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On efficient estimation of population mean under non-response
Communications for Statistical Applications and Methods 2019;26:11-25
Published online January 31, 2019
© 2019 Korean Statistical Society.

Shashi Bhushana, and Abhay Pratap Pandey1,b

aDepartment of Mathematics and Statistics, D. S. M. N. R. University, Lucknow, India, bRamanujan College, University of Delhi, India
Correspondence to: 1Abhay Pratap Pandey, Ramanujan College, University of Delhi, Kalkaji New Delhi-110019, India. E-mail: abhaypratap.pandey@gmail.com
Received June 5, 2018; Revised October 12, 2018; Accepted November 25, 2018.
 Abstract

The present paper utilizes auxiliary information to neutralize the effect of non-response for estimating the population mean. Improved ratio type estimators for population mean have been proposed and their properties are studied. These estimators are suggested for both single phase sampling and two phase sampling in presence of non-response. Empirical studies are conducted to validate the theoretical results and demonstrate the performance of the proposed estimators. The proposed estimators are shown to perform better than those used by Cochran (Sampling Techniques (3rd ed), John Wiley & Sons, 1977), Khare and Srivastava (In Proceedings-National Academy Science, India, Section A, 65, 195–203, 1995), Rao (Randomization Approach in Incomplete Data in Sample Surveys, Academic Press, 1983; Survey Methodology12, 217–230, 1986), and Singh and Kumar (Australian & New Zealand Journal of Statistics, 50, 395–408, 2008; Statistical Papers, 51, 559–582, 2010) under the derived optimality condition. Suitable recommendations are put forward for survey practitioners.

Keywords : auxiliary variable, non-response, ratio type estimator, mean square error, efficiency
1. Introduction

The problem of non-response is inevitable in most sample surveys because the information cannot be obtained from all units selected in the survey due to various reasons. An estimator based on such incomplete information is biased and the final outcome may be misleading, when the respondents differ from non-respondents. In their seminal paper Hansen and Hurwitz (1946) considered a technique of sub-sampling the non-respondents in order to adjust for the non-response bias in a mail survey.

In sampling theory, it is well known that the efficiency of the estimators of unknown population parameters of the study variable can be increased by suitably using known information on an auxiliary variable. The ratio, product and regression methods of estimation are good examples in this context. Non-response adversely affects the estimate of population mean and population variance; in addition, many authors have suggested a number of estimators to estimate population parameter and their variance under the non-response for various situations. Cochran (1977) and Rao (1986) suggested a ratio method to estimate the population mean of the study variate y with sub-sampling from non-respondents.

Khare and Srivastava (1995) suggested an estimation procedure of the population mean using an auxiliary character in the presence of non-response, Khare and Srivastava (1995) proposed the studying of a conventional and alternative two phase sampling ratio, product and regression estimators in the presence of non-response. Khare and Srivastava (1997) proposed transformed ratio type estimators for the population mean in the presence of non-response. Okafor and Lee (2000) proposed a double sampling scheme for ratio and regression estimation with sub sampling; in addition, the non-respondent also deal with the non-response problem. Khare and Srivastava (1993). Singh and Kumar (2008) proposed a general class of population mean estimators in survey sampling using auxiliary information with sub sampling for the non-respondent. Singh et al. (2010) also suggested a number of estimators to estimate population mean under non-response. Khare and Kumar (2011) proposed a method to estimate the population mean using known coefficient of variation of the study character in the presence of non-response. Singh and Bhushan (2012) proposed a generalized classes of two phase sampling estimators of population mean in presence of non-response. Shabbir and Khan (2013) also suggested a number of estimators to estimate the finite population mean using two auxiliary variables in two phase sampling in the presence of non-response. Sunil Kumar (Kumar, 2015) suggested an efficient use of auxiliary information in estimating the population ratio, product and mean in the presence of non-response.

An interesting finding of all these papers was that the regression (difference) estimators were found to be best in terms of mean squared error (MSE); in addition, any ratio type estimator can at best attain the MSE of these regression (difference) estimators. In this paper, we have proposed some improvement over regression as well as ratio estimators proposed by various authors in their earlier works.

2. Notations and existing results

Hansen and Hurwitz (1946) considered mail surveys in the first attempt-, and personal interviews in the second attempt. In the Hansen & Hurwitz method, the population of size of N is supposed to be composed of two strata, namely respondents and non-respondents; having sizes N1 and N2 (= N –N1). Thus we label the data as y1, . . . , yN1 for the response group, and as yN1+1, . . . , yN1+N2 for the non-response group. Let Y¯=i=1Nyi/N and Sy2=i=1N(yi-Y¯)2/(N-1) denote the population mean and variance, respectively. Let Y¯1=i=1N1yi/N1 and Sy12=i=1N1(yi-Y¯1)2/(N1-1) denote the mean and variance of the response group, respectively. Similarly, let Y¯2=i=N1N1+N2yi/N2 and Sy22=i=N1+1N1+N2(yi-Y¯2)2/(N2-1) denote the mean and variance of the non-response group, respectively. The population mean can be written as = W11 + W22, W1 = N1/N, and W2 = N2/N. Let us consider a random sample of size n are drawn by using simple random sampling without replacement (SRSWOR). The random sample n should be made of two strata, namely n1 respondents and (nn1) non-respondents. The sample mean y¯1=i=1n1yi/n1 is unbiased for 1, but has a bias equal to W1(12) in estimating the population mean . The sample mean y¯2r=i=1ryi/r is unbiased for the mean 2 of the n2 units. An unbiased estimator for the population mean is

y¯*=w1y¯1+w2y¯2,

where w1 = n1/n and w2 = n2/n. The variance of * is given by

Var(y¯*)=PSy2+QSy22,

where P = (1 – f )/n, Q = W2(k – 1)/n, and f = n/N.

Let xi (i = 1, 2, . . . , N) denote an auxiliary variate correlated with study variate yi (i = 1, 2, . . . , N). The population mean of the auxiliary variable x is X¯=i=1Nxi/N. Let 1 and 2 denote the means of the response and non-response groups. Let χ̄ denote the mean of all the n units. Let χ̄1 and χ̄2 denote the means of the n1 responding units and the n2 non-responding units. Further let x¯2r=i=1rxi/r denote the mean of the subsampled units. The population variances of x and y are denoted by Sx2 and Sy2, and the population covariance by S xy. The population correlation coefficient is ρ=Sxy/Sx2Sy2. The unbiased estimator of the population mean of the auxiliary variable x is

x¯*=w1x¯1+w2x¯2.

The variance of χ̄* is given by

Var(x¯*)=PSx2+QSx22,

where Sx22=i=N1+1N1+N2(xi-X¯2)2/(N2-1).

Similarly, In two phase sampling, we have n′ observations on x from the first-phase sample, n1 observations on y from the responding units of the n second-phase sample units, and r observations on y from the subsample units selected from the n2 non response units of the second-phase sample. Let χ̄′ be the sample mean of auxiliary variable x based on the first-phase sample. Using the information on the auxiliary variable x collected from the first-phase sample, Cochran (1977), Khare and Srivastava (1995), Rao (1986), Okafor and Lee (2000), and Singh and Kumar (2008, 2010) suggested ratio and regression type estimators under non-response. We have classified these estimators into seven different strategies depending upon the available auxiliary information under both single phase sampling and two phase sampling.

Strategy I

When *, χ̄*, and are used. If the auxiliary variable is known, the non-response occurs on the study variable y and information on the auxiliary variable x is not available from all the sample units along the population mean . The ratio and regression type estimators are

tr1=y¯*(X¯x¯*),tR1=y¯*(X¯x¯*)β1,t1=y¯*+b*(X¯-x¯*),

where b*=sxy*/sx*2 and the estimates sxy* and sx*2 are based on the available data. To the first order of approximation, the MSEs of estimators tr1 , tR1 , and t1 are given by

MSE(tr1)=P(Sy2+R2Sx2-2RSxy)+Q(Sy22+R2Sx22-2RSxy2),min.MSE(tR1)=PSy2+QSy22-{(PSxy+QSxy2)2PSy2+QSy22},min.MSE(t1)=PSy2(1-ρ2)+Q(Sy22+β2Sx22-2βSxy2),

where R = / , β=Sxy/Sx2, and βlopt=(PSxy+QSxy2)/(PSy2+QSy22).

Strategy II

When *, χ̄, and are used. The non-response occurs on the study variable y, and information on the auxiliary variable x is available from all the sample units along the population mean of the auxiliary variable is known. The ratio and regression type estimators are

tr2=y¯*(X¯x¯),tR2=y¯*(X¯x¯)β2,t2=y¯*+b(X¯-x¯),

where b=sxy*/sx2. The first order approximate MSEs of the estimators tr2 , tR2 , and t2 are given by

MSE(tr3)=P(Sy2+R2Sx2-2RSxy)+QSy22,min.MSE(tR2)=min.MSE(t2)=PSy2(1-ρ2)+QSy22,

where β2=Sxy2/Sx22 and β2opt=Sxy/Sx2.

Strategy III

When *, χ̄*, and χ̄ are used. The non-response occurs on the study variable y, and the information on the auxiliary variable x is obtained from all the sample units, but the population mean of the auxiliary variable is not known. The ratio and regression type estimators are

tr3=y¯*(x¯x¯*),tR3=y¯*(x¯x¯*)β3,t3=y¯*+b(2r)(x¯-x¯*),

where b(2r)=sxy(2r)/sx(2r)2. The MSEs to the first order of approximation, of the estimators tr3 , tR3 , and t3 are given by

MSE(tr3)=PSy2+Q(Sy22+R2Sx22-2RSxy2),min,MSE(tR3)=min.MSE(t3)=PSy2+QSy22(1-ρ22),

where β3opt=Sxy2/Sx22.

Strategy IV

When *, χ̄, χ̄*, and are used. Singh and Kumar (2008) suggested the estimators given below

tr4=y¯*(X¯x¯*)(X¯x¯),tR4=y¯*(X¯x¯*)β4(X¯x¯)β5,t4=y¯*+d1(x¯-x¯*)+d2(X¯-x¯).

To the first order of approximation, the MSEs of the estimators tr4 , tR4 , and t4 are given by

MSE(tr4)=P{Sy2+4RSx2(R-β)}+Q{Sy22+RSx22(R-2β(2))},min.MSE(t4)=min.MSE(tR4)=PSy2(1-ρ2)+QSy22(1-ρ22),

where optimum values of β4, β5, d1, and d2 are given by β4opt = β(2)/R, β5opt = ββ(2)/R, d1opt = β(2), and d2opt = β. Now, we shall consider three more strategies under the two-phase sampling scheme proposed by Okafor and Lee (2000) and Singh and Kumar (2010).

Strategy V

When *, χ̄*, and χ̄′ are used. Okafor and Lee (2000) proposed a double sampling scheme for ratio estimation for sub sampling the non-respondent that also deals with the non-response problem.

tr5=y¯*(x¯x¯*)tR5=y¯*(x¯x¯*)β6t5=y¯*+b*(x¯-x¯*)

To the first order of approximation, the MSEs of the estimators tr5 , tR5 , and t5 are given by

MSE(tr5)=TSy2+S(Sy2+R2Sx2-2RSxy)+Q(Sy22+R2Sx22-2RSxy2),min.MSE(tR5)=min.MSE(t5)=TSy2+SSy2(1-ρ2)+Q{Sy22+βSx22(β-2β(2))},

where β6opt={(P-T)Sxy+QSxy2}/(PSx2+QSx22), S = (1/n – 1/n′), and T = (1/n′ – 1/N).

Strategy VI

When *, χ̄, and χ̄′ are used. Khare and Srivastava (1995) and Okafor and Lee (2000) proposed a double sampling scheme for ratio estimation with sub sampling the non-respondent that also deals with the non-response problem.

tr6=y¯*(x¯x¯),tR6=y¯*(x¯x¯)β7,t6=y¯*+b**(x¯-x¯).

To the first order of approximation, the MSE’s of the estimators tr6 , tR6 , and t6 are given by

MSE(tr6)=TSy2+S(Sy2+R2Sx2-2RSxy)+QSy22,min.MSE(tR6)=min.MSE(t6)=TSy2+SSy2(1-ρ2)+QSy22,

where β7opt=Sxy/Sx2.

Strategy VII

When *, χ̄*, χ̄, and χ̄′ are used. Singh and Kumar (2008) gave estimators for the population mean by using a double sampling scheme under non-response, which are mentioned as:

tr7=y¯*(x¯x¯*)(x¯x¯),tR7=y¯*(x¯x¯*)β8(x¯x¯)β9,t7=y¯*+d3(x¯-x¯*)+d4(x¯-x¯).

To the first order of approximation, the MSEs of the above estimators are given by

MSE(tr7)=S{Sy2+4RSx2(R-β)}+Q{Sy22+RSx22(R-2β(2))}+TSy2,min.MSE(tR7)=min.MSE(t7)=TSy2+SSy2(1-ρ2)+QSy22(1-ρ22),

where optimum values of β8, β9, d3, and d4 are given by β8opt = β(2)/R, β9opt = β/R, d3opt = β(2), and d4opt = β.

Bhushan and Pandey (2017) proposed some improved regression type estimators under non-response in seven different strategies using Searls methodology (Searls, 1964). These estimators were an improvement over the corresponding regression estimators, which are BLUE, under non-response in seven different strategies stated as follow.

T1=γ1y¯*+δ1(X¯-x¯*),T2=γ2y¯*+δ2(X¯-x¯),T3=γ3y¯*+δ3(x¯-x¯*),T4=γ4y¯*+δ4(x¯-x¯*)+η1(X¯-x¯),T5=γ5y¯*+δ5(x¯-x¯*),T6=γ6y¯*+δ6(x¯-x¯),T7=γ7y¯*+δ7(x¯-x¯*)+η2(x¯-x¯),

where γi, δj, and ηk {i = j = 1, 2, . . . , 7; k = 1, 2} are suitably (optimally) chosen scalars to optimize MSE. The MSE of the estimators are given by

min.MSE(Ti)=Y¯2MSE(ti)Y¯2+MSE(ti),         i=1,2,,7

obviously,

min.MSE(Ti)<MSE(ti).

The optimal values of γi, δj, and ηk {i = j = 1, 2, . . . , 7; k = 1, 2} are given by

γ1=Y¯2{Y¯2+PSy2+QSy22-(PSxy+QSxy2)2(QSx2+QSx22)},         γ2=Y¯2{Y¯2+PSy2(1-ρ2)+QSy22},γ3=Y¯2{Y¯2+PSy2+QSy22(1-ρ22)},         γ4=Y¯2{Y¯2+PSy2(1-ρ2)+QSy22(1-ρ22)},γ5=Y¯2{Y¯2+PSy2+QSy22-(SSxy+QSxy2)2(SSx2+QSx22)},         γ6=Y¯2{Y¯2+TSy2+SSy2(1-ρ2)+QSy22},γ7=Y¯2{Y¯2+TSy2+SSy2(1-ρ2)+QSy22(1-ρ22)},δ1=Y¯2(PSxy+QSxy2)(PSx2+QSx22){Y¯2+PSy2+QSy22-(PSxy+QSxy2)2(PSx2+QSx22)},         δ2=Y¯2SxySx2{Y¯2+PSy2(1-ρ2)+QSy22},δ3=Y¯2Sxy2Sx22{Y¯2+PSy2+QSy22(1-ρ22)},         δ4=Y¯2Sxy2Sx22{Y¯2+PSy2(1-ρ2)+QSy22(1-ρ22)},δ5=Y¯2(SSxy+QSxy2)(SSx2+QSx22){Y¯2+PSy2+QSy22-(SSxy+BSxy2)2(SSx2+QSx22)},         δ6=Y¯2SxySx2{Y¯2+TSy2+SSy2(1-ρ2)+QSy22},δ7=Y¯2Sxy2Sx22{Y¯2+TSy2+SSy2(1-ρ2)+QSy22(1-ρ22)},η1=Y¯2SxySx2{Y¯2+PSy2(1-ρ2)+QSy22(1-ρ22)},         η2=Y¯2SxySx2{Y¯2+TSy2+SSy2(1-ρ2)+QSy22(1-ρ22)},

where P = (1/n – 1/N), Q = W2(k – 1)/n, S = (1/n – 1/n′), and T = (1/n′ – 1/N).

Bhushan and Pandey (2017) showed that the these estimators were better than conventional regression estimators. In this article, we propose some new ratio type estimators and compared these with the corresponding regression estimators given earlier. The proposed estimators are motivated by Cochran (1977), Khare and Srivastava (1995), Rao (1986), Okafor and Lee (2000), and Singh and Kumar (2008) under the one phase and two phase sampling using seven different strategies under non-response.

3. Proposed improved ratio type estimators under a non-response

We propose improved ratio type estimators using Searls methodology (Searls, 1964), Searls (1964) proposed a technique to improve the conventional estimators by multiplying a tuning constant term α whose optimum value depends on the coefficient of variation, which is a fairly stable quantity, and we refer this technique of multiplication by a tuning constant α as Searls type transformation (STT), under seven different strategies in single phase sampling and two phase sampling, as follows.

Ts=αy¯*

optimum value of α is given by

α=Y¯2Y¯2+PSy2+QSy22.

The proposed estimator under Strategy I, when *, χ̄*, and are used, is given by

Ts1=α1y¯*(X¯x¯*)β1.

The proposed estimator under Strategy II, when *, χ̄, and are used, is given by

Ts2=α2y¯*(X¯x¯)β2.

The proposed estimator under Strategy III, when *, χ̄*, and χ̄ are used, is given by

Ts3=α3y¯*(x¯x¯*)β3.

The proposed estimator under Strategy IV, when *, χ̄, χ̄*, and are used, is given by

Ts4=α4y¯*(X¯x¯*)β4(X¯x¯)β5.

The proposed estimator under Strategy V, when *, χ̄*, and χ̄′ are used, is given by

Ts5=α5y¯*(x¯x¯*)β6.

The proposed estimator under Strategy VI, when *, χ̄, and χ̄′ are used, is given by

Ts6=α6y¯*(x¯x¯)β7.

The proposed estimator under Strategy VII, when *, χ̄*, χ̄, and χ̄′ are used, is given by

Ts7=α7y¯*(x¯x¯*)β8(x¯x¯)β9,

where αj ( j = 1, 2, . . . 7) is a suitable chosen scalars. And the optimum values of αj are defined in appendix.

Theorem 1

The bias and minimum MSE of the new ratio type estimator Ts j ( j = 1, 2, . . . 7) is given by

Bias(Tsj)=Y¯(αj-1)

and

min.MSEαj(Tsj)=Y¯2(1-Bj2Aj),

j = 1, 2, . . . , 7.

Proof

See Appendix.

It is interesting to note that simultaneous optimization with respect to the characterizing scalars γi and δi of the expression (2.5) of MSE is possible for regression (difference) type estimators. But simultaneous optimization with respect to the characterizing scalars αi and βi of the expression (3.2) not possible for ratio type estimators.

Theorem 2

The proposed ratio type estimators Ts j , (j = 1, 2, . . . , 7) are better than difference type estimators Tk, (k = 1, 2, . . . , 7) iff

Bj2Aj>γkopt

and vice versa. Otherwise both are equally efficient in case of equality in (3.3).

Proof

It may be easily observed from (2.5) that the MSE of the difference type estimators Ti, (i = 1, 2, . . . , 7) are given by

min.MSE(Ti)=Y¯2(1-γiopt)

Comparing (3.4) with (3.2), we have the theorem.

The only way ascertain (3.3) if this holds in practice is through a computational study.

Theorem 3

The proposed ratio type estimators Ts j , j = 1, 2, . . . , 7 are better than the conventional ratio type estimators tRj , j = 1, 2, . . . , 7 iff

Bj2Aj>{1-min.MSE(tRj)Y¯2}

and vice versa. Otherwise both are equally efficient in case of equality in (3.5).

Proof

It may be easily observed from (3.2) and (2.2), we have the theorem.

Theorem 4

The proposed ratio type estimators Ts j , j = 1, 2, . . . , 7 are better than the conventional regression type estimators ti, i = 1, 2, . . . , 7 iff

Bj2Aj>{1-min.MSE(ti)Y¯2}

and vice versa. Otherwise both are equally efficient in case of equality in (3.6).

Proof

It may be easily observed from (3.2) and (2.4), we have the theorem.

4. Empirical study

In order to have a better understanding about the efficiency of the proposed estimators we have conducted a comprehensive empirical study on three populations and compared the proposed estimators with the existing estimators. The percentage relative efficiency (PRE) is calculated as

PRE=Var(y¯*)min.MSE(Tsj)×100,         j=1,2,,7.
  • The first population considered by Srivastava (1993, p.50) consists of a list of 70 villages in the administrative devision of Tehsil India that includes population and cultivated area (in acres) data from 1981. Here the cultivated area (in acres) is taken as the main study character and the population of village is taken as the auxiliary character. The parameters of the population are as follows: N = 70, n′ = 40, n = 25, = 981.29, = 1755.53, S y = 613.66, S x = 1406.13, 2 = 597.29, 2 = 1100.24, S y2 = 244.11, S x2 = 631.51, ρ = 0.778, ρ2 = 0.445, R = 0.5589, β = 0.3395, β2 = 0.1720, W2 = 0.20.

  • The second population considered by Khare and Kumar (2011). For the population of 96 villages of rural areas under Police Station, Singur, District Hooghly from district census Handbook (1981), published by the government of India, the data on the number of cultivators y, as a study character and the population of villages, as an auxiliary character x have been taken. The non-response rate in the population is considered to be 25%. The values of the parameters of the population are given as follows: N = 96, n′ = 65, n = 25, = 185.22, = 1807.23, S y = 195.03, S x = 1921.77, S y2 = 97.82, S x2 = 1068.44, ρ = 0.904, ρ2 = 0.895, R = 0.1025, W2 = 0.25.

  • Third population considered from Srivastava (1993, p.50). The data belongs to the data on physical growth of upper-socio-economic group of 95 school children of Varanasi under an ICMR study. The first 25% (i.e., 24 children) units have been considered as non-response units. The values of the parameters related to the study variate y (weight in kg) and the auxiliary variate x (chest circumference in cm) have been given below: N = 95, n′ = 70, n = 35, = 19.497, = 55.8611, S y = 3.0435, S x = 3.2735, S y2 = 2.3552, S x2 = 2.5137, ρ = 0.8460, ρ2 = 0.7290, R = 0.3490, β = 0.7865, β2 = 0.6829, W2 = 0.25, N2 = 24, N1 = 71.

From perusal of above results it is observed that the new ratio type estimators proposed Tsi are always better than the conventional ratio type counterparts tRi . Hence, we conclude that all proposed new ratio type estimators have higher efficiency in comparison to the conventional ratio type estimators. A comparison of STD (or regression) estimators Ti (i = 1, 2, . . . , 7) with new ratio type estimators Tsi (i = 1, 2, . . . , 7), we observe that the new ratio type estimators are always better than the estimators proposed by Bhushan and Pandey regression type estimators under optimality condition (3.3). Therefore, in population I and II proposed ratio type estimators Ts j are better than Bhushan and Pandey regression type estimators Ti, as (3.3) is satisfied. However, the case is reversed for population III.

5. Simulation study

In this section, simulation is conducted to evaluate the performance of the proposed class of estimators with respect to traditional estimators. For this study we have generated a population size N = 1,000 from standard normal distribution using MVRNORM package in software R, where study and auxiliary variable are correlated with correlation ρ = 0.7, draw sample of size n = 200 with 35% non-response. The whole simulation process starting from the drawing sample from variable Y and auxiliary variable X from normal population and calculating the estimates was repeated 50,000 times.

6. Conclusions

From the above computational results as shown in Tables 13, it may be concluded that the proposed estimators Ts j dominate the over conventional ratio type estimators tRi . The most interesting and noticeable finding of this paper is the vitiation of conventional thought, that the ratio type estimator can at most match upto its regression estimator. Consequently in such a manner, we proved that the proposed ratio type estimators Ts j ( j = 1, 2, . . . , 7) provides an improvement over traditional ratio type estimators counterpart tRi (i = 1, 2, . . . , 7) while satisfying (3.5); in addition, we also proved that the proposed ratio estimator can provide an improvement over both traditional regression estimators ti (i = 1, 2, . . . , 7) and proposed regression estimators Ti (i = 1, 2, . . . , 7) under the optimality conditions (3.3). Thus the proposed estimators are highly rewarding in terms of the increased precession of the estimates and negative impact of the non-response. Therefore, the proposed estimators may be recommended to survey practitioners for real-life applications.

TABLES

Table 1

Mean squared error and percentage relative efficiency of the estimators with respect to * for population 1

Estimatork

1234
*10160.17 (100.00)10636.88 (100.00)11113.60 (100.00)11590.32 (100.00)
Ts10054.08 (101.05)10520.67 (101.10)10986.79 (101.15)11452.47 (101.20)

Strategy ItR14288.77 (236.90)4745.94 (224.13)5195.20 (213.92)5637.74 (205.58)
T14269.75 (237.95)4722.66 (225.23)5167.32 (215.07)5604.92 (206.78)
Ts14247.51 (239.20)4696.51 (226.48)5137.05 (216.34)5570.35 (208.07)

Strategy IItR24298.93 (236.34)4775.65 (222.73)5252.36 (211.59)5729.08 (202.31)
T24279.82 (237.39)4752.08 (223.83)5223.87 (212.74)5695.19 (203.51)
Ts24259.43 (238.53)4729.86 (224.89)5199.82 (213.73)5669.32 (204.44)

Strategy IIItR310065.76 (100.94)10448.08 (101.81)10830.39 (102.61)11212.71 (103.37)
T39961.63 (101.99)10335.93 (102.91)10709.93 (103.77)11083.65 (104.57)
Ts39959.45 (102.01)10331.38 (102.96)10702.83 (103.84)11073.81 (104.66)

Strategy IVtR44204.53 (241.65)4586.84 (231.90)4969.16 (223.65)5351.47 (216.58)
T44186.25 (242.70)4565.09 (233.00)4961.16 (224.80)5321.89 (217.78)
Ts44164.88 (243.95)4540.74 (234.25)4916.11 (226.06)5291.00 (219.06)

Strategy VtR56727.54 (151.02)7182.84 (148.09)7638.14 (145.50)8093.44 (143.21)
T56680.87 (152.07)7123.29 (149.32)7554.66 (147.11)7977.40 (145.29)
Ts56662.06 (152.51)7104.86 (149.71)7546.58 (147.26)7987.23 (145.11)

Strategy VItR66741.11 (150.72)7217.83 (147.37)7694.55 (144.43)8171.26 (141.84)
T66694.24 (151.77)7164.13 (148.47)7633.55 (145.58)8102.50 (143.04)
Ts66677.77 (152.15)7146.59 (148.84)7614.95 (145.94)8082.86 (143.39)

Strategy VIItR76646.71 (152.86)7029.02 (151.33)7411.34 (149.95)7793.66 (148.71)
T76601.14 (153.91)6978.08 (152.43)7354.73 (151.10)7731.08 (149.91)
Ts76583.19 (154.33)6957.40 (152.88)7331.14 (151.59)7704.39 (150.44)

Table 2

Mean squared error and percentage relative efficiency of the estimators with respect to * for population 2

Estimatork

1234
*1220.94 (100.00)1316.63 (100.00)1412.31 (100.00)1508.00 (100.00)
Ts1178.98 (103.56)1267.96 (103.84)1356.47 (104.12)1444.51 (104.40)

Strategy ItR1225.71 (540.93)245.57 (536.15)265.30 (532.34)284.93 (529.25)
T1224.23 (544.49)243.83 (539.99)263.26 (536.46)282.58 (533.65)
Ts1223.29 (546.80)242.63 (542.65)261.79 (539.48)280.80 (537.03)

Strategy IItR2301.36 (405.14)397.05(331.61)492.74 (286.62)588.43 (256.28)
T2298.74 (408.69)392.51(335.44)485.76 (290.74)578.50 (260.67)
Ts2297.73 (410.08)391.21 (336.55)484.18 (291.69)576.64 (261.51)

Strategy IIItR31144.29 (106.70)1163.33 (113.18)1182.37 (119.45)1201.41 (125.52)
T31107.36 (110.26)1125.18 (117.01)1142.98 (123.56)1160.76 (129.91)
Ts31106.75 (110.32)1123.95 (117.14)1141.10 (123.77)1158.21 (130.20)

Strategy IVtR4224.72 (312.90)243.76 (291.08)262.79 (277.11)281.84 (267.40)
T4223.25 (546.88)242.04 (543.98)260.80 (541.53)279.54 (539.46)
Ts4222.32 (549.18)240.86 (546.63)259.36 (544.54)277.81 (542.82)

Strategy VtR5380.12 (226.28)400.02 (223.52)419.93 (221.57)439.83 (220.11)
T5375.95 (324.76)395.33 (333.05)414.54 (340.69)433.63 (347.76)
Ts5374.68 (325.86)393.68 (334.44)412.61 (342.28)431.48 (349.49)

Strategy VItR6455.79 (184.51)551.48 (162.44)647.16 (149.51)742.85 (141.01)
T6449.81 (271.43)542.75 (242.58)635.18 (222.35)727.11 (207.40)
Ts6448.60 (272.17)541.30 (243.23)633.50 (222.94)725.19 (207.94)

Strategy VIItR7379.14 (227.20)398.18 (224.87)417.22 (223.21)436.26 (221.97)
T7374.99 (325.59)393.61 (334.50)412.21 (342.62)430.78 (350.06)
Ts7373.74 (326.68)392.05 (335.83)410.31 (344.20)428.53 (351.90)

Table 3

Mean squared error and percentage relative efficiency of the estimators with respect to * for population 3

Estimatork

1234
*0.2067 (100.000)0.2464 (100.000)0.2860 (100.000)0.3256 (100.000)
Ts0.2066 (100.054)0.2462 (100.064)0.2857 (100.075)0.3253 (100.085)

Strategy ItR10.0664 (311.050)0.0853 (288.830)0.1040 (274.840)0.1227 (265.220)
T10.0664 (311.104)0.0852 (288.893)0.1040 (274.913)0.1227 (265.307)
Ts10.0664(311.060)0.0853 (288.840)0.1040 (274.860)0.1227 (265.250)

Strategy IItR20.0871 (237.290)0.1267 (194.380)0.1663 (171.900)0.2060 (158.070)
T20.0871 (237.342)0.1267 (194.441)0.1663 (171.977)0.2058 (158.158)
Ts20.0871 (237.310)0.1267 (194.410)0.1663 (171.950)0.2059 (158.130)

Strategy IIItR30.1857 (111.340)0.2043 (120.610)0.2228 (128.350)0.2414 (134.890)
T30.1856 (111.392)0.2041 (120.680)0.2227 (128.421)0.2412 (134.974)
Ts30.1856 (111.390)0.2042 (120.670)0.2227 (128.410)0.2412 (134.960)

Strategy IVtR40.0660 (312.890)0.0846 (291.080)0.1032 (277.110)0.1217 (267.400)
T40.0660 (312.950)0.0846 (291.144)0.1031 (277.186)0.1217 (267.486)
Ts40.0660 (312.910)0.0846 (291.090)0.1032 (277.130)0.1217 (267.430)

Strategy VtR50.0913 (226.280)0.1102 (223.520)0.1290 (221.570)0.1479 (220.110)
T50.0913 (226.331)0.1101 (223.692)0.1288 (221.952)0.1475 (220.722)
Ts50.0913 (226.300)0.1102 (223.550)0.1290 (221.600)0.1479 (220.150)

Strategy VItR60.1120 (184.510)0.1516 (162.440)0.1913 (149.510)0.2309 (141.010)
T60.1120 (184.568)0.1516 (162.503)0.1912 (149.581)0.2307 (141.098)
Ts60.1120 184.540)0.1516 (162.480)0.1912 (149.560)0.2308 (141.080)

Strategy VIItR70.0910 (227.200)0.1095 (224.870)0.1281 (223.210)0.1467 (221.970)
T70.0909 (227.259)0.1095 (224.934)0.1280 (223.286)0.1466 (222.057)
Ts70.0909 (227.230)0.1095 (224.890)0.1281 (223.240)0.1467 (222.010)

Table 4

Percentage relative efficiency (PRE) of the proposed estimators with respect to * using simulation

EstimatorPRE
*100
Ts100.086

Strategy ItR1165.163
T1166.775
Ts1167.016

Strategy IItR2142.838
T2143.0174
Ts2143.637

Strategy IIItR3120.771
T3121.753
Ts3120.795

Strategy IVtR4168.449
T4168.503
Ts4168.727

Strategy VtR5148.608
T5149.331
Ts5149.486

Strategy VItR6130.484
T6130.495
Ts6131.025

Strategy VIItR7153.491
T7153.893
Ts7154.491

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