On use of adaptive cluster sampling for variance estimation

Abstract

Adaptive cluster sampling is particularly helpful whenever the target population is unique, dispersed unevenly, concealed or difficult to find. In the current investigation, under an adaptive cluster sampling approach, we propose a ratio-product-logarithmic type estimator employing a single auxiliary variable for the estimation of finite population variance. The bias and mean square error of the proposed estimator are developed by using simulation as well as real data sets. The study results show that for estimating the finite population variance, the proposed estimator outperforms the competing estimators.

1. Introduction

In survey sampling, the auxiliary information is mostly used to improve the accuracy of survey estimates. Adaptive Cluster Sampling (ACS) can especially be helpful when the population of interest is uncommon, irregular, concealed or difficult to find. Assessing the prevalence of a rare disease, monitoring the spread of a rare infectious disease, such as a new strain of influenza, in a community, estimating the population of a rare animal species, such as the snow leopard, in a mountainous region are some examples. Firstly, Thompson [12] proposed the adaptive estimators and later on different statisticians like [2,5,7,8,11,13] and many others have contributed in this direction under different sampling designs. In our study, we reviewed some ACS estimators in simple random sampling such as [3,4,6]. Secondly, we have proposed an estimator in adaptive cluster sampling technique by using a simple random sampling design. Finally, we conclude that our proposed ACS estimator is highly efficient than competent mean estimators.

1.1. Adaptive cluster sampling

Consider a population, $K^{\star }=(K_1^{\star },K_2^{\star },\dots ,K_i^{\star },\dots ,K_N^{\star })$. Let $y_i$ and $x_i$ represent the study and the auxiliary variables for the $i^{\mathrm{th}}$ unit, respectively. Let population $K^{\star }$ is partitioned into $L^{\star }$ comprehensive networks and ${\varsigma }_i$ denotes the $i^{\mathrm{th}}$ network having $m_i$ units. The initial sample $s_o^{\star }$, where $s_o^{\star }=(i_1,i_2,i_3,\dots ,i_n)$ taken from the units that were sampled is used to choose the data set $d^{\star }$, such that, $d^{\star }=(s_o^{\star };y_{s_o^{\star }},x_{s_o^{\star }})$.

The pre-determined value C is chosen prior to the adaptive cluster sampling method. The spatially adjacent units (designated as North, South, East, West, Northwest, Northeast, Southwest, and Southeast) will be added to the sample and observed whether any of the first chosen sample satisfies or exceeds C, i.e. $y_i\ge \text{C}$. The units next to them will be chosen if any of these surrounding units meet this condition C, and so on. A network is a grouping of initially sampled units and nearby units that satisfy the predefined value C. Edge units are those units that are found below the value C. A cluster is made up of a network and its edge components. Any first sampled unit that does not satisfy the requirement is regarded as a network of size one (Table 1).

Table 1.

Sampling unit.

Northwest	North	Northeast
West	Sampling Unit	East
Southwest	South	Southeast

In order to better understand the adaptive cluster sampling strategy with the condition of adapt C, i.e. $y_i\ge 0$, let a population of size $9\times 13=117$ units with 6 networks (denoted as $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ and $B_6$) is taken into consideration in Table 2, where $B_i$ are the non-zero elements of the population. Any of the $i^{\mathrm{th}}$ network will select every element in the network $B_i$, if it is found in $B_i$ where $(i=1,2,3,4,5,6)$, according to the network's symmetry assumption. In Table 3, the selection of a sample with a size of 5 quadrates is taken using simple random sampling without replacement (SRSWOR). Asterisks(*) are used to represent the units in the initial sample. In Table 4, the initial units choosen in Table 3 are shown having the selected networks, ' ${\ast }_1$' represents the first sample unit, selected without replacement and we can say that the initial sample unit fulfills the condition ( $y_i\ge 0$) but it's neighboring units cannot satisfy the condition so it is regarded as a network of size 1. Here ' ${\ast }_2$', ' ${\ast }_3$', and ' ${\ast }_5$' represent the second, third and fifth random initial sample units and underline units represent the final sample units with network of sizes 4, 10 and 4, respectively. Here ' ${\ast }_4$' represent the fourth initial sample unit as it cannot satisfy the condition so it is regarded as a network of size one.

Table 2.

Population networks with edge units.

Table 3.

Population including the initial samples.

Table 4.

Population with neighborhood units and edge units.

1.2. Estimators in adaptive cluster sampling

Isaki [7], suggested the ratio type variance estimator in ACS having the form:

$$s_{10}=s_{w_y}^2\frac{S_{w_x}^2}{s_{w_x}^2}$$ (1)

The bias of $s_{10}$ is expressed as follows:

$$\mathrm{Bias}(s_{10})=S_{w_y}^2\phi [({\lambda }_{04}-1)-({\lambda }_{22}-1)]$$ (2)

where ${\lambda }_{\mathrm{rs}}=\frac{{\mu }_{\mathrm{rs}}}{{\mu }_{20}^{r/2}{\mu }_{02}^{s/2}}$, ${\mu }_{\mathrm{rs}}=\frac{1}{N-1}\sum _{i=1}^N(w_{y_i}-{\overline{W}}_y)(w_{x_i}-{\overline{W}}_x)$, and $\phi =\frac{1}{n}-\frac{1}{N}$

The MSE of $s_{10}$ is expressed as follows:

$$\mathrm{MSE}(s_{10})=S_{w_y}^4\phi [({\lambda }_{40}-1)+({\lambda }_{04}-1)-2({\lambda }_{22}-1)]$$ (3)

Singh et al. [9], proposed the exponential ratio type variance estimator as:

$$s_{30}=s_{w_y}^2\exp \left(\frac{S_{w_x}^2-s_{w_x}^2}{S_{w_x}^2+s_{w_x}^2}\right)$$ (4)

The bias of $s_{30}$ is expressed as follows:

$$\mathrm{Bias}(s_{30})=S_{w_y}^2\phi [\frac{3}{8}({\lambda }_{04}-1)-\frac{1}{2}({\lambda }_{22}-1)]$$ (5)

The MSE of $s_{30}$ is expressed as follows:

$$\mathrm{MSE}(s_{30})=S_{w_y}^4\phi [({\lambda }_{40}-1)+\frac{1}{4}({\lambda }_{04}-1)-2({\lambda }_{22}-1)]$$ (6)

Yasmeen and Thompson [14], proposed variance estimator using the parameters of an auxiliary variable, which can be defined as:

$$s_{40j}=s_{w_y}^2\frac{S_x^2\alpha +S_x^2\tau }{s_{w_x}^2\alpha +S_{w_x}^2\tau },j=1,2,3$$ (7)

where α and τ are known constants. The bias of $s_{40j}$ is expressed as follows:

$$\mathrm{Bias}(s_{40j})=S_{w_y}^2\phi [R^2({\lambda }_{04}-1)-R({\lambda }_{22}-1)]$$ (8)

where $R=\frac{S_x^2\alpha }{S_{w_x}^2\alpha +S_{w_x}^2\tau }$

The MSE of $s_{40j}$ is expressed as follows:

$$\mathrm{MSE}(s_{40j})=S_y^4\phi [({\lambda }_{40}-1)+R^2({\lambda }_{04}-1)-2R({\lambda }_{22}-1)]$$ (9)

The parameters, namely coefficient of correlation, coefficient of variation and median of the auxiliary variable are considered for different choices of α and τ to suggest some special cases of the estimator, $s_{40j}$ (See Table 5).

Table 5.

Choices of α and τ.

j	1	2	3
α	1	ρ	$M_d$
τ	$C_{w_x}$	$C_{w_x}$	$C_{w_x}$

2. Proposed adaptive variance estimator

We propose a ratio-product-logarithmic type estimator for estimating the finite population variance where constant $w_1$ needs to be determined and optimized.

$$s_{\mathrm{rpl}0}=s_{w_y}^2[w_1\frac{s_{w_x}^2}{S_{w_x}^2}+(1-w_1)\frac{S_{w_x}^2}{s_{w_x}^2}][1+\log \left(\frac{s_{w_x}^2}{S_{w_x}^2}\right)]$$ (10)

We utilize the relative error terms to obtain the bias and MSE of the proposed estimator as:

$$\begin{array}{rl}{s}_{w_y}^2& =S_{w_y}^2(1+{\overline{ϵ}}_{w_y})\\ s_{w_x}^2& =S_{w_x}^2(1+{\overline{ϵ}}_{w_x})\end{array}$$

Replacing the values of $s_{w_y}^2$ and $s_{w_x}^2$ in Equation (10), we get,

$$s_{\mathrm{rpl}0}=S_{w_y}^2(1+{\overline{ϵ}}_{w_y})[w_1(1+{\overline{ϵ}}_{w_x})+(1-w_1)(1+{\overline{ϵ}}_{w_x}{)}^{-1}][1+\log (1+{\overline{ϵ}}_{w_x})]$$ (11)

Applying the Taylor series on Equation (11) and leaving out terms that are closer to the second order of approximation, we have,

$$\begin{array}{rl}{s}_{\mathrm{rpl}0}& =S_{w_y}^2(1+{\overline{ϵ}}_{w_y})[w_1+w_1{\overline{ϵ}}_{w_x}+(1-w_1)(1-{\overline{ϵ}}_{w_x}+{\overline{ϵ}}_{w_x}^2)][1+{\overline{ϵ}}_{w_x}-\frac{{\overline{ϵ}}_{w_x}^2}{2}]\\ s_{\mathrm{rpl}0}& =(S_{w_y}^2+S_{w_y}^2{\overline{ϵ}}_{w_y})[1+2w_1{\overline{ϵ}}_{w_x}-{\overline{ϵ}}_{w_x}+{\overline{ϵ}}_{w_x}^2-w_1{\overline{ϵ}}_{w_x}^2\\ & +{\overline{ϵ}}_{w_x}+2w_1{\overline{ϵ}}_{w_x}^2-{\overline{ϵ}}_{w_x}^2-\frac{{\overline{ϵ}}_{w_x}^2}{2}]\\ s_{\mathrm{rpl}0}-S_{w_y}^2& =2w_1{S}_{w_y}^2{\overline{ϵ}}_{w_x}+w_1{S}_{w_y}^2{\overline{ϵ}}_{w_x}^2-\frac{S_{w_y}^2{\overline{ϵ}}_{w_x}^2}{2}+S_{w_y}^2{\overline{ϵ}}_{w_y}+2w_1{S}_{w_y}^2{\overline{ϵ}}_{w_x}{\overline{ϵ}}_{w_x}\end{array}$$ (12)

After applying the expectations on both sides of Equation (12), we get,

$$\mathrm{Bias}(s_{\mathrm{rpl}0})=S_{w_y}^2\phi ({\lambda }_{04}-1)[w_1-\frac{1}{2}+2w_{1}L]$$ (13)

where, $L=\frac{({\lambda }_{22}-1)}{({\lambda }_{04}-1)}$

Squaring both sides of Equation (12) and then taking expectations yields the expression as:

$$\mathrm{MSE}(s_{\mathrm{rpl}0})=S_{w_y}^4\phi [({\lambda }_{40}-1)+4w_1({\lambda }_{04}-1)(w_1+L)]$$ (14)

In order to obtain the optimum value of $w_1$, taking a partial derivative of Equation (14) with respect to $w_1$ and equating to zero, we get,

$$w_1^{\mathrm{opt}}=-\frac{({\lambda }_{22}-1)}{2({\lambda }_{04}-1)}$$ (15)

Substituting the optimum value of $w_1$ in Equation (14) yields the least MSE of the suggested estimator i.e.

$$\mathrm{MS}{E}_{\min }(s_{\mathrm{rpl}0})=S_{w_y}^4\phi [({\lambda }_{40}-1)-\frac{({\lambda }_{22}-1{)}^2}{({\lambda }_{04}-1)}]$$ (16)

which is equal to the MSE of the regression estimator.

3. Results and conclusions

We evaluated the performance of the proposed estimator in relation to the $s_{10}$ estimator of [7], the $s_{30}$ estimator of Ref. [9], and the $s_{40j}$ estimator of Ref. [14]. We performed the simulation experiment and also derived few results from the real data sets.

3.1. Simulation study

Zero-inflated Poisson cluster process is used to create a rarely flock artificial population (X) in the survey area, which is divided into $20\times 20=400$ squared units. Under this process, the average number of parents (λ) is chosen to be 5. The initial sample ranges from 45 to 65, with a 400 size of population. Using a linear and exponential model to simulate the values for the study variable (Y ) in order to assess how well the suggested estimator performs:

Model 1 $y_i=4x_i+{ϵ}_i$ where ${ϵ}_i\sim N(0,x_i)$

Model 2 $y_i={\mathrm{e}}^{0.4x_i}+{ϵ}_i$ where ${ϵ}_i\sim \exp (x_i)$

The absolute relative bias (ARB) and the mean square error(MSE) are expressed as:

$$(\mathrm{ARB}{)}_{s_i}=\left|\frac{\frac{1}{T}\sum _{t=1}^T((s_i{)}_t-S_{w_y}^2)}{S_{w_y}^2}\right|,i=10,30,40j,\mathrm{rpl}0,\mathrm{where},j=1,2,3$$

and

$$\mathrm{MSE}(s_i)=\frac{1}{T}\sum _{t=1}^T((s_i{)}_t-S_{w_y}^2{)}^2,i=10,30,40j,\mathrm{rpl}0,\mathrm{where},j=1,2,3$$

where T is the total number of iterations and $s_i$ is the value of the $i^{\mathrm{th}}$ estimator for the $t^{\mathrm{th}}$ sample. The proposed estimator ( $s_{\mathrm{rpl}0}$) is compared with the Equations (1), (4) and (7).

$$E_i=\frac{\mathrm{MSE}(s_i)}{\mathrm{MSE}(s_{\mathrm{rpl}0})}\times 100,i=10,30,40j,\mathrm{where},j=1,2,3$$

In order to obtain the precise estimates, 1000 iterations have been used to generate the samples from both artificial populations, respectively. The Absolute Relative Bias ( $\mathrm{AR}{B}^{\prime }s$) and Percent Relative Efficiencies ( $\mathrm{PR}{E}^{\prime }s$) of all the estimators are computed for different optimum values of $w_1$, ranging from 0.1 to 0.6.

3.2. Real data

We consider three real data sets for numerical comparisons of the proposed and existing estimators.

Population I: [Source: [10]]

The Blue-winged teal data, shown in Figures 1 and 2 is used to evaluate the performance of the suggested estimator. The data statistics are given in Table 6. Let auxiliary variable (X) represents the number of blue wings and the values of the study variable (Y) are generated by using the statistical Model 1, i.e.

Figure 1.

Blue Wing(X).

Figure 2.

Blue Wing(Y).

Table 6.

Data summary of Population I.

N = 50	$\overline{X}=282.42$	$\overline{Y}=759.58$
$\rho =0.999$	$C_{w_x}=1.61953$	$C_{w_y}=1.61932$

$y_i=4x_i+{ϵ}_i$, where ${ϵ}_i\sim N(0,x_i)$ Population II: [Source: [1]]

The effectiveness of the suggested estimate is assessed using the annual number of tornadoes in Lafayette Parish, as in Figures 3 and 4, Louisiana, US, from 1950 to 2012. The data statistics are given in Table 7. The number of tornado occurrences in Lafayette Parish is considered as an auxiliary variable (X) and are used to generate the values of study variable (Y ) using the statistical Model 1.

Figure 3.

Number of Tornado occurrences in Lafayette Parish(X).

Figure 4.

Number of Tornado occurrences in Lafayette Parish(Y).

Table 7.

Data summary of Population II.

N = 63	$\overline{X}=0.63$	$\overline{Y}=2.57$
$\rho =0.968$	$C_{w_x}=1.13197$	$C_{w_y}=1.14617$

$y_i=4x_i+{ϵ}_i$; where ${ϵ}_i\sim N(0,x_i)$

Population III: [Source: [1]]

The performance of the suggested estimator is assessed using the annual number of large US earthquakes data (those with magnitude at least 7.0) from 1950 to 2012, shown in Figures 5 and 6, respectively. The data statistics are given in Table 8. The number of major US earthquakes is considered as an auxiliary variable (X) and is used to generate the values of study variable (Y) using the statistical Model 1, i.e.

Figure 5.

US earthquake(X).

Figure 6.

US earthquake(Y).

Table 8.

Data summary of Population III.

N = 63	$\overline{X}=0.57$	$\overline{Y}=2.25$
$\rho =0.971$	$C_{w_x}=1.20582$	$C_{w_y}=1.21136$

$y_i=4x_i+{ϵ}_i$, where ${ϵ}_i\sim N(0,x_i)$

4. Concluding remarks

The results based on the simulated data are given in Tables 9 – 12. We obtain the percentage relative efficiencies of the proposed estimator with respect to the Equations (1), (4) and (7). Tables 9 and 11, based on T = 1000 iterations show the absolute relative biases of the suggested and the competent estimators for the simulated data. The bias of the suggested estimator decreases as the sample size increases for all cases of $w_1$. Table 10 based on Model 1 shows the percentage relative efficiencies of the suggested estimator for simulated data. The suggested estimator performs better as compared to all other competent estimators for all cases of $w_1$, especially suggested estimator is more efficient with a maximum efficiency of 13693.96 for a sample of size 60 when $w_1$ is 0.3. It is observed that by increasing the sample size, the efficiency of the proposed estimator increases for all cases of $w_1$. Table 12 based on Model 2, shows the percentage relative efficiencies of the suggested estimator for simulated data. The suggested estimator performs better as compared to all other competent estimators for all cases of $w_1$. Table 13, for Population I, considers the efficiencies of proposed estimator for five different sample sizes. Again it is more efficient than the considered estimators with a maximum efficiency of 744000006714 for a sample of size 30 when $w_1$ is 0.4. Tables 14 and 15 for Population II and Tables 16 and 17 for Population III, consider six different sample sizes for different optimum values of $w_1$. The suggested estimator is more efficient than the competent estimators (Table 18).

Table 9.

$\mathrm{AR}{B}^{\prime }s$ of proposed and competent estimators for varying sample sizes using simulated data (Model 1).

n	$(\mathrm{ARB}{)}_{s_{10}}$	$(\mathrm{ARB}{)}_{s_{30}}$	$(\mathrm{ARB}{)}_{s_{401}}$	$(\mathrm{ARB}{)}_{s_{402}}$	$(\mathrm{ARB}{)}_{s_{403}}$	$(\mathrm{ARB}{)}_{s_{\mathrm{rpl}0}}$
$w_1=0.1$
45	1.087438	1.023858	1.150881	1.141249	1.130281	1.018675
50	1.078438	1.022858	1.140881	1.140249	1.129281	1.017875
55	1.072296	1.024944	1.14689	1.146358	1.137002	1.020477
60	1.050103	1.016279	1.141234	1.14089	1.136879	1.014521
65	1.051101	1.020079	1.146705	1.146338	1.140642	1.017943
$w_1=0.2$
45	1.091378	1.026848	1.142017	1.141214	1.125323	1.015585
50	1.080323	1.030915	1.151883	1.151391	1.145201	1.028551
55	1.056838	1.018254	1.141948	1.141551	1.136238	1.015948
60	1.05886	1.023823	1.149337	1.148987	1.14501	1.022845
65	1.049255	1.015833	1.140963	1.140597	1.135485	1.013732
$w_1=0.3$
45	1.086384	1.027582	1.144586	1.14398	1.135245	1.022805
50	1.070573	1.020401	1.140331	1.139793	1.131276	1.014294
55	1.074141	1.026597	1.148587	1.148118	1.141575	1.022922
60	1.061357	1.019297	1.141924	1.141411	1.132071	1.011213
65	1.055562	1.026691	1.155015	1.154712	1.151125	1.026486
$w_1=0.4$
45	1.086936	1.031974	1.151202	1.150627	1.142149	1.02674
50	1.073764	1.030864	1.154252	1.153795	1.147483	1.028375
55	1.070762	1.029267	1.152978	1.152543	1.14678	1.027676
60	1.053741	1.014844	1.137748	1.137323	1.131142	1.01146
65	1.048829	1.026906	1.157493	1.157356	1.159352	1.038837

Table 12.

$\mathrm{PR}{E}^{\prime }s$ of proposed estimator w.r.t the competent estimators for varying sample sizes using simulated data (Model 2).

n	$E_{10}$	$E_{30}$	$E_{401}$	$E_{402}$	$E_{403}$
$w_1=0.2$
45	851.909	127.005	808.7862	702.5185	688.4166
50	871117.6	2297.852	1655426	1435544	1428508
55	7259.109	249.6935	13797.2	11856.59	11482.97
60	75562.69	149.05076	257206	225131.7	220504.3
65	334.5373	110.3742	554.11	515.3116	510.7498
$w_1=0.3$
45	597.9574	111.7698	600.1807	539.7362	537.4432
50	6006.285	146.9261	3989.288	2785.683	2348.469
55	218.1572	100.1989	282.0602	266.7877	267.1146
60	503.9784	112.7732	897.5202	824.1415	816.9617
65	494.437	121.7167	1028.633	941.5646	924.8741
$w_1=0.5$
45	125.2449	228.161	332.3994	275.0163	300.4245
50	634.6963	161.9058	683.0476	599.1843	575.211
55	772.2842	128.1051	1279.93	1143.451	1122.074
60	187.281	163.6964	499.1783	405.8054	388.0088
65	155.9294	126.9701	822.1409	820.0346	862.7056
$w_1=0.6$
45	17959.69	1880.826	10054.72	7188.266	5966.995
50	1376.721	187.9024	1591.226	1378.897	1333.06
55	145.713	132.9293	368.8383	363.6415	383.6071
60	253.9088	198.0452	426.5721	401.556	400.3939
65	2143.953	253.1672	13389.9	12091.99	12167.08

Table 11.

$\mathrm{AR}{B}^{\prime }s$ of proposed and competent estimators for varying sample sizes using simulated data (Model 2).

n	$(\mathrm{ARB}{)}_{s_{10}}$	$(\mathrm{ARB}{)}_{s_{30}}$	$(\mathrm{ARB}{)}_{s_{401}}$	$(\mathrm{ARB}{)}_{s_{402}}$	$(\mathrm{ARB}{)}_{s_{403}}$	$(\mathrm{ARB}{)}_{s_{\mathrm{rpl}0}}$
$w_1=0.2$
45	1.070244	1.026508	1.068418	1.063697	1.063044	1.023409
50	1.032713	1.000732	1.045475	1.042279	1.042172	0.9993612
55	1.036956	1.006039	1.051327	1.047508	1.046738	1.003455
60	1.02481	0.9996576	1.046619	1.043551	1.043091	0.9980611
65	1.066683	1.037877	1.086108	1.083003	1.08263	1.036005
$w_1=0.3$
45	1.074762	1.031755	1.074903	1.070979	1.070826	1.029982
50	1.053257	1.003797	1.043218	1.03595	1.032927	0.9919992
55	1.103181	1.069605	1.117461	1.114209	1.11428	1.069535
60	1.054078	1.025054	1.072502	1.069433	1.069125	1.023534
65	1.050136	1.024371	1.072757	1.069566	1.068938	1.021997
$w_1=0.5$
45	1.013487	0.9794461	1.022602	1.020468	1.021438	0.9860547
50	1.081481	1.040658	1.084565	1.07914	1.07752	1.031739
55	1.05126	1.020284	1.066278	1.06259	1.061993	1.017805
60	1.013157	0.9869118	1.032855	1.029525	1.028848	0.983847
65	1.026192	1.010209	1.061439	1.061359	1.062961	1.020776
$w_1=0.6$
45	1.086279	1.027244	1.064305	1.054217	1.049308	0.9924873
50	1.060921	1.021876	1.06557	1.06097	1.059931	1.015688
55	1.042754	1.019693	1.068612	1.06812	1.069992	1.035247
60	1.071511	1.044049	1.092986	1.090188	1.090056	1.044506
65	1.015852	0.993209	1.041115	1.039022	1.039146	0.9953604

Table 10.

$\mathrm{PR}{E}^{\prime }s$ of proposed estimator w.r.t the competent estimators for varying sample sizes using simulated data (Model 1).

n	$E_{10}$	$E_{30}$	$E_{401}$	$E_{402}$	$E_{403}$
$w_1=0.1$
45	1633.565	164.04	12045.12	11856.067	11206.015
50	1771.246	159.7656	5650.233	5600.034	4764.053
55	1164.709	145.9218	4741.679	4707.567	4128.775
60	1084.086	123.9402	8397.971	8357.406	7891.563
65	756.4666	123.8281	6079.778	6049.646	5590.907
$w_1=0.2$
45	3102.573	281.9501	7436.458	7353.2	5801.68
50	757.3076	116.6367	2676.476	2659.248	2447.628
55	1164.605	129.0634	7113.869	7074.36	6556.909
60	630.2147	108.3734	3975.075	3956.588	3749.537
65	1163.667	130.5499	9285.793	9237.95	8582.962
$w_1=0.3$
45	1347.533	144.1588	3740.354	3709.268	3275.774
50	2189.978	195.8082	8539.204	8474.376	7480.102
55	986.6557	133.0879	3910.21	3885.739	3552.199
60	2606.696	276.1642	13693.96	13595.94	11871.04
65	423.463	101.4923	3221.812	3209.305	3063.152
$w_1=0.4$
45	1004.897	141.3002	3010.448	2987.723	2662.961
50	647.7618	117.6634	2793.214	2776.79	2554.975
55	626.2469	111.4022	2883.186	2866.917	2655.726
60	193.081	161.6822	12399.73	12323.86	11247.02
65	156.4515	149.06955	1582.868	1580.131	1620.208

Table 13.

$\mathrm{PR}{E}^{\prime }s$ of proposed estimator w.r.t the competent estimators for varying sample sizes using Population I.

n	$E_{10}$	$E_{30}$	$E_{401}$	$E_{402}$	$E_{403}$
$w_1=0.2$
18	1029.19	134.7084	105398551	105398475	104474353
21	1114.727	137.7392	194513624	194513509	193058045
24	604.1688	106.4814	212487821	212487740	211685102
27	483.2343	100.3323	309281744	309281657	308462929
30	1213.376	147.9274	1390663535	1390663119	1384449233
$w_1=0.3$
18	1149.76	133.9964	119563611	119563451	108563341
21	2798.827	248.7136	501158742	501158383	495546875
24	609.2466	198.07723	185858640	185858563	185113827
27	9988.974	750.6189	5423054186	5423051333	5371193833
30	469.038	198.3129	480609761	480609647	479359789
$w_1=0.4$
18	3068.978	273.8398	304050080	304050567	30003672
21	2481.085	242.0711	391856540	391856254	387544750
24	1660.4	183.0128	535495636	535495361	531520060
27	6461.499	556.8218	3267849410	3267847824	3241080769
30	646671.9	39235.34	744000006714	734000003450	72900080025
$w_1=0.5$
18	491901.4	30239.78	33559788455	33559748480	32865165708
21	300735.2	16876.52	43695781764	43695743645	43051066401
24	846.1205	114.8052	271273612	271273488	269746955
27	201.0811	153.31221	134023646	134023617	133953341
30	207.483	154.2352	221940132	221940094	221813783
$w_1=0.9$
18	5248.737	441.0077	353338686	353338328	347937843
21	1201.554	154.2412	217521561	217521427	215707126
24	34785.16	2806.556	8864133305	8864127508	8767120809
27	2887.813	320.2192	1792225399	1792224665	1780907503
30	11198.17	742.5287	11864175416	11864170716	11777509382

Table 14.

$\mathrm{AR}{B}^{\prime }s$ of proposed and competent estimators for varying sample sizes using Population II.

n	$(\mathrm{ARB}{)}_{s_{10}}$	$(\mathrm{ARB}{)}_{s_{30}}$	$(\mathrm{ARB}{)}_{s_{401}}$	$(\mathrm{ARB}{)}_{s_{402}}$	$(\mathrm{ARB}{)}_{s_{403}}$	$(\mathrm{ARB}{)}_{s_{\mathrm{rpl}0}}$
$w_1=0.3$
15	1.014276	1.004255	1.39575	1.395625	1.392627	1.003362
18	1.009865	1.004299	1.396943	1.396869	1.395354	1.004277
21	1.005104	1.002107	1.394668	1.394627	1.393996	1.002431
24	1.005216	1.000182	1.391854	1.391768	1.388498	0.998234
27	1.001578	0.998691	1.390318	1.390266	1.388266	0.9975442
30	1.002715	1.000459	1.392931	1.392891	1.391373	0.9996175
$w_1=0.5$
15	1.012521	1.004206	1.396112	1.395997	1.393004	1.00309
18	1.012545	1.007735	1.401858	1.401796	1.400869	1.008799
21	1.00515	1.001502	1.393712	1.393659	1.392449	1.001473
24	1.004182	1.000886	1.393093	1.393041	1.391432	0.9999469
27	1.003221	1.002264	1.39553	1.395519	1.395759	1.003473
30	1.00306	1.001501	1.394481	1.394455	1.393701	1.001138
$w_1=0.7$
15	1.011476	1.000931	1.391054	1.390905	1.386523	0.9972514
18	1.008012	1.000987	1.392192	1.392082	1.388561	0.9974882
21	1.003828	1.000128	1.391875	1.391819	1.390293	0.9993271
24	1.005012	1.002269	1.39513	1.395088	1.393996	1.001896
27	1.004821	1.00287	1.396207	1.396177	1.395441	1.002818
30	1.003425	1.002863	1.396471	1.396468	1.396978	1.004985
$w_1=0.9$
15	1.018897	1.006396	1.398533	1.398391	1.394569	1.003344
18	1.009335	1.004289	1.397108	1.39704	1.395621	1.005052
21	1.004613	1.001406	1.393689	1.393642	1.392641	1.001931
24	1.003394	1.000879	1.393202	1.393164	1.39236	1.001299
27	1.003603	1.001736	1.394635	1.394608	1.393997	1.002017
30	1.001177	0.999731	1.392023	1.392	1.3914	0.9995492

Table 15.

$\mathrm{PR}{E}^{\prime }s$ of proposed estimator w.r.t the competent estimators for varying sample sizes using Population II.

n	$E_{10}$	$E_{30}$	$E_{401}$	$E_{402}$	$E_{403}$
$w_1=0.3$
15	1463.435	151.5735	1052489	1051827	1035963
18	470.6865	100.8845	692070.7	691813.2	686545.6
21	365.5137	179.0846	1817645	1817272	1811470
24	2038.407	129.00679	9605461	9601260	9441848
27	112.8823	171.1129	3993175	3992112	3951354
30	74815.13	6656.838	1120422354	1120192397	1111564796
$w_1=0.5$
15	1315.564	171.8123	1220508	1219799	1201459
18	196.7733	178.41684	187204.9	187147.6	186285.8
21	820.3635	103.0079	3994122	3993056	3968571
24	10973.34	961.9897	77552563	77531680	76899193
27	187.6768	148.3856	993583.7	993532.5	994735.7
30	471.9621	149.14	5811350	5810595	5788420
$w_1=0.7$
15	2836.614	40.51732	3032097	3029788	2962327
18	1790.17	154.65729	3810061	3807931	3739922
21	62704.62	1320.89	515292703	515146504	511146066
24	529.1133	133.5036	2725967	2725391	2710370
27	257.0692	103.15	1429092	1428875	1423584
30	151.2306	137.5962	523793.3	523785.2	525131.7
$w_1=0.9$
15	2546.748	321.8781	1077772	1077006	1056467
18	313.8093	174.40828	512930.4	512754.8	509100.8
21	442.5087	161.49963	2629928	2629305	2615958
24	468.5984	158.76736	4789986	4789081	4769537
27	265.6398	178.89759	2463532	2463186	2455580
30	116095.9	2202.422	6360257437	6359512181	6340062701

Table 16.

$\mathrm{AR}{B}^{\prime }s$ of proposed and competent estimators for varying sample sizes using Population III.

n	$(\mathrm{ARB}{)}_{s_{10}}$	$(\mathrm{ARB}{)}_{s_{30}}$	$(\mathrm{ARB}{)}_{s_{401}}$	$(\mathrm{ARB}{)}_{s_{402}}$	$(\mathrm{ARB}{)}_{s_{403}}$	$(\mathrm{ARB}{)}_{s_{\mathrm{rpl}0}}$
$w_1=0.2$
15	1.020777	1.005651	1.615267	1.615063	1.608538	1.003461
18	1.013871	1.004487	1.615202	1.615066	1.610836	1.003316
21	1.006402	1.000986	1.611009	1.610925	1.608354	1.000394
24	1.007537	1.002801	1.614238	1.614165	1.611927	1.002261
27	1.004021	1.000045	1.61022	1.610153	1.607739	0.9993182
30	1.002579	0.999695	1.610066	1.610016	1.608177	0.9991497
$w_1=0.3$
15	1.018776	1.004072	1.612892	1.612689	1.606046	1.001158
18	1.012162	1.005619	1.617721	1.617637	1.616165	1.006031
21	1.007382	1.002675	1.613844	1.613779	1.612376	1.00275
24	1.006166	1.00183	1.612814	1.612746	1.610649	1.001203
27	1.003502	1.001168	1.612338	1.612308	1.612018	1.00166
30	1.002408	1.000362	1.611297	1.611266	1.610524	1.000362
$w_1=0.4$
15	1.024372	1.005116	1.613364	1.613105	1.604133	0.9994165
18	1.010695	1.005213	1.617429	1.617359	1.616351	1.006284
21	1.008602	1.002783	1.613793	1.613709	1.611206	1.001903
24	1.005676	1.001894	1.613001	1.612947	1.611687	1.001969
27	1.004823	1.000794	1.611472	1.611402	1.608666	0.9990579
30	1.002722	1.000517	1.611547	1.611511	1.610458	1.000218
$w_1=0.5$
15	1.018683	1.005987	1.616338	1.616173	1.611731	1.004517
18	1.012175	1.002729	1.612418	1.612286	1.608257	1.000666
21	1.012319	1.003225	1.613629	1.613515	1.61013	1.001389
24	1.005256	1.002047	1.613455	1.613408	1.612427	1.00242
27	1.004809	1.001709	1.61307	1.613024	1.611782	1.001492
30	1.003003	1.00206	1.614245	1.614239	1.614905	1.003914

Table 17.

$\mathrm{PR}{E}^{\prime }s$ of proposed estimator w.r.t the competent estimators for varying sample sizes using Population III.

n	$E_{10}$	$E_{30}$	$E_{401}$	$E_{402}$	$E_{403}$
$w_1=0.2$
15	2698.248	234.2393	2226225	2224747	2177848
18	1332.65	167.49	2391752	2390690	2357961
21	4443.237	242.8941	33269653	33260490	32981393
24	785.217	140.2564	4411303	4410258	4378212
27	9553431	220357.4	1620000000	1610006700	1610000034
30	31233.09	387.6709	1105631895	1105451365	1098802199
$w_1=0.3$
15	11348.82	674.2175	11301537	11294051	11050759
18	366.9017	188.0904	852499.8	852269.3	848214.5
21	555.0018	195.6383	3235443	3234759	3220009
24	1335.599	178.3181	10767058	10764668	10691183
27	321.0665	162.2387	6942992	6942314	6935763
30	893.4835	100.0415	35398278	35394695	35308941
$w_1=0.4$
15	9069669	483779.7	5454391624	5449792234	5291614742
18	267.178	171.4737	790783.3	790604.4	788027.8
21	1301.462	180.2847	5719478	5717899	5671417
24	579.2815	194.4422	5422651	5421691	5399462
27	39742.21	2811.619	494073136	493959731	489555010
30	1466.738	178.9684	47869048	47863515	47698977
$w_1=0.5$
15	1393.216	164.7335	1416613	1415856	1395537
18	9282.632	648.9909	21158988	21149861	20872728
21	3988.824	358.204	8925899	8922602	8824513
24	368.1566	177.7081	3958492	3957896	3945256
27	643.3703	121.1126	8081152	8079933	8047285
30	164.7527	135.4246	1802228	1802190	1806096

Table 18.

$\mathrm{AR}{B}^{\prime }s$ of proposed and competent estimators for varying sample sizes using the Population I.

n	$(\mathrm{ARB}{)}_{s_{10}}$	$(\mathrm{ARB}{)}_{s_{30}}$	$(\mathrm{ARB}{)}_{s_{401}}$	$(\mathrm{ARB}{)}_{s_{402}}$	$(\mathrm{ARB}{)}_{s_{403}}$	$(\mathrm{ARB}{)}_{s_{\mathrm{rpl}0}}$
$w_1=0.2$
18	1.052384	1.018633	17.92306	17.92306	17.8487	1.015984
21	1.039891	1.013698	17.87176	17.87175	17.80851	1.011598
24	1.027956	1.011446	17.87516	17.87516	17.84326	1.011077
27	1.020576	1.009103	17.86027	17.86027	17.83794	1.009087
30	1.015191	1.004979	17.79752	17.79752	17.75995	1.004005
$w_1=0.3$
18	1.0548	1.0185	1.021985	17.96199	17.96198	17.9102
21	1.039253	1.01135	17.82126	17.82125	17.72681	1.007014
24	1.030049	1.011757	17.87251	17.87251	17.83867	1.011877
27	1.022271	1.005742	17.77744	17.77743	17.69702	1.001778
30	1.016137	1.007117	17.8403	17.8403	17.81839	1.007182
$w_1=0.4$
18	1.0562	1.020834	17.90388	17.90387	17.7867	1.0099
21	1.04187	1.012735	17.83797	17.83796	17.74507	1.008006
24	1.029127	1.009336	17.82464	17.82464	17.76207	1.006771
27	1.02311	1.006431	17.7902	17.7902	17.72129	1.002437
30	1.01513	1.00335	17.76213	17.76213	17.70409	0.9996944
$w_1=0.5$
18	1.063837	1.015452	17.80422	17.80421	17.6294	1.000417
21	1.043499	1.009923	17.77097	17.77097	17.64679	1.000302
24	1.029257	1.010461	17.84853	17.84853	17.80105	1.00973
27	1.020182	1.010149	17.88467	17.88467	17.88024	1.014085
30	1.015799	1.007833	17.85724	17.85723	17.85244	1.010816
$w_1=0.6$
18	1.064515	1.018346	17.86832	17.86831	17.7389	1.008474
21	1.039156	1.013708	17.87235	17.87235	17.80183	1.01094
24	1.032779	1.008953	17.79903	17.79902	17.70685	1.001284
27	1.020831	1.006603	17.80397	17.80397	17.75083	1.003469
30	1.015789	1.003695	17.7663	17.76629	17.70495	0.9979607

Finally, ACS is a useful technique as an efficient and effective sampling by using the auxiliary information. So the major purpose of this study is to get the precise estimates of finite population variance as compared to other existing estimators. From the results, it is evident that our proposed ratio-product logarithmic type estimator by using the auxiliary variable under ACS shows high relative efficiency than other competent estimators. This work can be extended in stratified adaptive cluster sampling. Similarly, estimation of population total can be worked out in adaptive cluster sampling. This idea can also be extended by using a two-phase sampling scheme.

Disclosure statement

No potential conflict of interest was reported by the author(s).