A Separate Exponential Ratio-Type Estimator of finite Population Mean under Power Transformation

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A Separate Exponential Ratio-Type Estimator of finite Population Mean under Power Transformation

1Ahmad Bandiya Jega and 2Ran Vijay Kumar Singh 1,2Kebbi State University of Science and Technology, Aliero, Nigeria

Abstract: In this paper, a separate ratio-type exponential estimator for estimating the finite population mean has been proposed. Mathematical expressions for the bias and mean square error (MSE) of the proposed estimator have been derived to the first order of approximation. Theoretical conditions have been obtained under which the proposed estimator is more efficient than the estimators under study. Numerical illustration has also been carried out to compare the efficiency of proposed estimator and found that the proposed estimator was more efficient.

Keywords:- Bias, Exponential estimator, Mean square error, Separate estimator, Stratified sampling.

  1. INTRODUCTION

    The problem of estimation of population parameters has been an important issue in sample survey and many methods have been used in order to improve the efficiency of the estimators. In survey sampling, it is well established that the use of auxiliary information results in substantial gain in efficiency over the estimators which do not use such information. In some cases, in addition to mean of auxiliary variable, various other parameters related to auxiliary variable such as coefficient variation, correlation coefficient etc are used to estimate the population parameter. Sisodia and Dwivedi (1981), Rao (1991), Upadhayaya and Singh (1999), developed various estimators to improve the ratio estimators in simple random sampling. Kadilar and Cingi (2003) modified the various estimators under stratified random sampling. Bahl and Tuteja (1991) introduced ratio and product- type exponential estimators which perform better than the classical ratio and product estimators respectively. Singh et al. (2008) proposed a ratio and product-type exponential estimators which were more efficient than the Bahl and Tuteja (1991) estimators. Upadhyaya et al. (2011), Singh and Ahmed (2014), Singh and Ahmed (2015a, 2015b), Singh (2016) did remarkable work in this direction. Singh et al. (2018) suggested combined ratio-type exponential estimator of population mean which was equally efficient as combine linear regression estimator.

    Let U U1,…..,UN

  2. NOTATIONS

    be a finite population of size N which are partitioned into K distinct strata with ith

    K

    stratum

    containing

    Ni units i 1, 2….., k such that Ni N . Let a sample of size ni units i 1, 2,…., k

    i1

    be drawn from

    k

    k

    the population using simple random sampling preferably without replacement (SRSWOR). Such that ni n .

    i1

    Let y x

    be the observed values of Y , X

    on the

    jth

    unit of the

    ith

    stratum j 1, 2,…, N

    Moreover, the

    ij , ij

    1 Ni

    i

    1 Ni

    population means of the variables Y and X in the ith stratum are Yi

    Yij , Xi

    N

    N

    i j1

    Xij and the corresponding

    N

    N

    i j1

    1 ni

    1 ni

    sample means of the variable Y and X in the ith stratum are

    yi

    yij

    n

    n

    i i1

    , xi

    xij

    n

    n

    i i1

    respectively.

    Estimators of the variable Y and X , in stratified random sampling are given by;

    k k k

    yst Wi yi i1

    k

    and

    xst Wi xi

    i1

    are the unbiased estimators of the population means

    N

    Y WiYi

    i1

    and

    X Wi Xi

    respectively. Where Wi

    i

    N

    denotes the stratum weight.

    i1

    Let be the correlation coefficient between the study variable and the auxiliary variable.

  3. EXISTING ESTIMATORS

    The separate ratio estimator for the population mean Y is defined as;

    k yi

    ysr Wi x X i

    (1)

    i1 i

    and its bias and MSE are given as;

    k 1 2

    B ysr Wii

    i1

    Ri Sxi SYXi

    Xi

    MSE y

    k

    W 2 S2 R2S2 2R S

    (2)

    sr i i yi i xi i YXi i1

    The separate product estimator for the population mean Y is define as;

    k yi . xi

    ysp Wi X

    (3)

    i1 i

    and its Bias and MSE are given as;

    k 1

    B ysp Wii

    i1

    SYXi

    X

    X

    i

    MSE y

    k

    W 2 S2 R2S2 2R S

    (4)

    sp i i yi i xi i YXi i1

    The separate linear regression estimator for the population mean Y is defined as;

    k

    k

    y W y

    • b X xi

      (5)

      sl i

      i1

      i i i

      And its MSE is given as;

      MSE( y

      ) W 2 S 2

      1 2

      (6)

      k

      k

      sl

      i1

      i i Yi YXi

  4. ADAPTED ESTIMATORS

    Bahl and Tuteja (1991) ratio-type exponential estimator for the population mean Y under stratified random sampling can be defined as:

    k X i xi

    ystBTR Wi yi exp X

    • x

    (7)

    i1

    i i

    its bias and MSE are given as;

    Bais y

    k

    W 1

    3 R S 2

    1 S

    stBTR i i X i 8

    i Xi

    2 XYi

    MSE y

    i1

    k

    W 2 S2 1 R2S2 R S

    (8)

    stBTR i i yi 4 i xi i YXi

    i1

    Bahl and Tuteja (1991) product-type exponential estimator for the population mean Y under stratified random sampling can be defined as:

    k xi X i

    (9)

    ystBTP Wi yi exp x X

    and bias and MSE are given as;

    i1

    i i

    k 1 1 1

    Bais y

    W S R S 2

    i1

    i1

    stBTP i i X i 2 XYi 8 i Xi

    MSE y

    k

    W 2 S2 1 R2S2 R S

    (10)

    stBTP i i yi 4 i xi i YXi

    i1

    Singh et al (2018) estimator

    y

    y

    A combined ratio-type exponential estimator for population mean under stratified random sampling using information on single auxiliary variable has been suggested by Singh et al ( 2018) given as:

    y

    x

    st

    st

    S st

    S st

    exp X x

    st

    where is a constant. (11)

    X X xst

    Bias and MSE of tS are obtained as

    2 1 1

    4 2 8 3 1

    B yS Y

    COV yst xst

    2 V xst

    2 XY

    k

    2 12

    8 X

    4

    4

    MSE y W 2 S 2

    R2 S 2

    2 1 RS

    S

    i1

    i i Yi

    Xi YXi

    For the optimum value of , the Minimum MSE of tS is obtained as

    k 1 1

    2 2 *2

    MSE yS

    i1 ni

    • Wi S Yi 1

      Ni

      YX

      (12)

      That is the MSE of combined linear regression estimator.

  5. THE SUGGESTED ESTIMATOR

    In stratified random sampling, we suggest a separate ratio-type exponential estimator as

    K x

    i

    X x

    i

    i

    i

    i

    t W y

    i i

    i i

    exp i

    P

    i1

    X i X i xi

    (13)

    Where i is a constant.

  6. BIAS AND MEAN SQUARE ERROR OF THE SUGGESTED ESTIMATOR In order to obtain the Bias and MSE to the first order of approximatin, let us define

    yi Yi (1 e0i ) and

    Therefore

    E(e0i ) E(e1i ) 0

    xi Xi (1 e1i )

    (14)

    1 1

    S 2

    E(e2 )

    Yi

    0 n N Y 2

    (15)

    i i i

    1 1 S 2

    E(e2 )

    • Xi

      1i n N X 2

      Now, by substituting the values of xi and y from (14) into (13) and on solving, the

      i i i

      1 1 S

      i

      bias of proposed estimator is obtained as;

      E(e0ie1i )

    • YXi

      K 1 1 1

      4 2 8

      3

      2

      1

      ni

      Ni Yi Xi

      Bais t

      W

      i

      i

      i i R S 2

      i S

      P

      (16)

      i1

      ni

      Ni Xi

      8 i Xi

      2 YXi

      Now, MSE of the proposed estimator tP can be obtained by squaring, simplify up to the first order of approximation and taking the expectation we will have;

      K 1 1 2 12

      MSE t

      W 2 S 2 i R2 S 2

      Yi

      Yi

      2

      1 R S

      (17)

      P i

      i1 ni Ni

      4 i Xi i i YXi

      To obtain the optimum mean squared error,

      MSEtP 0 gives 1 iCYi

      i

      iopt 2

      CXi

      and minimum MSE of tP is obtained as

      k 1 1

      2 2 2

      min

      min

      MSE tP

      i1 ni

    • Wi SYi 1 i

    Ni

    (18)

    That is the MSE of the separate linear regression estimator.

  7. THEORETICAL EFFICIENCY COMPARISON

    In this section, conditions have been found under which the proposed estimator is more efficient than existing estimators under study.

    1. Comparison of the separate ratio estimator with the proposed estimator

      MSE tP MSE ysr if and only if

      3

      C 1 3

      C 1

      min

      2i Yi

      i max

      2i Yi

      (19)

      2

      CXi ,

      2

      2

      CXi ,

      2

      Therefore the proposed estimator is more efficient than

      ysr

      if condition (19) is satisfied.

    2. Comparison of the separate product estimator with the proposed estimator

      MSE tP MSE ysp if

      3 1

      min 2

      CYi

      max 3 1 2

      CYi

      (20)

      2, 2

      i C i

      2, 2

      i C

      Xi

      Xi

      Therefore the proposed estimator is more efficient than

      y sp

      if condition (20) is satisfied.

    3. Comparison of the Bahl and Tuteja (1991) ratio-type exponential estimator under stratified random sampling with the proposed estimator

      MSE tP MSE ystBTR

      min 0,1 2

      CYi

      max 0,1 2

      CYi

      (21)

      i C i i C

      Xi Xi

      Therefore the proposed estimator is more efficient than

      ystBTR if condition (21) is satisfied.

    4. Comparison of the Bahl and Tuteja (1991) product-type exponential estimator under stratified random sampling with the proposed estimator

      MSE tP MSE ystBTP

      min1, 2

      CYi

      max1, 2

      CYi

      (22)

      i C i i C

      Xi Xi

      Therefore the proposed estimator is more efficient than

      ystBTP if condition (22) is satisfied.

  8. NUMERICAL EFFICIENCY COMPARISON

    For numerical illustration the data of Kadilar and Cingi (2003) is used which is given in table 1. Y represents apple production amount (as variable of interest) and X represents number of apple trees (as auxiliary variable) in 854 villages of Turkey in 1999 (Source: Institute of Statistics, Republic of Turkey). The data is stratified by regions of Turkey and from each stratum (region) the samples (villages) are selected by using Neyman allocation.

    Table 1: Data Statistics

    Ni

    ni

    X i

    Y i

    SXi

    SYi

    i

    i

    W 2

    i

    Stratum1

    106

    9

    24375

    1536

    49189

    6425

    0.82

    0.102

    0.015

    Stratum 2

    106

    17

    27421

    2212

    57461

    11552

    0.86

    0.049

    0.015

    Stratum 3

    94

    38

    72409

    9384

    160757

    29907

    0.90

    0.016

    0.012

    Stratum 4

    171

    67

    74365

    5588

    285603

    28643

    0.99

    0.009

    0.04

    Stratum 5

    204

    7

    26441

    967

    45403

    2390

    0.71

    0,138

    0.057

    Stratum 6

    173

    2

    9844

    404

    18794

    946

    0.89

    0.006

    0.041

    Table 2: Theoretical comparison supported numerically using data set provided in Table 1

    MSE tP MSE ysr

    1 = -1.1997

    -1.8994 < -1.1997 < -0.5

    Satisfied

    2 = -1.6433

    -2.7866 < -1.6433 < -0.5

    Satisfied

    3 = -0.7920

    -1.0839 < -0.7920 < -0.5

    Satisfied

    4 = -0.8213

    -1.1426 < -0.8213 < -0.5

    Satisfied

    5 = -0.5220

    -0.5439 < -0.5220 < -0.5

    Satisfied

    6 = -0.5916

    -0.6831 < -0.5916< -0.5

    Satisfied

    MSE tP MSE ysp

    1 = -1.1997

    -3.8994 < -1.1997 < -1.5

    Satisfied

    2 = -1.6433

    -4.7866 < -1.6433 < -1.5

    Satisfied

    3 = -0.7920

    -3.0839 < -0.7920 < -1.5

    Satisfied

    4 = -0.8213

    -3.1426 < -0.8213 < -1.5

    Satisfied

    5 = -0.5220

    -2.5439 < -0.5220 < -1.5

    Satisfied

    6 = -0.5916

    -2.6831 < -0.5916 < -1.5

    Satisfied

    MSE tP MSE ystBTR

    1 = -1.1997

    -2.3994 < -1.1997 < 0

    Satisfied

    2 = -1.6433

    -3.2866 < -1.6433 < 0

    Satisfied

    3 = -0.7920

    -1.5839 < -0.7920 < 0

    Satisfied

    4 = -0.8213

    -1.6426 < -0.8213 < 0

    Satisfied

    5 = -0.5220

    -1.0439 < -0.5220 < 0

    Satisfied

    6 = -0.5916

    -1.1831 < -0.5916 < 0

    Satisfied

    MSE tP MSE ystBTP

    1 = -1.1997

    -3.3994 < -1.1997 < 1

    Satisfied

    2 = -1.6433

    -4.2866 < -1.6433 < 1

    Satisfied

    3 = -0.7920

    -2.5839 < -0.7920 < 1

    Satsfied

    4 = -0.8213

    -2.6426 < -0.8213 < 1

    Satisfied

    5 = -0.5220

    -2.0439 < -0.5220 < 1

    Satisfied

    6 = -0.5916

    -2.1831 < -0.5916 < 1

    Satisfied

    Table 3: MSE and PRE of the Proposed Estimator with other estimators

    ESTIMATORS

    MSE

    PRE

    yst

    673477.7

    100

    ysr

    159137.4

    423.2

    y sp

    1790757

    37.6

    ystBTR

    340940

    197.5

    ystBTP

    1156750

    58.2

    yS

    202122.1

    333.2

    tp

    107065.8

    629

  9. CONCLUSION

In the present study, Singh et al (2018) estimator has been improved by modifying it as separate ratio-type exponential estimator and its properties have been studied. Theoretical conditions have been obtained under which the proposed estimator is more efficient than the estimators under study and supported numerically as mentioned in table 2. Table 3 reveals that the proposed

estimator tp

has smallest mean square than the conventional ratio, product and Bahl and tuteja (1991) estimators under stratified

random sampling as well as Singh et al (2018) estimator. Therefore, tp

estimating the finite population mean.

REFERENCES

is more efficient than the other existing estimators for

    1. Bahl, S. and Tuteja, R.K. (1991). Ratio and product type exponential estimator. Journal of information and Optimization Sciences, 12(1), 159- 163.doi:10.1080/02522667.1991.10699058

    2. Kadilar C. and Cingi H. (2003). Ratio estimators in stratied random sampling, Biometrica Journal. 45, pp . 218-225.

    3. Rao, T. (1991). On certain methods of improving ratio and regression estimators Communications in Statistics-Theory and Methods 20(10), 33253340.

    4. Singh, R., Chauhan, P. And Sawan, N. (2008).On linear combination of ratio-product type exponential estimator for estimating finite population mean. Statistic in transition, 9 ,105-115.

    5. Singh, R. V. K. and Ahmed, A. (2015). Improved Exponential Ratio and Product Type Estimators for Finite Population Mean International Journal of Engineering Science and innovative Technology, 4(3) , 317-322.

    6. Singh, R. V. K. and Ahmed, A. (2015). Improved Exponential Ratio and Product Type Estimators for Finite Population Mean Under Double sampling Scheme. International Journal of Scientific & Engineering Research, 6(4), 509-514.

    7. Singh, R. V. K. and Ahmed, A. (2014). Ratio Type Estimators in Stratified Random Sampling Using Auxiliary Attributes IAENG Conference IMECS, 389-393.

    8. Singh, R. V. K. (2016). Study of predictive approach product type estimator under polynomial regression model. International Journal of Scientific & Engineering Research, 7(1), 667-671.

    9. Singh, R.V. K., Mariam, M. and Sifawa, A. (2018). Combined exponential ratio-type estimator for estimating population mean. 37th Annual conference, Nigerian Mathematical Society, 8th -11th May 2018, Bayero University, Kano.

    10. Sisodia, B. V. S., and Dwivedi, V. K. (1981). A modified ratio estimator using coefficient of Variation of auxiliary variable. Journal of the indian of Agricultural Statistics,33(2),13-18

    11. Upadhyaya, L.N. and Singh, H. P. (1999).Use of transformed auxiliary variable estimating the finite population mean. Biometrical Journal.41 (5), 627-636.

    12. Upadhyaya, L.N., Singh, H.P., Chatterjee S., and Yadav, R. (2011). Improved ratio and product exponential type estimators. Journal of Statistical Theory and Practice,5(2): 285-302.

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