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Generalized Sine-Type Estimators for Population Median: Simulation-Driven Performance Analysis

This study proposes and evaluates a novel family of generalized sine-type and exponential-sine-type estimators for the finite population median under simple random sampling without replacement, demonstrating through theoretical analysis, simulations, and real-world data that these methods significantly outperform traditional estimators by leveraging auxiliary information to reduce mean squared error.

Original authors: Rajesh Singh, Shobh Nath Tiwari

Published 2026-10-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rajesh Singh, Shobh Nath Tiwari

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 a crowd by looking at a few people. Sometimes, the people you see are not a perfect mirror of the whole group; they might be taller, shorter, or skewed in a specific direction. This is especially true when dealing with data that is uneven, such as household incomes or production numbers, where a few extreme values can distort the picture. In these cases, the "median"—the middle value where half the data is higher and half is lower—often tells a truer story than the average. To get an even clearer picture, statisticians frequently use "auxiliary information," which means bringing in a second, related piece of data that is already known. For example, if you want to estimate the number of trees in a forest, you might also know the total area of the land. By combining the known area with a sample count, you can make a much sharper guess about the total number of trees. The goal is always to reduce the error in the guess, making the final estimate more reliable and precise.

A team of researchers at Banaras Hindu University in India has developed a new way to use this extra information to find the median more accurately. They created a family of mathematical tools that blend two familiar concepts: the smooth, repeating curve of a sine wave and the rapid growth or decay of an exponential function. While these terms sound complex, the researchers simply used them to build a flexible formula that can adapt to the messy, non-linear patterns found in real-world data. Unlike older methods that rely on straight lines or simple ratios, these new tools can bend and twist to fit the specific shape of the data, much like a flexible ruler that conforms to a curved surface rather than forcing a straight line onto it. The researchers tested these new formulas against traditional methods using both real-world data sets and computer simulations. They found that their new approach consistently produced guesses with less error than the standard methods, including the usual sample median and other existing techniques that try to use extra information.

To see how well this worked, the team first looked at two real-world scenarios. In one case, they examined data on the number of households and the total area of land in a specific region. In another, they looked at the number of teaching staff and student enrollment in educational institutions. When they applied their new sine-based formulas to these numbers, the results were striking. For the household data, the new method reduced the error of the guess by about 28 percent compared to the standard method. For the school data, the improvement was even more dramatic, with the new method achieving a relative efficiency of nearly 372 percent, which corresponds to a reduction in error of approximately 73 percent. This means that if a statistician were using the old method, their guess might be off by a significant margin, whereas the new method would land much closer to the true middle value. The researchers also ran thousands of computer simulations using different types of data distributions, including those that mimic natural growth patterns and others that follow a bell curve. In every single scenario, whether the data was slightly connected or strongly linked, the new sine-type estimator outperformed the competition.

The study also explored why some older methods failed while the new one succeeded. The researchers found that when the relationship between the two variables was positive, meaning they tended to rise and fall together, the traditional ratio-based methods worked reasonably well but still fell short of the new approach. However, when the data was more complex or the relationship was negative, the older methods often performed poorly, sometimes even becoming less accurate than just guessing the median from the sample alone. The new method, by incorporating the flexibility of sine and exponential functions, managed to handle these difficult situations with ease. It adjusted its shape to account for the specific quirks of the data, ensuring that the final estimate remained stable and accurate. The researchers confirmed that their method works best when the known information about the auxiliary variable is accurate and when the difference between the sample and the full population is relatively small, which is a standard condition for this type of statistical work.

Ultimately, this research offers a practical upgrade for anyone trying to understand the middle ground of a large group of numbers. By moving beyond rigid, straight-line assumptions and embracing formulas that can curve and adapt, the new estimators provide a more reliable way to find the median. The findings suggest that in fields ranging from economics to public health, where data is often skewed and uneven, using these advanced sine-based tools could lead to significantly better decisions. The study does not claim to solve every problem in statistics, but it does demonstrate that by refining how we combine known information with new samples, we can achieve a level of precision that was previously out of reach. For the curious observer, the takeaway is clear: sometimes, the best way to find the middle is to stop thinking in straight lines and start thinking in waves.

Technical Summary: Generalized Sine-Type Estimators for Population Median

Problem Statement
In survey sampling, while significant research has focused on improving estimators for population means, variances, and proportions using auxiliary information, the estimation of the finite population median has received comparatively less attention. Many real-world variables (e.g., income, production, poverty indices) exhibit skewed, non-normal distributions where the median is a more robust indicator of location than the mean. Although previous studies have introduced ratio, product, and exponential estimators for the median, the development of sine-type estimators for this specific parameter remains underexplored. Existing linear or purely exponential forms often fail to adequately model the intricate nonlinear, oscillatory, or periodic behaviors found in practical datasets. This study addresses the gap in estimating the finite population median by incorporating known auxiliary information through novel sine-type and exponential-cum-sine-type frameworks.

Methodology
The study operates within the framework of Simple Random Sampling without Replacement (SRSWOR). The authors utilize the known population median of an auxiliary variable ( $M_x$ ) to improve the estimation of the study variable's population median ( $M_y$ ).

Theoretical Derivation:
- The authors define relative error terms ( $\eta_y, \eta_x$ ) to derive the bias and Mean Squared Error (MSE) of proposed estimators using first-order approximation.
- They first review and adapt conventional estimators, including the usual sample median, sine-type ratio and product estimators, and exponential-cum-sine-type estimators (inspired by Bahl and Tuteja).
- A generalized family of exponential-cum-sine-type estimators ( $\hat{M}_y^*$ ) is proposed. This estimator combines the sample median of the study variable, the sample median of the auxiliary variable, and a scalar parameter ( $\beta$ ) optimized to minimize MSE. The formulation integrates sine functions and exponential terms to capture nonlinearity:
  $\hat{M}_y^* = \left[ \hat{M}_y \left\{ 1 + \sin\left(k_1 \frac{M_x - \hat{M}_x}{M_x}\right) \right\} + \beta(M_x - \hat{M}_x)\left(\frac{M_x}{\hat{M}_x}\right)^{k_2} \right] \exp\left(\frac{M_x - \hat{M}_x}{M_x + \hat{M}_x}\right) \left\{ 1 + \sin\left(k_3 \frac{\hat{M}_x - M_x}{M_x}\right) \right\}$
  where $k_1, k_2, k_3$ are scalar parameters taking values from $\{0, -1, 1\}$ .
Optimization:
- The optimal value of the scalar $\beta$ is derived to minimize the MSE of the proposed estimator.
- Specific members of the proposed family are generated by assigning different combinations of values to $k_1, k_2,$ and $k_3$ .

Key Contributions

Novel Estimator Class: The paper introduces a generalized family of sine-type and exponential-cum-sine-type estimators specifically designed for the population median, a domain where such hybrid functional forms were previously limited.
Theoretical Framework: It provides rigorous first-order approximations for the bias and MSE of these new estimators, establishing the theoretical conditions under which they outperform existing methods.
Comprehensive Evaluation: The study evaluates the proposed estimators against the usual sample median, sine-type ratio/product estimators, and exponential-cum-sine estimators using both real-world data and extensive simulation.

Results
The performance of the proposed estimator ( $\hat{M}_y^*$ ) was validated through two primary avenues:

Empirical Study (Real Data):
- Two real-world datasets were analyzed: one regarding household counts vs. area (Murthy [21]) and another regarding teaching staff vs. student enrollment (Koyuncu and Kadilar [22]).
- Findings: The proposed estimator consistently yielded the lowest MSE and the highest Percent Relative Efficiency (PRE) compared to all competing estimators.
  - For Population 1, the proposed estimator achieved a PRE of 128.045 (approx. 1.28 times more efficient than the sample median).
  - For Population 2, the PRE reached 371.781 (approx. 3.72 times more efficient).
- Notably, some existing sine-type product estimators performed worse than the simple sample median in these specific datasets, highlighting the importance of the proposed hybrid structure.
Simulation Study:
- Simulations were conducted using Bivariate Normal and Gamma distributions with a population size of $N=1000$ .
- The study varied sample sizes ( $n = 80, 150, 250$ ) and correlation coefficients ( $\rho_c = 0.3, 0.5, 0.7, 0.9$ ).
- Findings: Across all scenarios, the proposed estimator ( $\hat{M}_y^*$ $\hat{M}_{y}^{*}$ ) consistently outperformed the sample median and other sine-type estimators.
  - In the Gamma distribution simulation with $\rho_c = 0.9$ , the proposed estimator achieved a PRE of 514.06, significantly outperforming the ratio estimator (PRE 509.68) and the exponential ratio estimator (PRE 263.55).
  - The efficiency of the proposed estimator increased as the correlation between the study and auxiliary variables increased.

Significance and Claims
The authors claim that the proposed generalized sine-type estimator offers a superior alternative for estimating the finite population median when auxiliary information is available. The study asserts that:

The hybrid use of sine and exponential functions effectively addresses the limitations of linear or purely exponential estimators in modeling nonlinear and periodic data patterns.
The proposed estimator is robust, demonstrating improved efficiency and practical utility across diverse data distributions (Normal and Gamma) and varying correlation strengths.
Theoretical derivations and empirical evidence confirm that the proposed method reduces Mean Squared Error more effectively than conventional estimators, making it a valuable tool for enhancing the accuracy of population parameter estimation in simple random sampling.

The paper concludes that while the study assumes the population median of the auxiliary variable is known and relies on first-order approximations, the results strongly support the adoption of the proposed estimator for practical applications. Future work is suggested to extend these methods to other parameters (mean, CDF, variance) and sampling designs (stratified, double sampling).

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