improved ratio estimator in simple random sampling

Golden Research Thoughts ISSN 2231-5063 Impact Factor : 2.2052 (UIF) Volume-3 | Issue-12 | June-2014

Available online at www.aygrt.isrj.net

GRT IMPROVED RATIO ESTIMATOR IN SIMPLE RANDOM SAMPLING Asra Nazir1 , S.Maqbool2 , Aquil Ahmed1 and Rafia Jan1 1

Department of statistics, university of Kashmir,J&k,India. Division of Agricultural statistics, SKUAST-Kashmir, India.

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Abstract:- In the present paper, we proposed improved ratio estimator by adopting the estimator type of Ray and Singh(1981) in simple random sampling. Mean square error (MSE) of the proposed ratio estimator is obtained and compared with classical ratio estimator and existing ratio estimators. By this comparison, the condition which makes the proposed estimator more efficient than the others are found .Theoretical result is supported by a numerical illustration. Keywords: Ratio type estimator Simple random sampling, Mean square error, Auxilary variable, Efficiency;

1.INTRODUCTION The classical ratio estimate for the population mean

yr 

Y of variate y is defined by

y X  Rˆ X x

(1.1)

where it is assumed that the population mean X of auxiliary variate x is known .Here variate of interest and x is the sample mean of auxiliary variate .From (1.1), we have MSE of classical ratio estimate is given as: MSE ( y r ) 

1 f ( R 2 S x2  2 RS xy  S y2 ) n

y is the sample mean of

(1.2)

Where f=n\N, n is the sample size and N is the population size, R 

Y 2 is the population ratio ; S x is the X

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population variance of auxiliary variable and S Y is the population variance of variable of interest. When the population coefficient of variation of auxiliary variate C x is known , Sisodia and Dwivedi (1981) suggested a modified ratio estimator for

y SD  y

y as

X  CX Y  X SD  Rˆ SD X SD x  CX x SD

From (1.3),

( 1.3)

Y Y x SD  x  C X ; X SD  X  C X ; Rˆ SD  andRSD  x SD X SD

MSE of this estimate is given

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MSE ( (YSD ) 


1 f 2 2 Y [CY  C X2  (  2 K ] n

(1.4)

CY is the population coefficient of variation of variate of interest, C X  andK   Y X  CX CX Where  is the correlation coefficient between auxiliary and study variable in the population. Where

Motivated by Sisodia and Dwivedi (1981), Singh and Kakran (1993) developed ratio type estimator for Y as

X  2  X   y X SK  Rˆ SK X SK (1.5) x  2 x SK Where  2 x)  I s the population coefficient of kurtosis of auxiliary variable .From (1.5); y Y x SK  x   2 x ; X   2 x ; Rˆ SK  ; RSK  x SK X SK YSK  Y

Similar to (1.4);MSE of this estimator is given as

1 f 2 2 2 Y [CY  C X  (  2 K )] n X Where   X   2 X  MSE(

y SK )



( 1.6)

Motivated by Sisodia and Dwivedi (1981), we introduced median in existing estimator, then our proposed estimator takes the following form,

X  md Y  X AS 1  R AS 1 X AS 1 x  md x AS 1

YASI  Y

(1.7)

Where md is the median of population

Y Y x AS 1  x  md ; X AS 1  X  md ; Rˆ AS 1  ;R  x AS 1 X AS 1 Mean square error of (1.7) can be calculated by using the taylor series expansion method .





MSE YAS 1 

1 f 2 2 Y [CY  C X2  (  2 K ) n

(1.8)

where



X X  md

and

K=



CY CX

2 . THE SUGGESTED ESTIMATOR: Ray and Singh (1991) suggested estimator of type

y RS 

y  b( x   X  )  X x


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In the above estimator, we proposed to take   1 and median of the auxiliary variable takes the following form

  1 then the improved estimator which is based on

y  b( X  x ) (2.1) X  md  Rˆ PAS 1 X  md x  md s xy Where b= s x2 is the sample variance of auxiliary variate and s xy is the sample covariance between 2 ; sx y PAS 1 

auxiliary variable and variable of interest .From (2.1)

R pr 

y  b( X  x ) .Note that when b=0 in (2.1),the x

proposed estimator turns to be traditional ratio estimate given in (1.1) MSE of the proposed estimator can be found using Taylor series method defined as

h( x , y )  h( X , Y ) 

h(c, d ) c

Where h( x , y )  Rˆ pr and

 h(c, d ) ( x  X ) xy d

xy

(y Y )

(2.2)

h( X , Y )  R .

As per wolter(1985), equation (2.2) can be applied to the modified estimator in order to obtain MSE equation as follow ;

(( y  b( X  x )) \ x  md Rˆ PAS 1  R  x

X ,Y

(x  X ) 

(( y  b( X  x )) \ x  md y

X ,Y

(y Y ) (2.3)

 y bX b( x  md )  bx  1  X ,Y ( x  X )      (y Y ) 2 2 2 ( x  md ) ( x  md ) ( x  md ) 2 X ,Y   ( x  md )  Y b( X  md )  1 ( x  X )  E ( Rˆ PAS 1  R)    ( y  Y )2 2 2  ( X  md )  ( X  md ) 2  ( X  md ) 2

(2.4)

2

Y  B( X  md )  2Y B( X  md ) 1 E ( Rˆ PAS 1  R) 2   cov( x , y )  v( y )  v( x )  2 3 ( X  md ) ( X  md )  ( X  md )  (2.5) 2

Y  B( X  md )  1 2Y B( X  md )  cov( x , y )  v( y )  v( x )  2  ( X  md ) ( X  md )  ( X  md )  2 ˆ MSE YPAS 1   ( X  md ) E ( R PAS 1  R) (2.6)

2

Y  B( X  md )  Y  B( X  md )  cov( x , y )  v( y )  2 ( X  md )  ( X  md )  2

Y 2  B( X  md ) 2Y B( X  md ) Y  B( X  md )   v( x )  2( cov( x , y )  v( y ) 2 2 ( X  md ) ( X  md ) ( X  md ) _____________________________________________________________________________________ Golden Research Thoughts | Volume 3 | Issue 12 | June 2014

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 2 1 f  Y2 Y Y  B2  2 S XY  2 BS XY  SY2  S X  2 2 n  ( X  md ) ( X  md )  ( X  md ) 1 f 2  RPASI S X2  B 2 S X2  2 RS X2  2 RS XY  SY2  n 1 f 2 RPAS 1 S X2  SY2 (1   2 )} MSE (YAS 1 )  n (2.7)





3. EFFICIENCY COMPARISON: We compare the MSE of Existing estimator with the MSE of the proposed AS1 estimator using (1.2) and (2.7) MSE (YAS 1 )  MSE (YExisting )

YS Y S X2  x2 2 X2 ( X  md )

(3.1) It is evident from the above comparison that the condition given in (3.1) is satisfied in all conditions . 4. NUMERICAL ILLUSTRATION : We have used the data of Murthy (1967) in which fixed capital is denoted by Y and output of 80 factories are denoted by X Table 1 gives the data statistics about the population under consideration Table 1 Data statistics

N  80 n  20 X  51.8264 Y  11.2646 CY  0.7505 C X  0.3542  2 ( x)  0.06339 1 ( x)  1.05   0.9413 md  7.575 S X  18.3569 SY  8.4563


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MSE values of ratio estimator

Sisodia –Dwivedi Singh-Kakran AS1(proposed)

Proposed type 0.891 0.898

Classical type 1.29 1.32 2.69

0.757 From the numerical results ,we infer that the proposed estimator is more efficient in all situations .Hence it is recommended to use the proposed estimator in future. 5. CONCLUSION: We have derived a new ratio type estimator and obtain their MSE equations.MSE of proposed estimator is compared with each other and in the results of comparison ,the theoretical results have also been compared with the numerical.It is deduced that the proposed estimator is more efficient than the classical. we hope to extend the formulation presented here to ratio estimator in stratified random sampling. REFERENCES: 1.B.V.S.Sisodia,and V.K.Dwivedi, (1981). A modified ratio estimator using coefficient of variation of auxiliary variable. Journal of Indian society Agricultural statistics, 33,13-18. 2.C.kadilar and H.cingi (2003). Ratio estimator in stratified random sampling .Biometrical journal, 45(2), 218-225. K.M, Wolter(1985) .Introduction to variance estimation ,springer –verlag, 3.L.N.Upadhya and H.P.singh, (1999).Use of transformed auxiliary variable in estimating the finite population mean .Biometrical Journal, 41 (5), 627-636. 4.S.K.Ray and R.K. Singh (1981).Difference cum ratio type estimators. Journal of Indian statistical Association, 19, 147-151. 5.W.G.Cochran, (1977). Sampling Techniques. John wiley and sons ,New York..

Asra Nazir Department of statistics, university of Kashmir,J&k,India.

Rafia Jan Department of statistics, university of Kashmir,J&k,India.


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