An improved ratio estimator for population mean in the presence of non-response in two stage sampling using two auxiliary variables
Abstract
Nonresponse is a common challenge during sample surveys that tend to interfere with efficiency of population mean estimators. Nonresponse occurs when selected units fail to provide the required information. If ignored the resulting estimator maybe unbiased. The study proposes an improved class of ratio estimators for estimating the finite population mean using information from two auxiliary variables and two stage sampling where nonresponse occurs at the second stage of sampling. The estimator allows the two auxiliary variables to contribute simultaneously to the estimation of the population mean. Using first order Taylor approximation, expressions for bias and mean squared error (MSE) of the proposed estimators are derived. Theoretical efficiency conditions are established. Empirical comparison shows the proposed estimators outperforms existing by Singh et al. (2018) with a percentage relative efficiency (PRE) of over 120%
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Introduction
In survey sampling, auxiliary information is essential for improving precision and accuracy of estimators for population parameter of interest. Such information can be utilized both during sample selection and the estimation stage to enhance the design and yield more consistent estimates. For example, estimation becomes more precise when the auxiliary variable X is highly correlated with the study variable Y. in many practical survey, information on auxiliary variables may not be available for the entire population, necessitating two stage sampling. Furthermore, nonresponse is unavoidable in survey relating to health, education and industry. When nonresponse occurs, the Hansen and Hurwitz (1946) subsampling technique is widely adopted. Singh et al. (2012) proposed estimators under two-stage sampling with nonresponse using one auxiliary variable, Ongoma et al. (2021) [1] considered nonresponse in both study and auxiliary, limited work has addressed the use of two auxiliary variables that are also subject to nonresponse at the second phase. This gap motivates the present study. The objective of this paper is to propose a ratio estimator using two auxiliary variables under two stage sampling with nonresponse and derive its properties.
Methodology
Notation and sampling design
A finite population consisting N primary samples from 𝛽 1 , 𝛽 2 ……𝛽 𝑁 clusters of sizes 𝑀 𝑖 for 𝑖 = 1, 2…..𝑁. A sample of n clusters were considered from N clusters using SRSWOR as the ssu`s, assuming that m i units provide the required information on the two auxiliary variables; Due to nonresponse, only m i1 out of the m i units selected provide complete data of study variable and two auxiliary variables. Remaining m i2 units do not respond. Non response occurs only at the second stage and is observed both on study and auxiliary variables. By the use of Hansen and Hurwitz methods of taking a subsample of the units that fail to respond where another sample of h i2 units are taken out of the total m i2 that fail to respond in such a manner that 𝑚 𝑖2 = 𝑘 𝑖 ℎ 𝑖2 where k i ˃1. Let Y be study variables and 𝑋 1 , 𝑋 2 be the two auxiliary variables. Therefore, the samples picked on the second phase will have a total of the responding (𝑚 𝑖1 )and nonresponding (𝑚 𝑖2 ) that is, 𝑚 𝑖 = 𝑚 𝑖1 + 𝑚 𝑖 2 . Let ℎ 𝑖 2 units be a fraction of non-responding group that respond after callback, then the total responding units are; 𝑚 𝑖1 + ℎ 𝑖2 . with 𝑘 𝑖 as inverse sampling rate. For the responding units 𝑊 𝑖 = 𝑚 𝑖1 𝑚 𝑖 as their cluster weight and 𝑊 𝑖 = 𝑚 𝑖2 𝑚 𝑖 cluster weight for the non-responding units.
ISSN Online: 2456-1452 NAAS Rating (2026): 4.49
Proposed estimator
The proposed ratio-type estimator using two auxiliary variables is defined as: To obtain MSE and Bias we will use the large sample approximation and Tailor series expansion. Expanding to the second order of approximation, the bias is Bia We then find MSE by squaring both sides of the estimator and taking expectation neglecting powers that are ˃2 we have: The proposed estimator has the lowest MSE
Bias and mean squared error
3.Results: Efficiency comparison
The following estimators were computed in each application: Naïve estimator that uses only respondent data while ignoring auxiliary variables and nonresponse mechanism. One auxiliary ratio estimator that adjusts for the nonresponse while only making use of one auxiliary variable. Two auxiliary ratio estimators. The proposed estimator that listens to the nonresponse while in cooperating two auxiliary variables The typical results obtained from 1000 replications, with the population mean 89.4 and a 30% MAR nonresponse were as per the table below.
Conclusion
The aim of this study was to come up with an estimator that incorporates two auxiliary variables considering two stage sampling with an element of nonresponse. Non response led the Naïve estimator to decrease its bias but when we incorporated the application of two auxiliary variables the bias was reduced to as much as 91%. The use of two auxiliary variable as well lowered the MSE by 84% as compared to the Naive estimator.
The use of two auxiliary variables also lowered the MSE by 31% as compared to when using one auxiliary variable.
References
- Ongoma Jackson, Alila David Anekeya, Okuto Erick. Optimal Allocation in Small Area Mean Estimation Using Stratified Random Sampling in the presence of non-response. International Journal of Statistical Distributions and Application. 2021;7(1):1324.
- Rajesh Singh, Manoj kr Chaudhary, Anil Prajapati. two-phase sampling in estimation of population mean in the presence of non-response. Columbus, USA: Educational publisher; 2014:2841.
- Alila David Anekeya, Ouma Christopher Onyango, Nyongesa Kennedy. Domain Mean Estimation Using Double Sampling with Non-Linear Cost Function in the Presence of Non-Response. Science Journal of Applied Mathematics and Statistics. 2018;6(1):28-42.
- Nelson Kiprono Bii, Christopher Ouma Onyango, John Odhiambo. Estimating a finite population mean and random nonresponse in two stage cluster sampling with replacement. Journal of physics. conf series. 2018;1132:012071. International Journal of Statistics and Applied Mathematics https://www.mathsjournal.com 24561452, 2026, 11(10), Downloaded from https://www.doi.org/10.22271/maths.2026.v11.i10a.2714 by International Journal of Statistics and Applied Mathematics on [07/10/2026]. See the Terms and Conditions (https://www.mathsjournal.com) for rules of use; OA articles are governed by the applicable Creative Commons License
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