Vol 9, No 5 (2018)

Available online throug

ISSN 2229 – 5046

PREDICTIVE ESTIMATION OF FINITE POPULATION MEAN

USING COEFFICIENT OF KURTOSIS AND MEDIAN OF AN AUXILIARY VARIABLE

UNDER SIMPLE RANDOM SAMPLING SCHEME

1

_{Dharmendra Kumar Yadav,}

_{Ashish Kumar Shukla}

_,

_{Sachin Tomer and}

_{Birjesh Kumar}

1,2,3,4

_{Department of Statistics,}

Ramanujan College, University of Delhi, Kalaka Ji, New Delhi-110019, India.

(Received On: 31-01-18; Revised & Accepted On: 08-05-18)

ABSTRACT

T

Key words: Predictive estimation, Kurtosis, Median, Bias, and Mean squared error.

1. INTRODUCTION

It is notable that the use of auxiliary information is very common practice in sampling theory to improve the efficiency of estimators. The theory of basic sample survey as available in standard text books on sampling deals with the case which comprises linear estimators such as mean, variance, total and proportion using simple random sampling without replacement. It continues to supply new and improved procedures for estimation of population means assuming independence of observations. Ratio, difference and regression estimators utilize an auxiliary variable for more efficient estimation of the parameter in question. Such estimators take advantage of the correlation between the auxiliary variable (x) and the study variable y. In a similar manner, then, it seems reasonable that under suitable conditions efficient estimation of the mean of the estimator of the finite population total or mean of the characteristic y is also possible using such estimation techniques. With the increasing growth in the number and diverse uses of sample surveys worldwide, it is often desired to analyse and interpret the resulting voluminous data by swifter methods. A basic requirement of a good survey is that a measure of precision is provided for each estimate computed from survey data collected on the basis of the survey design. An important question is how to choose an appropriate mean estimator.

Many authors have come up with more precise estimators by employing prior knowledge of certain population parameter(s). [Searls [1964] for example attempted use of the coefficient of variation of study variable but prove inadequate for in practice, this parameter is unknown. Motivated by [Searls [1964] work, [Sen[1978]], [Sisodia and Dwivedi [1981]] and [Upadhyaya and Singh [1884]] used the known coefficient of variation but now that of the auxiliary variable for estimating population mean of study variable. Reasoning along the same path [Hirano et al.

[1973]] used the prior value of coefficient of kurtosis of an auxiliary variable in estimating the population mean of the study variable y.

Corresponding Author:

_{Ashish Kumar Shukla}

_,

2

_{Department of Statistics, Ramanujan College, University of Delhi,}

Kurtosis in most cases is not reported or used in many research articles, in spite of the fact that fundamentally speaking every statistical package provides a measure of kurtosis. This may be attributed to the likelihood that kurtosis is not well understood or its importance in various aspects of statistical analysis has not been explored fully. Median being the middlemost value in a distribution (when the values are arranged in ascending or descending order) has the advantage of being less affected by the outliers and skewed data, thus is preferred to the mean especially when the distribution is not symmetrical. We can therefore utilize the median and the coefficient of kurtosis of the auxiliary variable to derive a more precise ratio type population mean.

In the predictive approach a model is specified for the population values and is used to predict the non-sampled values. Prediction theory for sample surveys (or model-based theory) can be considered as a general framework for statistical inferences on the character of finite population. Well known estimators of population parameters encountered in the classical theory, as expansion, ratio, regression, another estimators can be predictors in the general prediction theory under some special model. Several authors have applied the predictive approach either to form new predictive estimators or to examine the existing estimators from the predictive view point. Srivastava [1983] has shown that if the usual product estimator is used as a predictor for the mean of the unobserved units of the population, the resulting estimator of the mean of the whole population is different from the customary (usual) product estimator. Agrawal and Roy [1999] and Nayak and Sahoo [2012] provided some predictive estimators for finite population variance. Sahoo and Panda [1999] developed the regression type estimator for two stage sampling procedure. Sahoo and Sahoo [2001] and Sahoo et al. [2009] introduced a class of estimators for the finite population mean availing information on two auxiliary variables in two stage sampling. Saini [2013] proposed a class of predictive estimators for two stage design consisting especially of two estimators namely ratio and regression. Singh et al. [2014] proposed exponential ratio and product type estimator of population mean based on Bahl and Tuteja [1991] estimator under predictive modeling approach. Yadav and Mishra [2015] also proposed improved predictive estimator of finite population mean using linear combination of Singh et al. [2014] estimators. Milton et al. [2017] have proposed a ratio type estimator of population variance using coefficient of kurtosis and median of the auxiliary variable under simple random sampling scheme. In the present article we have proposed a new modified ratio type predictive estimator of population mean by using Milton

et al. [2017] estimator as a predictor of the study variable.

2. NOTATIONS AND METHODOLOGY

Consider a finite population U = {U₁ , U₂, U₃… U_N} of N distinct and identifiable units. Let Y be our study and X be corresponding auxiliary variable. Suppose we take a random sample of size n from this bivariate population (Y, X) that is (yi, xi), for i = 1,2,3, … n using simple random sampling without replacement (SRSWOR) scheme. Let Y� and X� be the population means of the study and auxiliary variable and y� and x� be their respective sample means.

The present study is an attempt to improve the efficiency in the estimation of

Y

�= _N1∑Ni=1Yi (1)

Using the coefficient of kurtosis and median of the auxiliary variable x.

We define the following notations which we will use throughout the article.

For the population observations we have,

Y

�= _N1�Yi N

i=1

, X�= 1_N�Xi N

i=1

We also define the following from the sample observations

y�= 1_n�yi n

i=1

= mean of sample observations yi ,

x�= 1_n�xi n

i=1

= mean of sample observations xi

In general, we define the following parameters

μrs= _N−11 ∑Ni=1(yi−Y�)r (xi−x�)s

Cy= Sy 2

Y

�2=

μ20 Y

ρxy= _SxSySxy =coef�icient of correlation between x and y.

κx = μ_μ04 02

2 = coef�icient of kurtosis for the auxiliary variable x. Mx= Population median of the auxiliary variable.

3. PREDICTIVE ESTIMATION OF FINITE POPULATION MEAN

Let Y_i (i = 1, 2, … , N) be the real value taken by the variable under study from the finite population of U of size N. Here, the population parameter to be estimated is the population mean on the basis of observed values of y in an ordered sample of the finite population U of size N. Let S denote the collection of all possible samples from the finite population U. Let w(s) denote the effective sample size, for any given s∈S and s̅ denote the collection of all those units of U which are not in S.

We now denote:

y�s= _w(s)1 �yi i∈s

y�s�= _N−1_w(s)�yi i∈s�

We have,

Y

�= w(s)_{N y}�s+ N−_Nw(s)y�s�

Basu [1971], asserted that in the representation of Y� above the sample mean y�s being based on the observed y values on units in the sample s is known, therefore the statistician should attempt a prediction of the mean y�_s� of the unobserved units of the population on the basis of observed units in s.

For s∈S under simple random sampling without replacement (SRSWOR) with sample size w(s) = n and y�_s= y�, the population mean Y� is given by

Y

�= n_Ny�s+ (N−n)_N y�s� (2)

In view of equation (2) above, an appropriate estimator of population mean Y ���� is obtained as

t = _{N y}n�+ (N_N−n) T

where T is taken as the predictor of y�_s�.

Let xi (i = 1,2, … , N) denote the ithobservation of the auxiliary variable x and Xi(i = 1,2, … , N) be the values of x on the ith unit of the population U. Auxiliary variable x is correlated with the variable under study y.

Let

X

�₌ 1 N∑Ni=1Xi

and

x�= _N1∑i∈sxi

t =_Nny�+(N−n)_N T (3)

4. REVIEW OF EXISTING ESTIMATORS

(i) Bahl and Tuteja [1991] used the positively correlated auxiliary variable and proposed the following exponential ratio type estimator of population mean of the study variable using known population mean of the auxiliary variable as,

0

exp

X

x

t

y

X

x

−

=

+













The bias and the mean squared error of the above estimator up to the first order of approximation is given by,

0

2

1

( )

[3

4

]

8

f

B t

Y

C

C

n

−

=

−

2 2 2 0

( )

[

]

4

x

y yx

C

(ii) Singh et al.[2014], proposed the following ratio and product type exponential estimator of population estimator of population mean Y� using Bahl and Tuteja [1991], ratio and product types exponential estimator of population mean as the predictive estimator of Y_s� respectively as

t = tRe=�_{N y}n�+�N−_Nn�y�exp�X

�_s�−x�

X

�_{s�+ x�}��= _{N y}n�+�N−_Nn��yexp�_{N(X� −}N(X� −_x�)−x�)_2nx��

t = tpe =�_{N y}n�+�N_N−n�y�exp�_x�x� − _{+ X}X�_�s�

s���= n N y�+�

N−n

N �y�exp�

N(x� −X�) NX�+ (N−2n)x��

Bias(tRe) =ϕ₈Y�Cx2_{[3−4(C + f)], where ϕ=} 1 1−f Bias�tpe�=ϕ_{8 Y}�Cx2_�4C− 1

(1−f)� MSE(tRe) =ϕY�2_�Cy2₊Cx2

4 (1−4c)� (5) MSE�tpe�=ϕ�Y2�Cy2+Cx

2

4 (1 + 4c)� (6)

5. PROPOSED PREDICTIVE ESTIMATOR OF POPULATION MEAN

Motivated by work of Milton [2017] in the improvement of the performance of the population mean of the study variable using known population parameters of an auxiliary variable. We propose the following modified ratio type population mean estimator using a known value of population coefficient of kurtosis κx and median Mx of an auxiliary variable x.

y�dk= y� �X

�_{κx+ Mx}2 x

�κx+ Mx2� (7)

We will use y�_dk as a predictor for population mean. Hence using T = y�_dk in equation (3).

Now the proposed predictive estimator will be given as

t =_{N y}n�+(N_N−n)y�dk

= fy� + (1−f) y� t1 (8)

Where, t1= y� �X� κx+Mx2

x� κx+Mx2�

To calculate the Bias and MSE of t, we are using the following approximations,

y� = Y� (1 + e0), x�= X� (1 + e1), E(e0) = E(e1) = 0, E(e02) = λCy2, E(e12) = λCx2, E(e0e1) = λCyx

where λ= 1−f

n

= y� �X�_x�s� _κκx+ Mx2

x+ Mx2�

= Y � (1 + e0)�� NX�−nx�

N−n �κx+Mx2

X� (1+e1) κx+Mx2�

= Y � (1 + e0)��

NX� −nX� (1 + e1)

N−n � κx+ Mx2 X

� (1 + e1) κx+ Mx2 �

= Y � (1 + e0)�(X�κx+ Mx

2_{)−( n}

N−n)X�e1κx (X�_{κx+ Mx}2_{) + X}�κ_xe1 �

= Y � (1 + e0)

⎣ ⎢ ⎢

⎡1− �₁₋f _f� �_X�κ X�

x+ Mx2� κxe1 (1 + X � κx e1)

⎦ ⎥ ⎥ ⎤

= Y � (1 + e0)�1−f1θκxe1

1 + X�κxe1 � , where f1= f

1−f and θ= X

�

X

�_{κx+ Mx}2 = Y � (1 + e0) (1− f1θκxe1)(1 + X�κxe1)−1

= Y � (1 + e0)[1−f1θκxe1][1−X�κxe1+ X�2_κ x 2_e

1 2_] t1= Y � (1 + e0)[1−X�κxe1+ X�2_e12_κ

x

2_{−f1θκxe1+ f1θκ} x

Substituting value of t1 in (8),

t = fY � (1 + e0) + (1−f) Y � (1 + e0)[1−X�κxe1+ X�2_e 1 2_κ

x

2_{−f1θκxe1+ f1θκ} x 2_e

12]

t−Y� = Y�[ X�κxe1+ X�2_e 1 2_κ

x

2_{− f1θκx e1+ f1θkx}2_e 1

2_{+ f X�κxe1−f X�e} 1 2_κ

x

2_{+ ff1θκxe1−ff1θκ} x 2_e

12+ e0− X�κxe0e1

− f1θκxe0e1+ fX�κxe0e1+ ff1θκx e0e1] (10)

Taking expectation on both sides

E(t−Y� ) = Y�_{[ X�κx}E(e1) + X�2 κx2 E(e12)− f1θκx E(e1) + f1θkx2 E (e12) + f X�κx E(e1)−f X�κx2 E(e12) + ff1θκxE(e1)

−ff1θκx2 E(e12) + E(e0)− X�κxE(e0e1)− f1θκx E(e0e1) + fX�κx E(e0e1) + ff1θκxE( e0e1)]

using the values of E(e0), E(e1), E (e12_{), E(e}

0e1), we get the bias (t) Bias (t) = λY� [X�κx2Cx2− f1θCx2κx2−fX�κx2Cx2−ff1θκx2Cx2−X�κxCyx−f1θκxCyx+ fX��κxCyx+ ff1θκxCyx ]

Squaring (10) both sides, we get

(t−Y� )2_{= Y�}2_{[ X�κxe1+ X�}2_e 1 2_κ

x

2_{− f1θκx e1+ f1θkx}2_e

12+ f X�κxe1−f X�e12κx2+ ff1θκxe1−ff1θκx2e12+ e0− X�κxe0e1

− f1θκxe0e1+ fX�κxe0e1+ ff1θκx e0e1] 2

Using first order approximation, we have

= Y�2_{[ e0+ X�κ}

xe1−f1θκxe1+ fX�κxe1+ ff1θκxe1] 2 = Y�2_[e

0 2_{+ (X�κ}

xe1−f1θκxe1)2+ (fX�κxe1+ ff1θκxe1)2+ 2 e0(X�κxe1−f1θκxe1) + 2(X�κxe1−f1θκxe1)(fX�κxe1+ ff1θκxe1) + 2e0 (fX�κxe1+ ff1θκxe1) ]

On solving we get

= Y�2 _[e02_{+ X�}2_κ x

2_e12_{+ f1}2_θ2_κ x 2_−2X�κ

x

2_θe12_{+ f}2_X�2_κ x

2_e11_{+ f}2_f12_θ2_κ x

2_e12_{+ 2f}2_f1X�kx2_e12_{+ 2e0X�κxe1−2f1θκxe0e1} + 2X�2_fκ

x

2_e12_{+ 2ff1θX�κ} x

2_e12_−2f1fθκ x

2_e12_−2ff12_θ2_κ x

2_e12_{+ 2fX�κxe0e1+ 2ff1θκxe0e1]}

Taking expectation on both sides, we get MSE of proposed estimator t

E(t−Y� )2_{= λY�}2 _{[E (e} 0

2_{) + X�}2_κ x 2_{E (e}

1 2_{) + f}

12θ2κx2−2X�κx2θE (e12) + f2X�2κx2E (e12) + f2f12θ2κx2E(e12) + 2f2_f1X�kx 2_{E (e}

1

2_{) + 2X�κxE(e0e1)−2f1θκxE(e0e1) + 2X�}2_fκ x 2_E(e

12) + 2ff1θX�κx2E(e12)

−2f1fθκx2E(e12)−2ff12θ2κx2E(e12) + 2fX�κxE(e0e1) + 2ff1θκxE(e0e1)]

Substituting the values of E (e02), E (e12) and E(e0e1)

MSE (t) = λY�2_�Cy2_{+ X�}2_κ x

2 _Cx2 _{(1 + f}2_{) + f1}2_θ2_κ x

2_Cx2_{−2X �κ} x

2_θCx2_{+ f}2_f12_θ2_κ x

2_Cx2_{+ 2f}2_f1X�κ x

2_Cx2_{+ 2 X�κxCyx}

−2f1θκxCyx+ 2X��2fκx2Cx2+ 2ff1θX�κx2Cx2−2f1fθκx2Cx2−2ff12θ2κx2Cx2+ 2fX�κxCyx+ 2ff1θκxCyx�

MSE (t) = λY�2_�Cy2_{+ {X���}2_κ x

2 _{(1 + f}2_{) + f1}2_θ2_κ x 2_{−2X �κ}

x

2_{θ+ f}2_f12_θ2_κ x

2_{+ 2f}2_f1X�κ x

2 _{+ 2ff1θX�κ} x

2_{+ 2X��}2_fκ x

2_−2f1fθκ x 2

−2ff12θ2κx2}Cx2 + {2 X�κx−2f1θκx+ 2fX�κx+ 2ff1θκx}Cyx� (11)

6. THEORETICAL EFFICIENCY COMPARISON

In this section we have made a theoretical comparison of proposed estimator with Bahl and Tuteja [1991] estimator and the Singh et al. [2014] estimator and derive the optimal conditions under which proposed estimator performs better than these competing estimators.

i. Proposed estimator t will perform better than the Bahl and Tuteja [1991] estimator t0, iff

MSE (t) < 𝑀𝑆𝐸(𝑡0), from (11) and (4),

�Cy2+ {X���2κx2 (1 + f2) + f12θ2κx2−2X �κx2θ+ f2f12θ2κx2+ 2f2f1X�κx2 + 2ff1θX�κx2+ 2X��2fκx2 −2f1fθκx2 −2ff12θ2κx2}Cx2 + {2 X�κx−2f1θκx+ 2fX�κx+ 2ff1θκx}Cyx �< [𝐶𝑦

2₊𝐶𝑥2

4 − 𝐶𝑦𝑥]

ii. Proposed estimator t will perform better than the Singh et al.[2014] estimators, iff (a) MSE (t) < 𝑀𝑆𝐸(tRe), from (11) and (5),

�Cy2_{+ {X���}2_κ x

2 _{(1 + f}2_{) + f1}2_θ2_κ x 2_{−2X �κ}

x

2_{θ+ f}2_f12_θ2_κ x

2_{+ 2f}2_f1X�κ x

2 _{+ 2ff1θX�κ} x

2_{+ 2X��}2_fκ x

2_−2f1fθκ x 2

−2ff12_θ2_κ x

2_}Cx2 _{+ {2 X�κ}

x−2f1θκx+ 2fX�κx+ 2ff1θκx}Cyx�<𝜙 �Cy2+Cx 2

4 (1−4c)�

(b) MSE (t) < 𝑀𝑆𝐸�tpe�, from (11) and (6),

�Cy2_{+ {X���}2_κ x

2 _{(1 + f}2_{) + f1}2_θ2_κ x 2_{−2X �κ}

x 2_{θ+ f}2_f

12θ2κx2+ 2f2f1X�κx2 + 2ff1θX�κx2+ 2X��2fκx2−2f1fθκx2

−2ff12θ2κx2}Cx2 + {2 X�κx−2f1θκx+ 2fX�κx+ 2ff1θκx}Cyx�< 𝜙Y�2�Cy2+Cx 2

7. NUMERICAL ILLUSTRATION

To see the performance of the proposed estimator and to verify the conditions under which the proposed estimator performs better than mentioned estimators of the population mean in literature, the numerical example taken from Murthy [1967], pg. 228, in which

X= Fixed capital

Y= Output of 80 factories

Table-1: Values of parameters and constants computed from given population

Parameters/Constants Values

N 80

n 20

F 0.25

X

� 11.26

Y

� 51.82

κx 2.86

ρxy 0.94

Cy 0.35

Cx 0.75

Mx 10.30

𝜃

f1 0.33

𝜆

Φ 1.33

C 0.44

Cyx 0.25

Table-2: Mean squared Errors of existing and proposed estimators

S. No Estimators MSE

1 Bahl and Tuteja (1991) Estimator

₍

₎

0

t

MSE

= 2,70,383.86

2 Singh et al (2014) Estimator (i) MSE(tRe) = 9590029.95

(ii) MSE�tpe�= 9591799.89

3 Proposed Estimator (t) MSE(t) = 66, 982.32

Table-3: PRE of the proposed estimator with respect to existing estimators S. No Estimators PRE

1 t0 403.66

2 t_Re 14317.25

3 tpe 14319.89

8. CONCLUSIONS, DISCUSSIONS AND RECOMMENDATIONS

In the present article a modified ratio type predictive estimator has been proposed by using coefficient of kurtosis and median of auxiliary variable. Use of coefficient of kurtosis is significant in estimation process since it provides information about the nature of data. Median is also powerful statistical tool since it is unaffected by outliers present in the data. Table-1 describes the values of parameters and constants used for numerical study. From the table-2, it is seen that our proposed estimator has lesser MSE than that of competing estimators. Table-3 gives clear indication that proposed estimator is more efficient with respect to other existing estimators. Therefore proposed estimator is recommended to survey practitioners for estimation of population mean under predictive modelling approach especially in the situations where outliers are present in the data.

REFERENCES

1. Agrawal, M. C., Roy, D. C. (1999). Efficient estimators of population variance with regression-type and ratio-type predictor-inputs. Metron, 1999, 57(3), pp. 4–13

2. Basu, D. (1971). An Essay on the Logical Foundation of Survey Sampling, Part I. Foundations of Statistical Inference, eds. Godambe,V. P., Sprott, D. A. Toronto: Holt, Rinehart and Winston, pp. 203–242.

4. Cochran, W. G. (1994). Sampling Techniques. Third Edition, Wiley Eastern limted.

5. Cochran, W. G. (1940). The Estimation of the Yields of the Cereal Experiments by Sampling for the Ratio of Grain to Total Produce. The Journal of Agric. Science, 30, 262-275.

6. Hirano, K., Pandey, B.N., Singh, J. (1973). On the Utilization of a Known Coefficient of Kurtosis in the Estimation Procedure of Variance. Annals of the Institute of Statistical Mathematics, 25, 51-55.

7. Kadilar, G.O. (2016). A New Exponential Type Estimator for the Population Mean in Simple Random Sampling, Journal of Modern Applied Statistical Methods, 15, 2, 207-214.

8. Kadilar, C., Cingi, H. (2004). Ratio estimators in simple random sampling, Applied Mathematics and Computation, 151, 893–902.

9. Kadilar, C., Cingi, H. (2006a). An improvement in estimating the population mean by using the correlation co-efficient, Hacettepe Journal of Mathematics and Statistics, 35 (1), 103–109.

10. Kadilar, C., Cingi, H. (2006b). Improvement in estimating the population mean in simple random sampling, Applied Mathematics Letters, 19, 75–79.

11. Kadilar, C., Cingi, H. (2009). Advances in sampling theory - Ratio method of estimation, Bentham Science Publishers.

12. Khan, M. (2016). A ratio chain-type exponential estimator for finite population mean using double sampling, Springer Plus, DOI 10.1186/s40064-016-1717-4.

13. Koyuncu, N., Kadilar, C. (2009). Efficient estimators for the population mean, Hacettepe Journal of Mathematics and Statistics, 38(2), 217–225.

14. Milton, T.K., Odhiambo, R.O. Orwa, G.O. (2017). Estimation of Population Variance Using the Coefficient of Kurtosis and Median of an Auxiliary Variable under Simple Random Sampling. Open Journal of Statistics, 7, 944-955.

15. Misra, S., Singh, R. K, Shukla, A.K. (2013) Partial Predictive Estimation of Finite Population Variance Using Product Type Estimator” International Journal of Statistics and Analysis (IJSA) Number 2 (2013) pp. 211-219. 16. Murthy, M. N. (1967). Sampling theory and methods. Calcutta Statistical Publishing House, India.

17. Mukhopadhyay, P. (2005). Theory and methods of survey sampling, PHI Learning, 2nd edition, New Delhi. 18. Nayak, R., Sahoo, L (2012). Some alternative predictive estimators of population variance. Revista

Colombiana de Estadistica, 2012, 35(3), pp. 509–521.

19. Sahoo, L. N., Das, B. C., Sahoo, J. A (2009) Class of Predictive Estimators in Two stage sampling. Journal of the Indian Society of Agricultural Statistics, 63(2), pp. 175–180.

20. Sahoo, L. N., Panda P. (1999). A predictive regression-type estimator in two-stage sampling. Journal of the Indian Society of Agricultural Statistics, 52(3), pp. 303–308.

21. Sahoo, L. N., Sahoo, R. K. (2001). Predictive estimation of finite population mean in two phase sampling using two auxiliary variables. Journal of the Indian Society of Agricultural Statistics, 54(2), pp. 258–264. 22. Saini, M (2013). A class of predictive estimators in two-stage sampling when auxiliary character is estimated

at SSU level. International Journal of Pure and Applied Mathematics, 85(2), pp. 285–295.

23. Searls, D.T. (1964). The Utilization of a Known Coefficient of Variation in the Estimation Procedure. Journal of the American Statistical Association, 59, 1225-1226.

24. Sen, A.R. (1978). Estimation of the Population Mean When the Coefficient of variation is Known. Communications in Statistics—Theory and Methods, 7, 657-672.

25. Sisodia, B., Dwivedi, V. (1981) Modified Ratio Estimator Using Coefficient of Variation of Auxiliary Variable. Journal-Indian Society of Agricultural Statistics, 33, 13-18.

26. Upadhyaya, L.N., Singh, H.P. (1984) On the Estimation of the Population Mean With Known Coefficient of Variation. Biometrical Journal , 26, 915-922.

27. Singh, G.N. (2003). On the improvement of product method of estimation in sample surveys, Journal of Indian Society of Agricultural Statistics, 56 (3), 267–265.

28. Singh, D., Chaudhary, F. S. (1986). Theory and analysis of sample survey designs, New Age International Publisher, New Delhi.

29. Singh, H. P., Tailor, R. (2003). Use of known correlation coefficient in estimating the finite population means, Statistics in Transition, 6 (4), 555–560.

30. Singh, H. P., Tailor, R. (2005). Estimation of finite population mean with known co-efficient of variation of an auxiliary variable, Statistica, anno LXV, 3, 301–313.

31. Singh, H. P., Solanki, R. S., Singh, A. K. (2014). Predictive Estimation of Finite Population Mean Using Exponential Estimators. Statistika: Statistics and Economy Journal, 94 (1), pp. 41–53.

32. Srivastava, S. K. (1983). Predictive estimation of finite population mean using product estimator. Metrika, 30, pp. 93–99.

33. Yadav, S. K., Mishra, S. S (2015). Developing Improved Predictive Estimator for Finite Population Mean Using Auxiliary Information, Statistika, 95(1).

Source of support: Nil, Conflict of interest: None Declared.