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General Parameter(s) Estimation using Sine Function-Based Estimators

This paper proposes a new sine-based difference estimator for general population parameters under simple random sampling without replacement, deriving its theoretical bias and mean squared error while demonstrating through empirical and simulation analyses that it offers improved accuracy over existing competing estimators.

Original authors: Shobh Nath Tiwari, Rajesh Singh, Madhulika Mishra

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shobh Nath Tiwari, Rajesh Singh, Madhulika Mishra

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of statistics, researchers often face a challenge similar to trying to guess the average height of everyone in a city by measuring only a few people. To make this guess more accurate, they frequently use a "helper" piece of information, known as an auxiliary variable. Imagine trying to estimate the total weight of a shipment of fruit; if you already know the exact number of crates in the shipment, you can use that known count to refine your weight estimate, even if you haven't weighed every single crate. This technique of combining a known fact with a sample measurement is a standard way to reduce error. However, the mathematical formulas used to blend these pieces of information are often rigid, relying on straight lines or simple ratios that may not capture the full complexity of how the data behaves. When the relationship between the known helper and the unknown target is subtle or curved, these traditional tools can miss the mark, leaving the final estimate less precise than it could be.

A team of researchers from Banaras Hindu University and Symbiosis International University in India has proposed a new way to handle this problem by introducing a mathematical curve based on the sine function. While sine waves are most famous for describing rhythmic patterns like sound, light, or tides, the researchers realized that the gentle, oscillating shape of this function could be used to fine-tune statistical estimates. Instead of forcing the data into a straight line, their new method allows the estimate to "bend" slightly to fit the natural variations in the sample. They developed a new class of estimators that incorporates this sine-shaped adjustment along with a difference-based correction, creating a tool designed to be more sensitive to the nuances of the data than the standard methods currently in use.

The core of their work involves testing this new tool against established methods using both real-world data and computer simulations. In the real-world tests, they examined two distinct scenarios. The first involved daily stock prices for a major car manufacturer, looking at how the opening price could help predict the daily high. The second scenario involved a census-style dataset counting the number of people living in different city blocks, using the number of rooms in those blocks as the helper information. In both cases, the researchers compared their new sine-based estimator against the traditional average and other existing variations. The results were clear: the new method consistently produced a smaller margin of error. For the stock price data, the new estimator reduced the error significantly compared to the standard approach, and for the housing data, it showed a similar improvement, proving that the sine-based adjustment could extract more accuracy from the same amount of information.

To ensure these findings were not just a lucky break with specific datasets, the researchers ran thousands of computer simulations. They created vast populations of data that followed different statistical patterns, including bell-shaped curves and skewed distributions, and then drew samples from them. They tested their new estimator across various sample sizes and different levels of connection between the helper data and the target data. In the majority of simulation scenarios, the new sine-based method outperformed the traditional estimators. The computer models showed that as the connection between the known and unknown variables strengthened, the advantage of the new method grew even larger, sometimes reducing the error by more than half compared to the standard technique. However, the results were not uniform across all conditions; in cases where the correlation between variables was weak, some competing estimators actually performed better, and the error reduction was less dramatic. This suggests that while the method is robust and reliable for many data types, its superiority is most pronounced when the relationship between the variables is strong.

The researchers emphasize that their new estimator is a general tool that can be adapted to measure different things, such as the average value, the spread of the data, or the variability relative to the average. By adjusting a few simple settings within the formula, the same underlying method can be used for these different statistical goals. The study concludes that by allowing the estimation process to follow a curved, sine-based path rather than a rigid straight line, statisticians can achieve a higher degree of precision. While the method relies on knowing certain characteristics of the helper variable in advance, the authors suggest it offers a practical and superior alternative for anyone looking to make the most accurate predictions possible from a limited sample. They also note that this approach opens the door for future work, such as combining multiple helper variables or applying the technique to more complex sampling designs, but for now, the primary finding stands: a simple mathematical curve can make a significant difference in how well we understand the world through data.

Technical Summary: General Parameter(s) Estimation using Sine Function-Based Estimators

Problem Statement
In sampling theory, the estimation of population parameters (such as the mean, variance, standard deviation, and coefficient of variation) is a fundamental task. While the integration of auxiliary variables has long been recognized as a method to enhance estimation precision, the specific application of sine-type functions to estimate general parameters remains underexplored. Although sine functions are powerful tools for modeling periodic behaviors in various fields, their utility in constructing efficient estimators for general population parameters under Simple Random Sampling Without Replacement (SRSWOR) has not been fully utilized. This study addresses the gap in developing a new class of sine-based difference estimators to improve the accuracy of estimating general parameters by leveraging known auxiliary information.

Methodology
The authors propose a new class of estimators, denoted as $t^*$ , designed to estimate a general parameter $t_{(p,q)} = \bar{Y}^p S_Y^q$ , where $\bar{Y}$ is the population mean and $S_Y$ is the population standard deviation of the study variable $Y$ . The estimator incorporates an auxiliary variable $Z$ and utilizes a sine function transformation.

The proposed estimator is formulated as:
$t^* = [\hat{t}_{(p,q)} + k(\bar{Z} - \bar{z})] \left[1 + \sin\left\{\frac{w_1(\bar{Z} - \bar{z})}{\bar{Z}}\right\}\right] \left[1 + \sin\left\{\frac{w_2(S_Z^2 - s_Z^2)}{S_Z^2}\right\}\right]$
where:

$\hat{t}_{(p,q)}$ is the conventional estimator ( $\bar{y}^p s_y^q$ ).
$\bar{Z}$ and $S_Z^2$ are known population parameters of the auxiliary variable.
$\bar{z}$ and $s_Z^2$ are the corresponding sample statistics.
$k$ , $w_1$ , and $w_2$ are scalars, with $w_1, w_2 \in (0, 1)$ and $k$ chosen to minimize the Mean Squared Error (MSE).

The theoretical properties of the estimator, specifically the Bias and MSE, are derived using first-order approximation methods, ignoring finite population correction (fpc) terms and higher-order error components. The study also derives specific cases of the estimator for estimating the population mean ( $\bar{Y}$ ), population variance ( $S_Y^2$ ), population standard deviation ( $S_Y$ ), and coefficient of variation ( $C_Y$ ) by adjusting the parameters $p$ and $q$ .

Key Contributions

Generalized Framework: The paper introduces a unified class of sine-based estimators capable of estimating various population parameters (mean, variance, standard deviation, coefficient of variation) by simply varying the scalar parameters $p$ and $q$ .
Novel Estimator Structure: The proposed estimator combines a difference component with two sine-function components, one utilizing the difference in means and the other the difference in variances of the auxiliary variable.
Theoretical Derivation: The authors provide closed-form expressions for the Bias and MSE of the proposed estimator and determine the optimal values for the scalar weights ( $w_1$ and $w_2$ ) that minimize the MSE.
Efficiency Conditions: Theoretical conditions are established demonstrating that the proposed estimator $t^*$ (and its specific forms like $t_1^*$ and $t_2^*$ ) yields a lower MSE compared to the conventional estimator $\hat{t}_{(p,q)}$ and other competing sine-type estimators.

Results
The performance of the proposed estimator was evaluated through two distinct approaches:

Empirical Illustration: Two real-world datasets were analyzed:
- Population 1: Daily stock prices of Suzuki Motor Corporation (2025).
- Population 2: A dataset from Cochran regarding individuals and rooms in blocks.
  In both cases, the proposed estimator ( $t^*$ or $t@$ ) demonstrated the lowest MSE and the highest Percentage Relative Efficiency (PRE) compared to the conventional mean estimator and other sine-based variants. For Population 1, the proposed estimator achieved a PRE of 212.99%, and for Population 2, it achieved 195.16%.
Simulation Study: Extensive simulations were conducted using R software with $N=1000$ observations drawn from Bivariate Normal and Gamma distributions. The study varied correlation coefficients ( $\rho_{zy} = 0.3, 0.5, 0.7, 0.9$ ) and sample sizes ( $n = 40, 50, 60$ ).
- The results consistently showed that the proposed estimator outperformed the conventional estimator and competing sine estimators across all tested scenarios.
- The efficiency gain (PRE) increased significantly as the correlation between the study and auxiliary variables increased. For instance, under the Gamma distribution with $\rho=0.9$ , the proposed estimator achieved PRE values exceeding 500% in certain configurations.

Significance and Claims
The paper claims that the proposed sine-based difference estimator offers a significant improvement in estimation accuracy over existing methods. The authors assert that by incorporating known auxiliary information through a sine-function transformation, the estimator effectively reduces the Mean Squared Error. The study concludes that the proposed class of estimators is superior to the traditional sample mean and other competing sine estimators, making it a recommended tool for practical applications in statistical sampling. The findings are supported by both theoretical derivations and robust empirical and simulation evidence.

Future Directions
The authors suggest that future research could extend this methodology by:

Utilizing multiple auxiliary variables.
Incorporating measurement errors and non-response effects.
Adapting the approach to alternative sampling designs such as stratified, ranked set, and double sampling.
Applying two-phase and stratified two-phase sampling techniques.

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