Log-ratio type estimation for the finite population mean under simple random sampling without replacement with theory, simulation and application

Sep 29, 2026

Proposes two logarithmic ratio–type estimators for the finite-population mean under simple random sampling without replacement (SRSWOR). The estimators use a logarithmic transformation of an auxiliary variable to stabilize variance, mitigate outlier influence, and model nonlinear relationships between study and auxiliary variables. Derives closed-form first-order bias and mean squared error (MSE), and obtains optimal tuning constants via minimization of the approximate MSE. Numerical evaluation covers five engineering datasets and extensive Monte-Carlo simulations across multivariate normal, log-normal, and gamma populations, showing consistent MSE reduction and large PRE gains versus the classical sample mean and common competitors, especially for skewed or heavy-tailed populations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

AccessFree to download

Abstract

Proposed estimators and theoretical results

Bias and MSE derivations

Optimal tuning constants

Numerical study: datasets and Monte-Carlo simulations

Practical implications and robustness

Future extensions

[www. nature.com/scientificreports](www. nature.com/scientificreports) OPEN Log-ratio type estimation for the finite population mean under simple random sampling without replacement with theory, simulation and application Fazal Shakoor1, Muhammad Atif1, Hameed Ali2, Abdulrahman Obaid Alshammari3, Bilal Himmat4􀀍 & Khaled Kefi5􀀍 We propose two novel logarithmic ratio–type estimators for the finite-population mean under simple random sampling without replacement (SRSWOR). The estimators integrate a logarithmic transformation of the auxiliary variable to stabilize variance, reduce the influence of outliers, and better capture nonlinear relationships between study and auxiliary variables. We derive closed-form expressions for first-order bias and mean squared error (MSE) and obtain analytic expressions for the optimal tuning constants by direct minimization of the approximate MSE. A comprehensive numerical study, comprising five real engineering datasets and extensive Monte-Carlo simulations from multivariate normal, log-normal and gamma populations, evaluates finite-sample behavior across a range of sample sizes and correlation structures. The proposed estimators consistently reduce MSE and deliver large percent-relative-efficiency (PRE) gains relative to the classical sample mean and common competitors (empirical PREs ≈ 283; simulation PREs up to ≈ 670), with especially large and stable improvements under skewed or heavy-tailed populations. Theoretical formulas and simulation evidence align closely, showing robustness to nonlinearity and skewness while retaining simple implementation for practitioners. Results are derived under SRSWOR using first-order approximations; extensions to higher-order corrections, stratified and two-phase designs, and uncertainty in auxiliary means are recommended for future work. Keywords Estimation, Auxiliary information, Bias, Efficiency, Logarithmic Estimator, PRE, Ratio Estimator, Simple Random Sampling List of symbols Nn S2y S2x sy x Cy Cx Cy x αandβ e0 , e1 Size of the population Size of the sample Variance of y Variance of x Covariance Coefficient of variation (y) Coefficient of variation (x) Coefficient of covariance x, y Generalizing constants Relative error in ¯y = ¯yµyy~~ ~~ and ¯x = ~~¯~~xµx~~x~~ respectively 1Department of Statistics, University Peshawar, Peshawar, KP, Pakistan. 2Higher Education, Archives and Libraries Department, Government of Khyber Pakhtunkhwa, Peshawar, Pakistan. 3Department of Mathematics, College of Science, Jouf University, 72388 Sakaka, Saudi Arabia. 4Department of Software Engineering, Faculty of Computer Science, Sayed Jamaluddin Afghani University(SJAU), Asadabad, Kunar, Afghanistan. 5Center for Scientific Research and Entrepreneurship, Northern Border University, 73213 Arar, Saudi Arabia. 􀀍 email: [bilalhimmat@sjau.edu.af](bilalhimmat@sjau.edu.af); [Khaled_kefi@yahoo.fr](Khaled_kefi@yahoo.fr) [www. nature.com/scientificreports/](www. nature.com/scientificreports/) S2y S2x ρyx ¯ Y ¯ X Rf Tln1 Tln2 k1 , k2 , k3 , k4 Population variance of (y) Population variance of (x) Population correlation coefficient Mean of the population of y Population Mean ofx Population ratios Sampling fraction First proposed estimator Second proposed estimator Optimizing constants Accurate estimation of population means from sampled data lies at the heart of survey statistics and many applied fields, from official statistics and environmental monitoring to engineering quality control and experimental sciences1. When using simple random sampling (SRS) to collect data on system performance or material properties, utilizing supplementary information, such as known historical measurements or operating conditions, can enhance estimator efficacy. Estimation methods such as the ratio estimator and logarithmicratio type estimator utilize this auxiliary data to adjust the primary estimates, reducing the mean squared error (MSE) compared to ba

FAQs

What problem do the authors address in the paper?

They address estimating the finite-population mean under simple random sampling without replacement using auxiliary information.

How do the proposed estimators improve over classical approaches?

They apply a logarithmic transformation to the auxiliary variable, which helps stabilize variance, reduce outlier effects, and capture nonlinear relationships, leading to lower MSE and higher PRE.

What evidence supports the performance of the estimators?

A numerical study uses five real engineering datasets and extensive Monte-Carlo simulations across multivariate normal, log-normal, and gamma populations, comparing MSE and PRE against the classical sample mean and competing estimators.

Sep 29, 2026

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