Adaptive cluster sampling (ACS) is an efficient sampling technique for studying populations where the characteristic of interest is rare or spatially clustered. This method is widely applied in fields such as ecological studies, epidemiology, and resource management. ACS initially selects sampling units using simple random sampling without replacement. However, in some cases, selected networks may overlap, leading to multiple networks being included in the sample. To address this issue, a modified version of ACS was developed to ensure sampling without replacement at the network level, maintaining sampling symmetry and preventing the inclusion of overlapping networks. Despite this adjustment, asymmetry may still occur when network formation is highly irregular. This issue can be mitigated by incorporating auxiliary variables, which help correct distortions in the sampling process. In many situations, auxiliary variables related to the variable of interest can be utilized to enhance the precision of population parameter estimates. This research proposes multiplicative generalization for an estimator with two auxiliary variables using adaptive cluster sampling with networks selected without replacement. The bias and mean square error (MSE) are derived using a Taylor series expansion to determine the optimal conditions for minimizing MSE. A simulation study is conducted to support the theoretical findings. The results show that the proposed estimator under the optimal values of $T_1$ and $T_2$ is the most efficient to minimize MSE.

Adaptive cluster sampling is a methodology designed for data collection in contexts where the population is rare and spatially clustered. This approach has been effectively applied in various disciplines, including epidemiology and resource management. The present study introduces novel estimators that incorporate auxiliary variable information to improve estimation efficiency. These estimators were developed using the jackknife resampling technique to improve the performance of ratio-type estimators. Theoretical properties, including bias and mean square error (MSE), were derived, and a simulation study was conducted to validate the theoretical findings. The results demonstrated that the proposed estimators consistently outperformed conventional estimators that do not utilize auxiliary variables across all network sample sizes. Furthermore, in several scenarios, the proposed estimators also exhibited superior efficiency to existing ratio estimators that do incorporate auxiliary information.

Adaptive cluster sampling is a methodology designed for data collection in contexts where the population is rare and spatially clustered. This approach has been effectively applied in various disciplines, including epidemiology and resource management. The present study introduces novel estimators that incorporate auxiliary variable information to improve estimation efficiency. These estimators are developed using the Jackknife resampling technique, which is employed to enhance the performance of ratio-type estimators. Theoretical properties, including bias and mean square error (MSE), are derived, and a simulation study is conducted to validate the theoretical findings. Results demonstrate that the proposed estimators consistently outperform conventional estimators that do not utilize auxiliary variables across all network sample sizes. Furthermore, in several scenarios, the proposed estimators also exhibit superior efficiency compared to existing ratio estimators that do incorporate auxiliary information.

This research is motivated by the unbiased ratio estimator proposed by Tin (1965), the unbiased product estimator proposed by Robson (1957), and the unbiased estimator proposed by Mahanty and Mishra (2020), which were developed under the auxiliary attribute in simple random sampling without replacement. It was found that if the data analyzed met the conditions, the proposed estimators will be more efficient than the mean per unit estimator. Among them, the estimator based on Mahanty and Mishra’s is the most efficient, both theoretically and in numerical illustration.