New Estimator Sharpens Population Mean Estimates in Stratified Surveys

Statisticians have proposed a new estimator for estimating the mean of a population in stratified sampling that combines several classical estimation strategies into a single flexible formula. The method, described in a study published in Advanced Modeling and Simulation in Engineering Sciences, integrates auxiliary attribute information — characteristics that are not the primary variable of interest but are connected to it — through a unified nonlinear framework. According to the authors, Mukesh Kumar Verma and Saif Ali Khan, the estimator merges ratio, product, and exponential-type approaches within one formulation, allowing it to adapt to different correlation structures between the study variable and the auxiliary attribute.

Stratified sampling is a widely used survey technique in which a heterogeneous population is divided into relatively homogeneous groups, called strata, before sampling. Precision can be pushed further when surveyors exploit auxiliary information that is strongly associated with the variable being measured. In agricultural surveys, for example, attributes such as the number of fruit trees in an orchard are often readily available and closely related to crop yield, so using them can substantially improve estimation accuracy without increasing survey costs. Despite this practical advantage, the authors note that auxiliary attributes remain underutilized in stratified sampling, which motivated the new work.

The proposed estimator builds on earlier work by Akintunde and colleagues and takes the form of the stratified sample mean plus two correction terms: one weighted ratio-type term involving the population and sample proportions of the auxiliary attribute, and one exponential-type term. Two constants, K1 and K2, control the contribution of each component. The authors derived the mean squared error of the estimator up to the first order of approximation, a standard technique in sampling theory in which higher-order terms of the error expansion are ignored. This yields simple, tractable expressions for bias and mean squared error that make analysis and comparison of estimators easier, and the approximation is considered sufficiently accurate for moderate to large samples.

A key feature of the method is that the optimal values of the two constants are obtained analytically, by minimizing the mean squared error expression. The resulting formulas depend on quantities that summarize the variability of the study variable, the variability of the auxiliary attribute, and the covariance between them across strata, each weighted by the stratum sizes and sampling fractions. The authors also worked out special cases: setting one constant to zero recovers a ratio-augmented estimator, while setting the other to zero recovers an exponential-augmented version, each with its own optimal constant.

The study derives explicit conditions under which the new estimator outperforms a series of existing competitors. These include the usual stratified sample mean, the ratio estimator for auxiliary attributes proposed by Naik and Gupta, the exponential ratio-type estimator of Bahl and Tuteja, estimators examined by Zaman and by Zaman and Kadilar, the double-exponential family of Akintunde and colleagues, and a combined estimator due to Verma and coauthors. In each case the comparison reduces to an inequality involving the mean squared errors, and the proposed estimator is more efficient whenever its optimized constants satisfy the corresponding condition. The authors also report a robustness analysis examining sensitivity to variations in the auxiliary attribute and potential model misspecification, stating that performance remains stable under moderate deviations from the underlying assumptions, and an asymptotic analysis establishing consistency as sample size grows.

To test the method on real data, the researchers used an apple production dataset originally reported by Akintunde and colleagues, obtained from the Turkish Republic Institute of Statistics. The quantity of apples produced in 1999 served as the study variable, while the number of apple trees in 1999 served as the auxiliary attribute. Turkey was divided into six regions — Marmara, Aegean, Mediterranean, Central Anatolia, Black Sea, and East and Southeast Anatolia — and random samples were drawn from each stratum, with stratum sample sizes determined by Neyman allocation. A second real dataset, taken from Cochran, used the number of teachers as the study variable and the number of students as the auxiliary attribute, again with Neyman allocation across strata.

Performance was assessed using two standard measures: mean squared error, which quantifies the average squared deviation of an estimate from the true population mean, and percentage relative efficiency, which compares an estimator against a reference estimator, with lower mean squared error and higher relative efficiency indicating better performance. Across both real datasets, the authors report that the proposed estimator consistently attained the highest percentage relative efficiency among all estimators considered, indicating a substantial improvement in precision. In their discussion of the empirical results, the authors note that the modified special-case estimators showed only marginal gains, with relative efficiency values slightly above one hundred percent, while among the traditional estimators some performed moderately better than the usual stratified estimator and others exhibited substantially higher mean squared errors, with relative efficiency below forty-five percent. This, they argue, shows that efficiency gains from auxiliary attributes depend critically on choosing an appropriate estimator form.

The evaluation was extended with Monte Carlo simulation. The authors generated a non-normal stratified population of 10,000 units divided into four unequal strata, drawing a stratified random sample of 6,000 units using non-proportional allocation and simple random sampling without replacement within each stratum. The study variable was generated from a Gamma distribution with shape parameter 4 and scale parameter 8, while the auxiliary variable followed a Lognormal distribution with mean parameter 3.5 and variance 0.6, deliberately introducing skewness into the data. A sensitivity analysis systematically altered the skewness parameters of both variables as well as the allocation schemes and stratum sample sizes. According to the authors, the proposed estimators maintained lower mean squared error and higher efficiency across a wide range of these scenarios, suggesting robustness even under departures from normality and changes in population structure.

The authors are careful to state the limitations of their approach. Because the estimator is developed under a first-order approximation, the derived bias and mean squared error expressions are approximate and may be less accurate for small sample sizes or complex population structures. The methodology also assumes that the auxiliary attribute is measured without error and is strongly associated with the study variable, and the empirical evaluation rests on a limited set of datasets. The simulation results, while extensive, are model-based rather than observations from an operational survey, so their findings describe estimator behavior under the specified generated conditions. The authors suggest that future research could consider second-order approximations, measurement errors, non-response, and broader simulation studies to further assess robustness.

Within those boundaries, the study’s conclusion is that incorporating auxiliary attribute information can substantially improve the estimation of population means in stratified sampling, particularly in agricultural surveys of the kind illustrated by the apple production data. The authors report strong agreement between their theoretical derivations and the empirical findings, which they interpret as evidence that the estimator is both mathematically sound and practically applicable. They point to potential extensions to more complex sampling designs, multiple auxiliary attributes, and applications in environmental, health, and socio-economic surveys. For survey practitioners, the message is that when a cheaply available attribute correlates strongly with the variable of interest, folding it into an optimally weighted combination of ratio and exponential terms can deliver more precise estimates at no additional sampling cost — provided the sample is large enough for the first-order approximations on which the method relies to hold.

Subject of Research: A new ratio-type estimator for population mean estimation in stratified sampling using auxiliary attributes, evaluated with real and simulated populations

Article Title: A robust population mean estimation method in stratified sampling using auxiliary attributes under real and simulated populations

Article References: Verma, M. K., & Khan, S. A. (2026). A robust population mean estimation method in stratified sampling using auxiliary attributes under real and simulated populations. Advanced Modeling and Simulation in Engineering Sciences, 13(1), Article 20. https://doi.org/10.1186/s40323-026-00339-4

Image Credits: AI Generated

DOI: 10.1186/s40323-026-00339-4

Keywords: stratified sampling, population mean estimation, auxiliary attributes, mean squared error, percentage relative efficiency, ratio estimator, exponential estimator, Monte Carlo simulation, survey sampling, Neyman allocation, apple production data, first-order approximation