K 21 ) 2( K 02  K 01K 12 )  a10 ( say ), a 2    a 20 ( say ). (1  K 12 K 21 ) (1  K 12 K 21 )

307

(6.4)

Substitution of (6.4) in (6.1) yields asymptotically optimum estimator (AOE) in ( a1 , a 2 ) as the class of estimators t RPe   a ( X  x 1 )   a 20 ( x 2  X 2 )  ( a10 , a 20 ) t RPe  y exp   10 1   .   ( X 1  x 1 )   ( x 2  X 2 ) 

(6.5)

( a1 , a 2 ) is given by The MSE of AOE t RPe ( a10 , a 20 ) 2 2 MSE( t RPe )  ( / 2) S02 [1  ( 01   02  2  01  02 12 )/(1  122 )] .

(6.6)

( a10 , a 20 ) can be used in practice only when the It is to be noted that the AOE t RPe optimum values a10 and a 20 of the scalars a1 and a 2 respectively are known. However it may happen in some practical situations that the optimum values a10 and a 20 are not known. In such situation it is worth advisable to estimate them from the sample data at hand, let

aˆ10 

2( Kˆ 01  Kˆ 02 Kˆ 21 ) 2( Kˆ 02  Kˆ 01Kˆ 12 ) , aˆ20   ; (1  Kˆ 12 Kˆ 21 ) (1  Kˆ 12 Kˆ 21 )

(6.7)

be consistent estimators of a10 and a 20 respectively, where s Cˆ Cˆ Cˆ Cˆ Kˆ 01  ˆ 01 0 ; Kˆ 21  ˆ12 2 ; Kˆ 02  ˆ 02 0 ; Kˆ 12  ˆ12 1 ; ˆ 0 i  0 i s 0s i Cˆ 1 Cˆ 2 Cˆ 1 Cˆ 2 ˆ12 

and

s 12 . s 1s 2

Replacing a10 and a 20 by their consistent estimators aˆ10 and aˆ20 respectively ( aˆ10 , aˆ20 ) (based esin (6.5) we get a ratio-cum-product type exponential estimator t RPe

timated optimum values) of the population mean Y as  aˆ ( X  x 1 )   aˆ 20 ( x 2  X 2 )  ( aˆ10 , aˆ20 ) t RPe  y exp  10 1  exp    ( X1  x1 )   (x2  X2 ) 

(6.8)

( a1 , a 2 ) is given by The MSE of AOE t RPe ( aˆ10 , aˆ 20 ) ( a10 , a 20 ) 2 2 MSE( t RPe )  ( / 2)S02 [( 01   02  2  01  02 12 )/(1  122 )]  MSE( t RPe ) (6.9)

308


Thus if the optimum values a10 and a 20 of a1 and a 2 respectively are not ( aˆ10 , aˆ20 ) (based on estiknown, then it is worth advisable to prefer the estimator t RPe ( a10 , a 20 ) in practical research. mated optimum value) over the AOE t RPe

7. EMPIRICAL STUDY To have tangible idea about the performance of the various estimators of Y , we consider the two natural population data. Population –I: [Source : Steel and Torrie (1960, p. 282)] Y: Log of leaf burn in sec., X 1 : Potassium percentage, X 2 : Chlorine percentage. Y  0.6860 , X 1  4.6537 , X 2  0.8077 , C 0  0.4803 , C 1  0.2295 , C 2  0.7493 ,  01 =0.1794, 02  0.4996 and 12  0.4074 Population –II: [Source: Singh (1965, p. 325)] Y: Females employed, X 1 : Females in service, X 2 : Educated females Y  7.46 , X 1  5.31 , X 2  179.00 , C 02  0.5046 , C 12  0.5757 , C 22  0.0633 ,  01 =0.7737,  02  0.2070 and 12  0.0033 We have computed the percent relative efficiencies (PRE(s)) of different estimators of population mean Y with respect to usual unbiased estimator y and findings are complied in Table 7.1. TABLE 7.1

Percent relative efficiencies (PRE’s) of different estimators of population mean Y with respect to usual unbiased estimator y Estimators

PRE (  , y )

y

Population –I 100.00

Population –II 100.00

tR

94.62

208.23

tP

53.34

102.16

t RP

75.50

216.66

t Re

102.95

217.95

t Pe

121.25

104.38

(u ) t RPe

155.23

241.81

t R Re( u )

157.30

276.25

( aˆ10 ,aˆ20 ) t RPe

174.04

278.10

t R Re or

Table 7.1 exhibits that the proposed ratio-cum-product type exponential esti( aˆ10 , aˆ20 ) mator t RPe is more efficient than all the estimators y , t R , t P , t Re , t Pe , t RP ,

309


(u ) t RPe or t RPe and t RPe ( u ) . It is interesting to note that the ratio-cum-product type (u ) is more efficient than Singh’s (1967) ratioexponential estimator t RPe or t RPe cum-product estimator t RP , y , t R , t P , t Re , t Pe with substantial gain in efficiency in both the populations I and II. Thus the proposed ratio-cum-product ( aˆ10 , aˆ20 ) (u ) type exponential estimators t RPe , t RPe , t RPe ( u ) and t RPe are to be preferred over

Singh’s (1967) estimator t RP in practical research. ACKNOWLEDGEMENT

Authors are thankful to the referee for his valuable suggestions regarding improvement of this paper. The research of the second author was supported by the grant of U.G.C., New Delhi. School of Studies in Statistics Vikram University, Ujjain, M.P., India

HOUSILA P. SINGH

Department of Applied Mathematics Indian School of Mines, Dhanbad, Jharkhand, India School of Studies in Statistics Vikram University, Ujjain, M.P., India

L.N. UPADHYAYA RAJESH TAILOR

REFERENCES

(1987), A class of almost unbiased ratio and product type estimators for finite population mean applying Quenouilles method, “Journal of Indian Society of Agricultural Statistics” , 39, pp. 280-288. H. P. SINGH, R. TAILOR (2005), Estimation of finite population mean using known correlation coefficient between auxiliary characters, “Statistica” , Anno LXV, 4, pp. 407-418. M. P. SINGH (1965), On the estimation of ratio and product of population parameters, “Sankhya”, 27,B, pp. 321-328. M. H. QUENOUILLE (1956), Notes on bias in estimation, “Biometrika”, 43, pp. 353-360. R. G. D. STEEL, M. P. TORRIE (1956), Principles and Procedures of Statistics, McGraw Hill Book Co. New York. S. BAHL, R. K. TUTEJA (1991), Ratio and product type exponential estimator, “Information and Optimization Sciences”, 12, 1, pp. 159-163. T. JOHNSTON (1972), Econometrics Methods, McGraw Hill, New York. H. P. SINGH

SUMMARY

Ratio-cum-product type exponential estimator This paper addresses the problem of estimating the population mean Y of the study variate Y using information on two auxiliary variables X 1 and X 2 . A ratio-cum-product

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type exponential estimator has been suggested and its bias and mean squared error have been derived under large sample approximation. An almost unbiased ratio-cum-product type exponential estimator has also been derived by using Jackknife technique envisaged by Quenouille (1956). A generalized version of the ratio-cum-product exponential estimator has also been given along with its properties. Numerical illustration is given in support of the present study.