THE EFFICIENT CLASSES OF ESTIMATORS FOR THE PRODUCT OF TWO POPULATION MEANS IN THE EXISTENCE OF NON-RESPONSE UNDER THE STRATIFIED POPULATION-A SIMULATION STUDY Текст научной статьи по специальности «Фундаментальная медицина»

THE EFFICIENT CLASSES OF ESTIMATORS FOR THE PRODUCT OF TWO POPULATION MEANS IN THE

EXISTENCE OF NON-RESPONSE UNDER THE STRATIFIED POPULATION-A SIMULATION STUDY

Manish Mishra, B. B. Khare & Sachin Singh

•

School of Liberal Studies, UPES, Dehradun-248001, India.

Department of Statistics, Banaras Hindu University, Varanasi-221005, India.

Department of Mathematics, Sharda University, Greater Noida-201306, India. manish.mishra@ddn.upes.ac.in, bbkhare56@yahoo.com, singhat619@gmail.com

Abstract

This paper focuses on estimating the product of two population means. Within this paper, we have introduced three distinct classes of estimators for product of two population means. These estimators take into account the known population mean of an auxiliary variable under the framework of stratified random sampling and the presence of non-response in the study variable. Basically, for case (I) we assume the non-response on the study variable and utilize the auxiliary information corresponding to the responding units of the study variable and in case (II), we utilize the complete dataset from the auxiliary variable while also accounting for non-response in the study variable. In case (III) we combined both the information of the auxiliary variable and assumed the non-response on the study variable. Expressions for bias and mean square error have been derived, extending up to the first-order derivative. We have also pinpointed some specific members of the proposed estimator. We have conducted a simulation study to evaluate the valuable insights into the performance of the suggested classes of estimators with the conventional estimator.

Keywords: Product of two population means, Auxiliary variable, Stratified Sampling, Non-response, Mean square error

1. Introduction

The product of two population means is a common parameter used in various areas, including agriculture, economics, social sciences, public health and other scientific investigations. The estimation problem of the product of two population means is very useful in practice. For instance, suppose we want to estimate the total production in a farm having N plants then we select some plants and observe the number of pods and the seeds in pods in a plant then from selected units we will calculate the average number of pods/plant and average number of seeds/plant. Multiplying the average number of seeds/plant and the number of pods/plant with the total number of plants we get the total number of seeds.

In the case of finite population utilizing the auxiliary information(s), the estimation problems related to the product of population means have been examined by several researchers such as Singh[14], Singh[15], Singh[16], Singh[17], Ray and Singh[11], Srivastava et al.[20] and Khare[3] Kumar and Srivastava [8].

Any researcher or statistician may typically encounter the phenomena of the non-response in a scientific investigation or in a sample survey. The reasons for the occurrence of the non-response

are the respondent's unwillingness to respond to some questions that are sentimental, the person not at home, Lack of enthusiasm, etc.

To cope with the complications of non-response, Hansen and Hurwitz [2] have put forward a method of sub-sampling from non-responding units. Using information from responding and sub-sampling units picked from non-responding units in the sample, he suggested a population mean estimator.

In the existence of non-response, the estimation of the product of population means utilizing auxiliary information has been considered by Khare and Sinha[4, 5] and Khare et al.[6].

For stratified populations, Khare and Jha[7], Singh et al.[18] and Mishra et al.[9] have suggested precise estimators of population parameters in the situation of non-response on study character using known and unknown population mean of auxiliary variable.

Motivated by Khare and Jha[7], Three classes of estimators are proposed to estimate the product of two population means under the stratified random sampling with the known population mean of auxiliary character having the non-response on study character. The expressions for bias and mean square error (MSE) of the suggested estimators have been obtained. For the numerical study, the data generated by simulation using R-software has also been given to validate the supremacy of the suggested estimators.

2. Notations and Sampling procedure:

In the study of the population n : (n1, n2, ^-^N) of the size N. The population is divided into L strata. We denote yh (h = 1,2) as study characters of the population having population means Y^ (h = 1,2) in the ¿th stratum, (i = 1,2,3..L) and for the auxiliary character (x), the population mean is denoted by X, which is known for each stratum. For each stratum, the stratum proportion and size is also known. Let N;1 and N;(2) are the number of units belonging to the responding and

non-responding part of the ¿th stratum such that N;1 + N;(2) = N;. For the ¿th stratum, W;1 = ^

and Wi2 = —N(2- represent the response and non-response rates.

Here, we are considering the problem in stratified population using non-response in each stratum, the estimation of P (= Y1Y2) for known X has been considered. The sampling procedure we use for the study is as:

we select U, units from ¿th stratum, u,1 units are selected from N¿1 units using simple random sampling without replacement (SRSWOR) method of sampling, Here, we obtain u;1 units as respondent and U;(2) units as non-responding units out of U; selected units. We select r; units,

(r, = t2-, k > 1) from Uj(2) units of the ¿th stratum.

Using the information available on (u;1 + r;) units, the estimator yh;, (h = 1,2) is given by Hansen and Hurwitz[2] method as follows:

v* = U;1 y + U;(2) y (1)

where, ^¿(1) denotes the sample means of Vh f°r U;1 responding units in the ¿th stratum and Vhi(2) is the sample mean of r; units drawn from non-responding units (u;(2)) in the ¿th stratum. The estimator yh (h = 1,2) is unbiased and has variance given by:

VyM — (1 - /') c*2 , (ki - 1) S*2 ns

V (yhi) = U Syhi + U Syhi(2) (2)

where, Syh, and Syh;(2) are the population mean square for u;1 responding units and non-

responding units of ¿th stratum of the population.

The Hansen and Hurwitz [2] estimator for the auxiliary variable x, we have

r* = Ui1 r + Ui(2) x (3)

where, xi(1) and x,(2) are the sample means of x for ni1 responding units and based on r units sub-sampled from n^) non-responding units of the ith stratum.

In the presence of non-response, the stratified sample means for Yh (h = 1,2) and X are given as follows:

L L

yhst = E Wifhi and x*st = E Wx

(4)

i=1

i=1

In each stratum utilizing the information on n units, the stratified sample mean to estimate X is given by:

L

xst = E Wix (5)

i=1

where, W = N.

3. Proposed Classes of Estimators:

Let Pst = y1sty2st denotes an estimator for P using stratified random sampling in the existence of non-response. We propose three different estimators for P under different situations of non-response which are given as follows:

Case (A): Utilizing Incomplete information on yh, (h = 1,2) and corresponding values of x when X is known.

11 = d(i)(Pst, mi ) (6)

Case (B): Incomplete information on yh, (h = 1,2) and complete information on x for the known X.

12 = d(2) (Pst, m2 ) (7)

Case (C): Utilizing Incomplete information on yh, (h = 1,2) and corresponding values of x and complete information on auxiliary variable.

¡3 = h(Pst, mi, m.2 )

(8)

such that,

d{j)(P,1) = P, d1(j)(P,1) = 1, h(P, 1,1) = P and h1 (P,1,1) = 1 where,

d1(j)(P,1) = iJKtd(J)(Pst,m' , d2(j)(PA) = ш-^)^, m)

(P,1)

h1 (P,1,1)= h(Pst, m1, m2 ) ldPst

_ с

hs (P,1,1) = Ymh^' m\, mi )

, hi (P,1,1)= -— h(Pst, m1, mi)

(P,1,1) Umi

(P,1,1)

(P,1,1)

for j=1,2.

(9)

We denote, Pst = ylsty*2st, mi = ^t and m2 = X.

We assume that the function d(1)(P^st,m1), d(2)(Pst,m2) and h(Pst,m1,m2) are fulfilling the regularity conditions which are given as follows:

• For any sample chosen from any design, the selected functions dj (Pst, mj), and h(Pst, m1, m2) take the values in two-dimensional and three-dimensional real space G1 and G2 having the points (P,1) and (P,1,1).

• The partial derivatives of the first-order and second-order with respect to P, m1 and m2 for the functions d(j)(Pst,mj), and h(Pst,m1,m2) are continuous and bounded in G1 and G2.

(10)

Using the regularity conditions (10) and expanding the function dy (Pst, my), for j = 1,2 and h(Pst, m1, m2) about the point (P, 1) and (P, 1,1) respectively by utilizing Taylor's series up to the second-order partial derivatives, we get

1 r

h = d(j)(01) + (Pst - P)d1O-)(01) + (my - 1)d2(j)(01) + - |_(Pst - P)2d11(j) (01) + (my - 1)2d22a)(01) + 2(Pst - P)(my - 1)d12a)(01)] (11)

l3 = h(02) + (Prf - P)h1(02) + (m1 - 1)h2(02) + (m2 - 1)h3(02) + 1 r

2 [(Pst - P)2hu(0*) + (m1 - 1)2h22(02) + (m2 - 1)2h^) + 2(Pst - P)(m1 - 1)h12(02) + 2(Pst - P)(m2 - 1)hu(0*)

+ 2(m1 - 1)(m2 - 1)h23(02*)] (12)

Where, 01 = (P, 1), 01 = (Ps*t, m**), 02 = (P, 1,1), 02* = (Ps*t, m1, m1), P * = P + a1 (Pst - P) and m* = 1 + a2 (my - 1), 0 < ay > 1 for j = 1,2.

Here, d1(y)(01) and d11(y) (01) are the first and second-order partial derivatives with respect to Pst and d2(y)(01) and d22(y)(01) are the first and second-order partial derivatives with respect to my for the function d(y)(01). Similarly h1 (02), h2(02) and h3(02) are the first-order partial derivatives with respect to Pst, m1 and m2 respectively and using the condition given in equation (9) and presume that the second-order derivatives are very small in equations (11) and (12), the expression for ly and I3 are given as follows:

ly = Pst + (my - 1)d2(y)(01) + P-1(Pst - P)(my - 1)d2(y)(01) (13)

l3 = Pst + (m1 - 1)h2(02) + (m2 - 1)h3(02) (14)

For the function d(y)(Pst, my), for y = 1,2 and h(Pst, m1, m2), under the conditions given in equation (10) the Bias(1y), bias(l3), MSE(1y) and MSE()13) are always exist.

4. Properties of the Proposed Classes of Estimators Utilizing the large sample approximation, we assume that:

y1st = Y1 (1 + £0),V2st = Y2(1 + 61),x**t = X(1 + £2) and Xst = X(1 + 63) (15)

Such that £ | < 1 and E(e;) = 0 V i = 0,1,2,3.

We also have,

e(«8 ) = Vf1 = Y2 £ + ^ W^'

^) = = Y"£ {f + ^Wi2Sj2i(2) E(%2) = £(^3) = = ^ £ iwfd,

X X i=i

£<**) = VStl = X £ if + ^ WaS* J,

X X i=i I "i )

£(pp ) = Cov(KstKst) = 1 £W2i fS . (ki -1) WS E(£°ei) = -YlY- = YY2 £ Wi IfiSyl''y2i + m Wi2Syiiy2i(2)

)= Cov(yist^st) = 1 £W2i fS . (k - 1) WS

£(e°62) = —YX— = YX £ Wi \fiSyi;*; + m Wi2V*i(2)

^) = ^fx51 = fix £ W{ f.s,,i + ikin-li W.2 V, (2) E<- ) = * = yk£ W2{ >'S-},

E(W3)= ^ = f1_£ w?{f,Si2,i) (16)

Where, (S^., Sj^.) represent the population mean square error of (y1, y2) for ith stratum and

(S^ <2), S^ <2)) are the population mean square of r. units sub-sampled from ni<2) units of the

study variable for i'th stratum and (SX;, SX;(2)) represent the population mean square error of x for

i'th stratum and r. units sub-sampled from ni<2) for i'th stratum respectively.

(Syiiy2i, SyiiXi, Sy2iXi) and (Syi.y2i(2), Syi,.x,.(2), Sy2iXi(2)) are the covariance between ^y^,., ^x,., Sy2ix,.) and (Syi.y2.<2), SyiiX.<2), Sy2.x.<2))stratum of the population and r. units sub sampled from

n.(2) units of the population for the i'th stratum respectively and f = ^n — jr).

The Bias and MSE of /y and /3 by using the equation (8i) and (82) respectively up to the n—i terms of order are given as below:

Bias(/y) = £(/y - P) = E [Psf + (my - i)^)^) + P—i(Psf - P)(my - i)d2O-)(0i) - P

= £(Psf - P) + P-1E(Psf - P)(my - i)d2<y)<0i) (i7)

MS£(/y) = E(1y - P)2 = E(Psf - P)2 + E(my - i)2d2(y)(0i)

+ 2E(Psf - P)(my - i)d2O-)(0i) (i8)

Bias(/3) = E(l3 - P) = £[Pst + (mi - i)h2 (02) + (m2 - i)hs (02) - P

= £(Pst - P) (i9)

MS£(/a) = £(/3 - P)2 = £(Psf - P)2 + E(mi - i)2h2^) + £(«2 - i)2h2(02) - 2£(Pst - P)(mi - i)h2<02) + 2£(Pst - P)(m2 - i)^) + 2£(mi - i)(m2 - i)h20M02) (2°)

The equation (86) can be written as:

MSE(l3) = E(l3 - P)2 = MSE(Pst) + Ah2(02) + Bh3(02) + 2Ch2(02) + 2Dh3(02)+ 2Eh2 (02 )h3(02)

(21)

Where, MSE(Pst) = E(Pst - P)2, A = E(m1 - 1)2, B = E(m2 - 1)2, C = E(Pst - P)(m1 - 1), D = E(Pst - P)(m2 - 1) and E = E(m1 - 1)(m2 - 1).

Now, to get the optimum value of <2(1) (01), <^(2) (01), h2(02) and h3(02) We partially differentiate the expression (18) w.r.t. <2(1)(01), <2(2)(01) and equation (21) w.r.t h2(02) and h3(02) and equating to zero. Assuming that the partial derivatives of order second are positive, we get

d2(1) (01) (opt)

X

vXt)

Y2Cov(УÍst, xtt) + Y1Cov(У*2st, x*st)

(22)

X

<2(2)(01 )(oVt) = - Vxst)

Y2Cov(У\st, xst) + Y1Cov(У2st, xst)

(23)

) = (DE -BC) h2(02)oPt = (AB - E2)

(24)

h3(02)opt

(CE - AD) (AB - E2)

(25)

The minimum mean square errors after putting the optimum values of <2(1) (01), <2(2) (01), h2(02) and h3(02) in equation (18) and (21) are given by:

MSE(l1 )mm = MSE(pst) -

V(xs t)

Cov(y1st, x*st) + Cov(yy2st, xst) Y1 Y2

(26)

MSE(l2)min = MSE(pst) - vx^y

Cov(yist, xst) + Cov(y2st, xst)

Y1

Y2

(27)

MSE(l3)min = MSE(Pst) -

BC2 + AD2 - 2CDE AB — E2

(28)

Members of the proposed classes l\, l2 and l3:

For the given condition in (9), any parametric function d(1)(i'st, m\), d(2)(Pst, m2) and h(Pst, m2) can produce a class of asymptotic estimators. Such types of estimators have a very large number of classes. Some of the members are given below for the proposed classes of estimators:

l,1 = Pst (Ax + (1 - Ax )mj), lj2 = (PPst + «1(m, - 1))mf

lj3 = Pst(mjr, lj4 = Pst(2 - ma3), lj5 = Pst exp

-my - 1 Lmj +1

l31 = PPst(71 + m172 + (1 - 71 - 72)m2),l32 = P^st exp[«4(m1 - 1)

1

<-2 , l32 = Pst ^

+ «5(m2 - 1)], l33 = Pstm16m^7, /32 = Pst 2 [1 + «8m73 + (1 - «8)m274]

(29)

The MSEs of the suggested classes of the estimators will attain the minimum value of the MSE given in equation (26), (27), (28) for the optimum value <2(1) (01), <2(2) (01), h2(02) and h3(02) given in equation (22), (23), (24) and (25). The member of the proposed classes of the estimators l\, l2 and l3 given in equation (29) will also attain the same minimum MSE. The optimum values are occasionally in the form of some unknown parameters and occasionally in the form of the value of unknown constants these values can be obtained from past data (Reddy [13]) or can be estimated by sample values that do not affect the minimum MSE of the estimator up to the term of order (n) (Srivastva and Jhajj [19]).

2

2

P

2

2

Manish Mishra and B.B. Khare OT^ No 2 (78) PRODUCT OF TWO POPULATION MEANS_Volume 19, June, 2024

5. Comparison of (l\, l2 and l3) with Pertinent Estimators [Pst]

In stratified random sampling, the estimator for P in the case of non-response is defined as:

Pst = y1sty2st (30)

The MSE of the Pst is given as:

MSE(P ) = P2

V (VU ) + V (y2st ) + 2 CoviVlsU tist ) y2 Y2 YJYI

(31)

6. Simulation Study:

To evaluate the characteristics of suggested classes of estimators, we perform a simulation study to artificially generate the population using normal distribution. For this study, we generate a population of size 4500. The following table (1) shows the distributions of the population generated to perform the study.

Table 1: Sample Size and Distribution

Strata No. Stratum Size(Nj) Sample Size (ni) Distribution of yih. ~ a2) Distribution of y2h ~ a2) Distribution of Xh ~ a2)

1 1800 600 N(500,81) N(900,121)i N(600,100)i

2 1200 400 N(300,64) N(500,100) N(300,121)

3 900 300 N(400,100) N(400,64) N(500,81)

4 600 200 N(500,121) N(300,81) N(300,64)

Here, we employ some transformations suggested by Reddy et al.[12]. To generate the population for study variables and auxiliary variable with some association. The following table (2) shows the required transformation and correlation to generate the random variables. Now, to estimate the approximate mean square error of the suggested classes of the estimators and estimator Pst. We average the outcomes after 6000 iterations of the loop.

To calculate the approximate mean square error (AMSE) the formula is given as follows:

1 6000

AMSE(m1 ) = 6000 £ (m- P)2

where, m^ = Pst, l-y l2 and l3.

Table 2: Transformation and correlation:

Strata no. cor(V1hi ' xhi ) cor(V1h; ' xhi ) Transformed auxiliary variable xh using yih. Transformed auxiliary variable xh, using y2h

I II III IV -0.58 -0.62 -0.70 -0.65 -0.60 -0.72 -0.58 -0.65 x11i = ry\xi yili + x1i V1 - ryj xj x12i = ryiX2yi2i + x2i-J1 - ry1 x2 x13i = ryiX3yi3i + x3i-J1 - r21 x3 x14i = ryiX4yi4i + x4i\j1 - r21 x4 x21i = rV2 X1 y21i + x1i V1 - ry2 x1 x22i = ry2 x2 y22i + X2i*J1 - ry2 x2 x23i = ry2X3y23i + X3i-J1 - ry2x3 x24i = ry2 X4 y24i + x4i\/1 - ry2 x4

Table 3: Percentage re/ative efficiency (PR£) of the proposed estimator with respect to relevant estimators.

Rate of Non-response Estimators i/k

i/2 i/3 i/4

i°% Pst i00(399i.83) i00(4484.76) i00(5022.54)

/i ii7.97(3383.59) ii8.25(3792.38) ii8.49(4238.5i)

/2 ii5.26(3463.26) ii3.39(3954.87) iii.83(449i.2i)

/3 ii8.22(3376.57) ii8.80(3774.79) ii9.45(4204.63)

20% Pst i00(4509.60) i00(5508.47) i00(66i6.83)

/i ii8.06(38i9.70) ii8.i0(4664.08) ii8.08(5603.45)

/2 ii3.2i(3983.25) ii0.62(4979.58) i08.69(6087.46)

/3 ii8.26(38i3.2i) ii8.40(4652.04) ii5.57(5580.08)

30% Pst i00(5034.i8) i00(6578.39) i00(8i46.68)

/i ii7.98(4266.66) ii8.24(5563.5i) ii8.54(6872.50)

/2 iii.72(4505.72) i08.73(6049.82) i06.93(76i8.i8)

/3 ii8.i4(426i.ii) ii8.47(5552.53) ii8.85(6854.30)

(Note: The AMSE of the estimators are shown in parenthesis.)

Table (3) depicts the AMS£ of the estimators Pst, /i, /2 and /3 for different values of k and non-response rates.

From the above Table (3) we can see that as we increase the value of k and non-response rates from i°% to 3°% the approximate means square error of the estimators Pst, /i, /2 and /3 increases.

7. Conclusion

The proposed estimators /i, /2 and /3 are found to be more efficient than the estimator Pst . And it is also observed that from Table (3) the proposed classes of estimators /i and /3 are found to be almost equally efficient for the different choices of sub-sampling fraction and non-response rates.

The AMS£(/i) less than or greater than AMS£(/2) depending upon the situation of correlation between study variables yi, y2 and auxiliary variable x. Rao[i°] has shown the situation when the conventional estimator using X* and alternate estimator using X will be less than each other depending upon the situations given by him. This theory also works here in the case of /i, /2 and /3.

Here, some times AMS£(/i) < AMS£(/2) then AMS£(/3) will be less than AMS£(/2) and almost equal or less than AMS£(/i). But if AMS£(/i) > AMS£(/2) then AMS£(/3) will be less than AMS£(/i) and almost equal to the AMS£(/2). So the use of AMS£(/3) is advisable if the condition when AMS£(/i) < AMS£(/2) or AMS£(/i) > AMS£(/2) is unknown.

Hence, for the estimation of P under the stratified random sampling in the existence of non-response for the Known X. We suggest to use the estimators /i, /2 and /3 depending on the situations discussed in the results.

References

[1] El-Badry, M. A. (i956).A sampling procedure for mailed questionnaires. Journa/ of the American Statistica/ Association , 5i(274), 209-227.

[2] Hansen, M. H., Hurwitz, W. N. (i946). The problem of non-response in sample surveys. Journa/ of the American Statistica/ Association, 4i(236), 5i7-529.

[3] Khare, B. B. (i992). On class of estimators for product of two population means using multi-auxiliary characters with known and unknown means, .Jour. App/. Stat, i, 56-67.

[4] Khare, B. B., and R. R. Sinha(20i0). On class of estimators for the product of two population means using auxiliary character in presence of non-response."Inter. Trans. App/. Sci 2.4 84i-846.

[5] Khare, B. B., & Sinha, R. R. (20i2). Combined class of estimators for ratio and product of two population means in presence of non-response. Internationa/ Journa/ of Statistics and £conomics, 8(i2), i2-20.

Khare, B. B., Jha, P. S., & Kumar, K. (2014). Improved generalized chain estimators for ratio and product of two population means using two auxiliary characters in the presence of non-response. International J. Stats & Economics, 13(1), 108-121.

Khare, B. B., Jha, P. S. (2017). Classes of estimators for population mean using auxiliary variable in stratified population in the presence of non-response. Communications in Statistics-Theory and Methods, 46(13), 6579-6589.

Kumar, K. Srivastava, U. (2018). Estimation of ratio and product of two population means using exponential type estimators in sample surveys. International Journal of Mathematics and Statistics, 19"3, pp. 102"109.

Mishra, M., Khare, B. B., Singh, S. (2023). TWO-CLASSES FOR REGRESSION TYPE OF ESTIMATORS FOR THE RATIO OF TWO POPULATION MEANS IN TWO-PHASE SAMPLING IN THE PRESENCE OF NON-RESPONSE FOR STRATIFIED POPULATION. Reliability: Theory Applications, 18(3 (74)), 610-620.

Rao, P. S. R. S. (1986). Ratio estimation with sub sampling the non-respondents. Survey methodology, 12(2), 217-230.

Ray, S. K., & Singh, R. K. (1985).Some estimators for the ratio and product of population parameters. Journal of the Indian Society of Agricultural Statistics, 37(1), 1-10.

Reddy, M. K., Rao, K. R., & Boiroju, N. K. (2010). Comparison of ratio estimators using Monte Carlo simulation. International Journal of Agriculture and Statistical Sciences, 6(2), 517-527. Reddy, V. N. (1978). A study on the use of prior knowledge on certain population parameters in estimation. Sankhya C, 40, 29-37.

Singh, M. P. (1965). On the estimation of ratio and product of the population parameters. Sankhy': The Indian Journal of Statistics, Series B, 321-328.

Singh, M. P. (1967). cum product method of estimation.Metrika, 12(1), 34-42.

Singh, M. P. (1969). of some ratio-cum-product estimators. Sankhy': The Indian Journal of Statistics, Series B, 375-378.

Singh, R.K. (1982). the estimating ratio and product of population parameters. . Stat. Assoc. Bull, 31, 69-76.

Singh, R. , Khare, S. , Khare, B.B. Jha , P.S. (2020). The general classes of estimators for population mean under stratified two phase random sampling in the presence of non-response. Int. J. Agricult. Stat. Sci., Vol. 16, No. 2, pp. 557-565, 2020.

Srivastava, S. K., Jhajj, H. S. (1983). A class of estimators of the population mean using multiauxiliary information. Calcutta Statistical Association Bulletin, 32(1-2), 47-56.

Srivastava, S. R., Khare, B. B., & Srivastava, S. R. (1988).On generalized chain estimator for ratio and product of two population means using auxiliary characters. Assam Statistical Review, 2(1), 21-29.