International Journal of Statistical Distributions and Applications 2021; 7(1): 13-24 http://www.sciencepublishinggroup.com/j/ijsda doi: 10.11648/j.ijsd.20210701.13 ISSN: 2472-3487 (Print); ISSN: 2472-3509 (Online) Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response Ongoma Jackson 1 , Alilah David Anekeya 1 , Okuto Erick 2 1 Department of Mathematics Masinde Muliro University of Science and Technology, Kakamega-Nairobi, Kenya 2 Departments of Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Siaya-Nairobi, Kenya Email address: To cite this article: Ongoma Jackson, Alilah David Anekeya, Okuto Erick. Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response. International Journal of Statistical Distributions and Applications. Vol. 7, No. 1, 2021, pp. 13-24. doi: 10.11648/j.ijsd.20210701.13 Received: November 3, 2020; Accepted: December 1, 2020; Published: March 12, 2021 Abstract: Sample survey provides reliable current statistics for large areas or sub-population (domains) with large sample sizes. There is a growing demand for reliable small area statistics, however, the sample sizes are too small to provide direct (or area specific) estimators with acceptable and reliable accuracy. This study gives theoretical description of the estimation of small area mean by use of stratified sampling with a linear cost function in the presence of non-response. The estimation of small area mean is proposed using auxiliary information in which the study and auxiliary variable suffers from non-response during sampling. Optimal sample sizes have been obtained by minimizing the cost of survey for specific precision within a given cost using lagrangian function multiplier lambda and Partial Differential Equations (PDEs). Results demonstrate that as the values of the respondent sample increases sample units that supply information to study and auxiliary variable tends to small area population size, the non-response sample unit tends to sample units that supply the information as the sampling rate tends to one. From theoretic analysis it is practical that the Mean Square Error will decrease as the sub-sampling fraction and auxiliary characters increase. As the sub-sampling fraction increases and the value of beta increases then the value of large sample size is minimized with a reduction of Lagrangian multiplier value which minimizes the cost function. Keywords: Stratified Sampling for Ratio Estimation, Small Area Mean, Auxiliary Variable, Linear Cost Function and Non-response 1. Introduction 1.1. Small Area Small Area refers to a population for which reliable statistics of interest could not be computed using standard methods because of small or even zero sample sizes in the area. Some of the perceived small areas include geographical regions such as county, sub-county and wards, and demographic regions such as age, sex and race. In sampling the units are divided into two strata for homogeneity, the first strata represent respondents while the second strata represent non-respondents. 1.2. Small Area Estimation According to Rahman [13] small area estimation has received much attention in recent decades due to increasing demand for reliable small area estimates for both public and private sectors. Sample data on small areas is inadequate to provide statistical estimates with high precision. This therefore makes it necessary to borrow strength from data on related auxiliary variables using appropriate models. Small area estimation is therefore any statistical technique that involves the estimation of parameters for small sub- populations. Methods used in small area estimation are categorized as design based and model based. According to Rahman [13] design-based method reference was made for particular sampling design used whereas model-based method involves statistical method based on Bayesian approaches. Among the models used in small area estimation and prediction is Linear Mixed Model that has found wide range of applications particularly for its ability to predict linear combination of fixed and random effects. Henderson [10]