Enhanced Ratio-Type Estimators in Adaptive Cluster Sampling Using Jackknife Method

1. Introduction

Sampling methods are employed when it is impractical to collect data from an entire population. For statistical inferences to be valid, the selected sample must be obtained through probability-based sampling techniques. One of the most commonly used probability sampling methods is simple random sampling. The population mean for a variable of interest, denoted by${\mu }_y$, is typically estimated using the sample mean $\overline{y}={\displaystyle {\sum }_{i=1}^n{y}_i}/n$, where n is the sample size. This estimator is known to be unbiased. A primary objective in sampling theory is to improve the efficiency of such estimators.

In many cases, the estimation efficiency can be enhanced by utilizing an auxiliary variable that is correlated with the variable of interest. When both variables exhibit a high positive correlation, the ratio estimator, which incorporates the auxiliary variable’s mean, is widely adopted. Numerous studies have proposed improvements to the ratio estimator in the context of simple random sampling. For example, Sisodia and Dwivedi [1] introduced a modified ratio estimator based on the coefficient of variation of the auxiliary variable. Singh and Tailor [2] proposed an estimator that incorporates the population correlation coefficient between the target and auxiliary variables. Yadav et al. [3] developed ratio-cum-product estimators, while Jerajuddin and Kishun [4] enhanced ratio estimators by considering sample size. Soponviwatkul and Lawson [5] proposed further refinements by incorporating the coefficient of variation, correlation coefficient, and regression coefficient.

However, in situations where the population is both rare and clustered, simple random sampling may be suboptimal. To address this, Thompson [6] introduced adaptive cluster sampling (ACS) in 1990. In ACS, an initial sample is selected using simple random sampling without replacement. If a unit in this initial sample satisfies a pre-specified condition for the variable of interest, its neighboring units are added to the sample. This expansion continues iteratively until no additional units meet the condition. The collection of initial and subsequently added units forms a network. Units that do not meet the condition are referred to as edge units. The union of a network and its edge units constitutes a cluster. If the initial unit fails to satisfy the condition, it remains a singleton network. For this study, neighborhoods are defined as the four orthogonally adjacent units (up, down, left, and right), with mutual neighborhood relationships assumed.

Thompson also proposed an estimator and demonstrated that ACS yields improved efficiency in clustered populations. Analogous to simple random sampling, incorporating auxiliary variable information in ACS can further improve estimator performance. Chao [7] introduced a ratio estimator for ACS, while Dryver and Chao [8] introduced modified ratio estimators. Chutiman and Kum-phon [9] presented a ratio estimator using two auxiliary variables. Chutiman[10] and Yadav et al. [11] proposed ratio estimators based on population parameters, including the coefficient of variation, kurtosis, skewness, and correlations with auxiliary variables. Chaudhry and Hanif [12,13] proposed generalized exponential-type estimators, while Bhat et al. [14] developed a generalized class of ratio-type estimators.

Increasing the auxiliary variable information is the primary focus of the development of the parameter estimators discussed above. This study presents the development of Chao's ratio-type estimator in adaptive cluster sampling, which uses the Jackknife method to leverage data from a single auxiliary variable, specifically the auxiliary variable's mean. Section 2 outlines relevant estimators in ACS, Section 3 introduces the proposed Jackknife-based estimators, Section 4 presents simulation results, and Section 5 concludes the study.

2. Adaptive Cluster Sampling

In adaptive cluster sampling, an initial sample of units is selected using simple random sampling without replacement.

Let $n$ represent the initial sample size and $\nu$ be the final sample size. Let ${\psi }_i$ denote the network that includes unit $i$, and let $m_i$ represent the number of units in that network.

Let ${\left(w_y\right)}_i$ be the average of the y-value in the network that includes the initial sample unit $i$, that is, ${\left(w_y\right)}_i={\displaystyle \frac{1}{m_i}}{\displaystyle \sum _{j\in {\psi }_i}{y}_j}$.

The Hansen-Hurwitz estimator of the population mean for the variable of interest is [15]:

$${\overline{w}}_y={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(w_y\right)}_i}.$$

(1)

The mean square error (MSE) of ${\overline{w}}_y$ is:

$$MSE\left({\overline{w}}_y\right)=\left(1-f_n\right){\displaystyle \frac{S_{wy}^2}{n}},$$

(2)

where $f_n={\displaystyle \frac{n}{N}}$ and $S_{wy}^2={\displaystyle \frac{1}{\left(N-1\right)}}{\displaystyle \sum _{i=1}^n{\left[{\left(w_y\right)}_i-{\mu }_y\right]}^2}$.

Let $x$ be the auxiliary variable. The population mean of $x$ is ${\mu }_x$ and ${\left(w_x\right)}_i$ is the average of the auxiliary variable in the network that includes the initial sample unit $i$, that is, ${\left(w_x\right)}_i={\displaystyle \frac{1}{m_i}}{\displaystyle \sum _{j\in {\psi }_i}{x}_j}$. The modified Hansen-Hurwitz estimator of the population mean of the auxiliary variable is:

$${\overline{w}}_x={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(w_x\right)}_i}$$

(3)

Let $R$ be the population ratio between $y$ and $x$, $R={\displaystyle \frac{{\mu }_y}{{\mu }_x}}$ . Chao [7] introduced the ratio estimator of the population mean, which is

$${\overline{w}}_R=\left({\displaystyle \frac{{\overline{w}}_y}{{\overline{w}}_x}}\right){\mu }_x=\widehat{R}{\mu }_x,$$

(4)

where ${\overline{w}}_R$ is a biased estimator of ${\mu }_y$. The bias of ${\overline{w}}_R$ is:

$$Bias\left({\overline{w}}_R\right)={\displaystyle \frac{{\mu }_y}{n}}\left(1-f_n\right)C_{wx}\left(C_{wx}-{\rho }_{wx\cdot wy}{C}_{wy},\right)$$

(5)

where ${\rho }_{wx\cdot wy}={\displaystyle \frac{S_{wx\cdot wy}}{S_{wx}{S}_{wy}}}$, $C_{wx}={\displaystyle \frac{S_{wx}}{{\mu }_x}}$, $S_{wy}^2={\displaystyle \frac{1}{\left(N-1\right)}}{\displaystyle \sum _{i=1}^n{\left[{\left(w_y\right)}_i-{\mu }_y\right]}^2}$, $S_{wx}^2={\displaystyle \frac{1}{\left(N-1\right)}}{\displaystyle \sum _{i=1}^n{\left[{\left(w_x\right)}_i-{\mu }_x\right]}^2}$ and $S_{wx\cdot wy}={\displaystyle \frac{1}{\left(N-1\right)}}{\displaystyle \sum _{i=1}^n\left[{\left(w_x\right)}_i-{\mu }_x\right]\left[{\left(w_y\right)}_i-{\mu }_y\right]}$.

The mean square error (MSE) of ${\overline{w}}_R$ is:

$$MSE\left({\overline{w}}_R\right)={\displaystyle \frac{{\mu }_y^2}{n}}\left(1-f_n\right)\left(C_{wx}^2+C_{wy}^2-2{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}\right)$$

(6)

3. Proposed Estimators in Adaptive Cluster Sampling Using the Jackknife Method

Motivated by Banerjee and Tiwari [16], Quenouille’s Jackknife method [17] was applied to propose the estimators. The sample network of size $n$ is randomly partitioned into two groups, each of size m = n/2.

The proposed estimators are:

1) ${{\overline{w}}^{\prime }}_R={\displaystyle \frac{{\overline{w}}_R^{\left(1\right)}+{\overline{w}}_R^{\left(2\right)}}{2}}$, (7)

where ${\overline{w}}_R^{\left(i\right)}={\displaystyle \frac{{\overline{w}}_y^{\left(i\right)}}{{\overline{w}}_x^{\left(i\right)}}}{\mu }_x$ and $i=1,2$, where ${\overline{w}}_y^{\left(i\right)}$ and ${\overline{w}}_x^{\left(i\right)}$ are the sample means based on group $i$ of size $m$, for the y-variable and x-variable, respectively.

2) ${{\overline{w}}^{″}}_R={\displaystyle \frac{{\overline{w}}_R-K{{\overline{w}}^{\prime }}_R}{1-K}}$, where $K={\displaystyle \frac{Bias\left({\overline{w}}_R\right)}{Bias\left({{\overline{w}}^{\prime }}_R\right)}}$ . (8)

3) ${\overline{w}}_{JK}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\overline{w}}_{R\left(i\right)}}$, (9)

where ${\overline{w}}_{R\left(i\right)}={\displaystyle \frac{{\overline{w}}_{y\left(i\right)}}{{\overline{w}}_{x\left(i\right)}}}{\mu }_x$ is the ratio estimator of the population mean for the y-variable in the delete network $i$, and ${\overline{w}}_{y\left(i\right)}$, ${\overline{w}}_{x\left(i\right)}$ are the modified Hansen-Hurwitz estimators of the population mean in the delete network $i$ for the y-variable and x-variable, respectively.

The bias and MSE of each estimator are as follows:

The first estimator: ${{\overline{w}}^{\prime }}_R={\displaystyle \frac{{\overline{w}}_R^{\left(1\right)}+{\overline{w}}_R^{\left(2\right)}}{2}}$

Let ${\epsilon }_0^{\left(i\right)}={\displaystyle \frac{{\overline{w}}_y^{\left(i\right)}-{\mu }_y}{{\mu }_y}}$ and ${\epsilon }_1^{\left(i\right)}={\displaystyle \frac{{\overline{w}}_x^{\left(i\right)}-{\mu }_x}{{\mu }_x}}$.

Where i = 1:

$${\overline{w}}_R^{\left(1\right)}={\displaystyle \frac{{\overline{w}}_y^{\left(1\right)}}{{\overline{w}}_x^{\left(1\right)}}}{\mu }_x=\left(1+{\epsilon }_0^{\left(1\right)}\right){\left(1+{\epsilon }_1^{\left(1\right)}\right)}^{-1}{\mu }_y$$

Assuming $\left|{\epsilon }_1^{\left(1\right)}\right|<1$, the term ${\left(1+{\epsilon }_1^{\left(1\right)}\right)}^{-1}$can be expanded as an infinite series.

$${\overline{w}}_R^{\left(1\right)}={\mu }_y\left(1+{\epsilon }_0^{\left(1\right)}\right)\left[1-{\epsilon }_1^{\left(1\right)}+{\left({\epsilon }_1^{\left(1\right)}\right)}^2-\dots \right]$$

$$={\mu }_y\left[1+{\epsilon }_0^{\left(1\right)}-{\epsilon }_1^{\left(1\right)}+{\left({\epsilon }_1^{\left(1\right)}\right)}^2-{\epsilon }_0^{\left(1\right)}{\epsilon }_1^{\left(1\right)}\dots \right]$$

Therefore, ${\overline{w}}_R^{\left(1\right)}-{\mu }_y\approx {\mu }_y\left[{\epsilon }_0^{\left(1\right)}-{\epsilon }_1^{\left(1\right)}+{\left({\epsilon }_1^{\left(1\right)}\right)}^2-{\epsilon }_0^{\left(1\right)}{\epsilon }_1^{\left(1\right)}\right]$.

The bias of ${\overline{w}}_R^{\left(1\right)}$ is given by:

$$E\left[{\overline{w}}_R^{\left(1\right)}-{\mu }_y\right]={\mu }_{y}E\left[{\epsilon }_0^{\left(1\right)}-{\epsilon }_1^{\left(1\right)}+{\left({\epsilon }_1^{\left(1\right)}\right)}^2-{\epsilon }_0^{\left(1\right)}{\epsilon }_1^{\left(1\right)}\right],$$

where $E\left({\epsilon }_0^{\left(1\right)}\right)=E\left({\epsilon }_1^{\left(1\right)}\right)=0$, $E\left[{\left({\epsilon }_1^{\left(1\right)}\right)}^2\right]={\displaystyle \frac{\left(1-f_m\right)}{m}}{C}_{wx}^2$

$$E\left({\epsilon }_0^{\left(1\right)}{\epsilon }_1^{\left(1\right)}\right)={\displaystyle \frac{\left(1-f_m\right)}{m}}{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}\mathrm{and}{f}_m={\displaystyle \frac{m}{N}}$$

Therefore, $Bias\left({\overline{w}}_R^{\left(1\right)}\right)=E\left[{\overline{w}}_R^{\left(1\right)}-{\mu }_y\right]={\displaystyle \frac{\left(1-f_m\right)}{m}}{\mu }_y{C}_{wx}\left(C_{wx}-{\rho }_{wx\cdot wy}{C}_{wy}\right)$.

The bias of ${\overline{w}}_R^{\left(2\right)}$ is derived in the same way as that of ${\overline{w}}_R^{\left(1\right)}$, and the $Bias\left({\overline{w}}_R^{\left(2\right)}\right)$ is equal to the $Bias\left({\overline{w}}_R^{\left(1\right)}\right)$ .

$$E\left({{\overline{w}}^{\prime }}_R-{\mu }_y\right)=E\left[{\displaystyle \frac{{\overline{w}}_R^{\left(1\right)}+{\overline{w}}_R^{\left(2\right)}}{2}}-{\mu }_y\right]$$

$$={\displaystyle \frac{1}{2}}\left[E\left({\overline{w}}_R^{\left(1\right)}-{\mu }_y\right)+E\left({\overline{w}}_R^{\left(2\right)}-{\mu }_y\right)\right]$$

Therefore, the bias of ${{\overline{w}}^{\prime }}_R$ is:

$$Bias\left({{\overline{w}}^{\prime }}_R\right)=E\left({{\overline{w}}^{\prime }}_R-{\mu }_y\right)={\displaystyle \frac{\left(1-f_m\right)}{m}}{\mu }_y{C}_{wx}\left(C_{wx}-{\rho }_{wx\cdot wy}{C}_{wy}\right)$$

(10)

For the MSE of ${{\overline{w}}^{\prime }}_R$,

$$MSE\left({{\overline{w}}^{\prime }}_R\right)=E{\left({{\overline{w}}^{\prime }}_R-{\mu }_y\right)}^2=E{\left[{\displaystyle \frac{{\overline{w}}_R^{\left(1\right)}+{\overline{w}}_R^{\left(2\right)}}{2}}-{\mu }_y\right]}^2$$

$$={\displaystyle \frac{1}{4}}\left\{E{\left({\overline{w}}_R^{\left(1\right)}-{\mu }_y\right)}^2+E{\left({\overline{w}}_R^{\left(2\right)}-{\mu }_y\right)}^2+2E\left[\left({\overline{w}}_R^{\left(1\right)}-{\mu }_y\right)\left({\overline{w}}_R^{\left(2\right)}-{\mu }_y\right)\right]\right\}$$

where $E{\left({\overline{w}}_R^{\left(1\right)}-{\mu }_y\right)}^2=MSE\left({\overline{w}}_R^{\left(1\right)}\right)={\displaystyle \frac{{\mu }_y^2}{m}}\left(1-f_m\right)\left(C_{wx}^2+C_{wy}^2-2{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}\right)$

$$E{\left({\overline{w}}_R^{\left(2\right)}-{\mu }_y\right)}^2=MSE\left({\overline{w}}_R^{\left(2\right)}\right)={\displaystyle \frac{{\mu }_y^2}{m}}\left(1-f_m\right)\left(C_{wx}^2+C_{wy}^2-2{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}\right)$$

$$E\left[\left({\overline{w}}_R^{\left(1\right)}-{\mu }_y\right)\left({\overline{w}}_R^{\left(2\right)}-{\mu }_y\right)\right]=COV\left({\overline{w}}_R^{\left(1\right)},{\overline{w}}_R^{\left(2\right)}\right).$$

The MSE of ${{\overline{w}}^{\prime }}_R$ is:

$$MSE\left({{\overline{w}}^{\prime }}_R\right)={\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{{\mu }_y^2}{m}}\left(1-f_m\right)\left(C_{wx}^2+C_{wy}^2-2{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}\right)+COV\left({\overline{w}}_R^{\left(1\right)},{\overline{w}}_R^{\left(2\right)}\right)\right\}$$

(11)

The second estimator: ${{\overline{w}}^{″}}_R={\displaystyle \frac{{\overline{w}}_R-K{{\overline{w}}^{\prime }}_R}{1-K}}$, where $K={\displaystyle \frac{Bias\left({\overline{w}}_R\right)}{Bias\left({{\overline{w}}^{\prime }}_R\right)}}$

From$Bias\left({\overline{w}}_R\right)={\displaystyle \frac{{\mu }_y}{n}}\left(1-f_n\right)C_{wx}\left(C_{wx}-{\rho }_{wx\cdot wy}{C}_{wy}\right)$ and $Bias\left({{\overline{w}}^{\prime }}_R\right)={\displaystyle \frac{\left(1-f_m\right)}{m}}{\mu }_y{C}_{wx}\left(C_{wx}-{\rho }_{wx\cdot wy}{C}_{wy}\right)$, it follows that $K={\displaystyle \frac{\left(1-f_n\right)/n}{\left(1-f_m\right)/m}}={\displaystyle \frac{N-n}{2N-n}}$, and ${{\overline{w}}^{″}}_R$ is an unbiased estimator of ${\mu }_y$to the first order of approximation (based on Banerjee and Tiwari [15]).

For the MSE of ${{\overline{w}}^{″}}_R$,

$$MSE\left({{\overline{w}}^{″}}_R\right)=E{\left({{\overline{w}}^{″}}_R-{\mu }_y\right)}^2=E{\left({\displaystyle \frac{{\overline{w}}_R-K{{\overline{w}}^{\prime }}_R}{1-K}}-{\mu }_y\right)}^2$$

$$={\displaystyle \frac{1}{{\left(1-K\right)}^2}}\left\{E{\left({\overline{w}}_R-{\mu }_y\right)}^2+K^{2}E{\left({{\overline{w}}^{\prime }}_R-{\mu }_y\right)}^2-2KE\left[\left({\overline{w}}_R-{\mu }_y\right)\left({{\overline{w}}^{\prime }}_R-{\mu }_y\right)\right]\right\},$$

where $E{\left({\overline{w}}_R-{\mu }_y\right)}^2=MSE\left({\overline{w}}_R\right)={\displaystyle \frac{{\mu }_y^2}{n}}\left(1-f_n\right)\left(C_{wx}^2+C_{wy}^2-2{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}\right)$,

$$E{\left({{\overline{w}}^{\prime }}_R-{\mu }_y\right)}^2=MSE\left({{\overline{w}}^{\prime }}_R\right)={\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{{\mu }_y^2}{m}}\left(1-f_m\right)\left(C_{wx}^2+C_{wy}^2-2{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}\right)+COV\left({\overline{y}}_R^{\left(1\right)},{\overline{y}}_R^{\left(2\right)}\right)\right\},$$

and

$$E\left[\left({\overline{w}}_R-{\mu }_y\right)\left({{\overline{w}}^{\prime }}_R-{\mu }_y\right)\right]={\displaystyle \frac{1}{2}}\left\{COV\left({\overline{w}}_R,{\overline{w}}_R^{\left(1\right)}\right)+COV\left({\overline{w}}_R,{\overline{w}}_R^{\left(2\right)}\right)\right\}.$$

Therefore, The MSE of ${{\overline{w}}^{″}}_R$ is

$$MSE\left({{\overline{w}}^{″}}_R\right)={\displaystyle \frac{1}{{\left(1-K\right)}^2}}\left\{MSE\left({\overline{w}}_R\right)+K^{2}MSE\left({{\overline{w}}^{\prime }}_R\right)-K\left[COV\left({\overline{w}}_R,{\overline{w}}_R^{\left(1\right)}\right)+COV\left({\overline{w}}_R,{\overline{w}}_R^{\left(2\right)}\right)\right]\right\}.$$

(12)

The Third estimator: ${\overline{w}}_{JK}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\overline{w}}_{R\left(i\right)}}$

Let ${\epsilon }_{0\left(i\right)}={\displaystyle \frac{{\overline{w}}_{y\left(i\right)}-{\mu }_y}{{\mu }_y}}$ and ${\epsilon }_{1\left(i\right)}={\displaystyle \frac{{\overline{w}}_{x\left(i\right)}-{\mu }_x}{{\mu }_x}}$ .

Where i =1:

$${\overline{w}}_{R\left(1\right)}={\displaystyle \frac{{\overline{w}}_{x\left(1\right)}}{{\overline{w}}_{y\left(1\right)}}}{\mu }_x=\left(1+{\epsilon }_{0\left(1\right)}\right){\left(1+{\epsilon }_{1\left(1\right)}\right)}^{-1}{\mu }_y$$

Assuming $\left|{\epsilon }_{1\left(1\right)}\right|<1$, the term ${\left(1+{\epsilon }_{1\left(1\right)}\right)}^{-1}$can be expanded as an infinite series.

$${\overline{w}}_{R\left(1\right)}={\mu }_y\left(1+{\epsilon }_{0\left(1\right)}\right)\left[1-{\epsilon }_{1\left(1\right)}+{\left({\epsilon }_{1\left(1\right)}\right)}^2-\dots \right]$$

$$={\mu }_y\left[1+{\epsilon }_{0\left(1\right)}-{\epsilon }_{1\left(1\right)}+{\left({\epsilon }_{1\left(1\right)}\right)}^2-{\epsilon }_{0\left(1\right)}{\epsilon }_{1\left(1\right)}\dots \right]$$

$${\overline{w}}_{R\left(1\right)}-{\mu }_y\approx {\mu }_y\left[{\epsilon }_{0\left(1\right)}-{\epsilon }_{1\left(1\right)}+{\left({\epsilon }_{1\left(1\right)}\right)}^2-{\epsilon }_{0\left(1\right)}{\epsilon }_{1\left(1\right)}\right].$$

The bias of ${\overline{w}}_{R\left(1\right)}$ is given by:

$$E\left[{\overline{w}}_{R\left(1\right)}-{\mu }_y\right]={\mu }_{y}E\left[{\epsilon }_{0\left(1\right)}-{\epsilon }_{1\left(1\right)}+{\left({\epsilon }_{1\left(1\right)}\right)}^2-{\epsilon }_{0\left(1\right)}{\epsilon }_{1\left(1\right)}\right],$$

where $E\left({\epsilon }_{0\left(1\right)}\right)=E\left({\epsilon }_{1\left(1\right)}\right)=0$, $E\left[{\left({\epsilon }_{1\left(1\right)}\right)}^2\right]={\displaystyle \frac{\left(1-f_{\left(n-1\right)}\right)}{n-1}}{C}_{wx}^2$, $f_{\left(n-1\right)}={\displaystyle \frac{n-1}{N}}$,

and $E\left({\epsilon }_{0\left(1\right)}{\epsilon }_{1\left(1\right)}\right)={\displaystyle \frac{\left(1-f_{\left(n-1\right)}\right)}{n-1}}{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}$.

$$Bias\left({\overline{w}}_{R\left(1\right)}\right)=E\left[{\overline{w}}_{R\left(1\right)}-{\mu }_y\right]={\displaystyle \frac{\left(1-f_{\left(n-1\right)}\right)}{n-1}}{\mu }_y{C}_{wx}\left(C_{wx}-{\rho }_{wx\cdot wy}{C}_{wy}\right).$$

The bias of ${\overline{w}}_{R\left(2\right)}$ is derived in the same way as that of ${\overline{w}}_{R\left(1\right)}$, and the $Bias\left({\overline{w}}_{R\left(2\right)}\right)$ is equal to the $Bias\left({\overline{w}}_{R\left(1\right)}\right)$. Therefore, the bias of ${\overline{w}}_{JK}$ is:

$$Bias\left({\overline{w}}_{JK}\right)=E\left[{\overline{w}}_{JK}-{\mu }_y\right]={\displaystyle \frac{1}{n}}\left[{\displaystyle \sum _{i=1}^{n}E\left({\overline{w}}_{R\left(i\right)}-{\mu }_y\right)}\right]$$

$$={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left\{\frac{\left(1-f_{\left(n-1\right)}\right)}{n-1}{\mu }_y{C}_{wx}\left(C_{wx}-{\rho }_{wx\cdot wy}{C}_{wy}\right)\right\}}$$

$$Bias\left({\overline{w}}_{JK}\right)={\displaystyle \frac{\left(1-f_{\left(n-1\right)}\right)}{n-1}}{\mu }_y{C}_{wx}\left(C_{wx}-{\rho }_{wx\cdot wy}{C}_{wy}\right).$$

(13)

For the MSE of ${\overline{w}}_{JK}$,

$$MSE\left({\overline{w}}_{JK}\right)=E{\left({\overline{w}}_{JK}-{\mu }_y\right)}^2=E{\left[{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\overline{w}}_{R\left(i\right)}}-{\mu }_y\right]}^2$$

$$={\displaystyle \frac{1}{n^2}}E{\left[{\displaystyle \sum _{i=1}^n\left({\overline{w}}_{R\left(i\right)}-{\mu }_y\right)}\right]}^2$$

$$={\displaystyle \frac{1}{n^2}}\left\{{\displaystyle \sum _{i=1}^{n}E{\left({\overline{w}}_{R\left(i\right)}-{\mu }_y\right)}^2}+{\displaystyle \sum _{i\ne j=1}^n{\displaystyle \sum _{j=1}^{n}E\left[\left({\overline{w}}_{R\left(i\right)}-{\mu }_y\right)\left({\overline{w}}_{R\left(j\right)}-{\mu }_y\right)\right]}}\right\},$$

where $E{\left({\overline{w}}_{R\left(i\right)}-{\mu }_y\right)}^2=MSE\left({\overline{w}}_{R\left(i\right)}\right)={\displaystyle \frac{{\mu }_y^2}{n-1}}\left(1-f_{\left(n-1\right)}\right)\left(C_{wx}^2+C_{wy}^2-2{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}\right)$and $E\left[\left({\overline{w}}_{R\left(i\right)}-{\mu }_y\right)\left({\overline{w}}_{R\left(j\right)}-{\mu }_y\right)\right]=COV\left({\overline{w}}_{R\left(i\right)},{\overline{w}}_{R\left(j\right)}\right)$.

Therefore, the MSE of ${\overline{w}}_{JK}$is :

$$MSE\left({\overline{w}}_{JK}\right)={\displaystyle \frac{1}{n^2}}\left\{{\displaystyle \frac{{\mu }_y^2}{n-1}}\left(1-f_{\left(n-1\right)}\right)\left(C_{wx}^2+C_{wy}^2-2{\rho }_{wx\cdot wy}{C}_{wx}{C}_{wy}\right)+{\displaystyle \sum _{i\ne j=1}^n{\displaystyle \sum _{j=1}^{n}COV\left({\overline{w}}_{R\left(i\right)},{\overline{w}}_{R\left(j\right)}\right)}}\right\}.$$

(14)

4. Simulation Study and Discussion

The populations for both the auxiliary variable and the variable of interest, as used in Chao [7], were generated using a linked-pairs process in conjunction with a bivariate Poisson cluster process. The resulting population comprised a 20 × 20 grid, yielding a total of 400 units. The mean of the variable of interest (y) in the population was 0.635, and the Pearson correlation coefficient between the auxiliary variable (x) and y was 0.707035. For each simulation iteration, initial sample units were selected via simple random sampling without replacement. The expansion criterion for adaptive cluster sampling was defined by the condition $C=\left\{y:y>0\right\}$. A total of 10, 000 iterations were conducted for each estimator under investigation. The number of initial networks n was varied across the values 4, 8, 10, 16, 20, 26, 30, 40, 50, 100, and 200. The expected final sample size $E\left(\nu \right)$ was computed as follows: $E\left(\nu \right)={\displaystyle \frac{1}{10,000}}{\displaystyle \sum _{i=1}^{10,000}{\nu }_i}$.

The estimated absolute relative bias was defined as:

$$ARB\left(\overline{w}\right)={\displaystyle \frac{1}{10,000}}{\displaystyle \sum _{i=1}^{10,000}\frac{\left|{\overline{w}}_i-{\mu }_y\right|}{{\mu }_y}}$$

The estimated mean square error of the estimator was defined as:

$$M\widehat{S}E\left(\overline{w}\right)={\displaystyle \frac{1}{10,000}}{\displaystyle \sum _{i=1}^{10,000}{\left({\overline{w}}_i-{\mu }_y\right)}^2}$$

The percentage relative efficiency of the proposed estimator, compared with ${\overline{w}}_y$, was defined as: $PRE{\left(\overline{w}\right)}_{pro}={\displaystyle \frac{M\widehat{S}E\left({\overline{w}}_y\right)}{M\widehat{S}E\left({\overline{w}}_{pro}\right)}}\times 100$ .

The estimated absolute relative bias, estimated mean square error (MSE), and percentage relative efficiency of the estimators are presented in Table 1, Table 2 and Table 3.

5. Conclusions

Adaptive cluster sampling (ACS) is particularly effective for studying rare and spatially clustered populations. This research proposed three enhanced ratio-type estimators for ACS, building on Chao’s [7] original ratio estimator and employing the Jackknife method to reduce bias and improve efficiency. Analytical derivations of bias and MSE were provided for each estimator, and their performance was evaluated through extensive simulation. The simulation results demonstrated that all three proposed estimators outperformed conventional estimators that do not utilize auxiliary variable information. Specifically, ${{\overline{w}}^{\prime }}_R$ proved to be more efficient than Chao’s estimator for small network sample sizes, while ${\overline{w}}_{JK}$ exhibited superior efficiency for both small and moderate sample sizes. In large-sample settings, the efficiency of ${\overline{w}}_{JK}$ became comparable to that of the traditional ratio estimator. Although ${{\overline{w}}^{″}}_R$ is an unbiased estimator, its efficiency was the lowest among the estimators that incorporate auxiliary variable information.