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An Efficient Estimator of Population Mean Using a Poisson Regression-Based Approach: Simulation and Theoretical Efficiency Analysis

This paper proposes and validates a new Poisson regression-based estimator for the population mean under simple random sampling, demonstrating through theoretical analysis, empirical studies in environmental health and transportation safety, and simulations that it outperforms traditional ratio estimators in terms of mean square error.

Original authors: Rajesh Singh, Shobh Nath Tiwari, Yashveer Singh

Published 2026-09-28

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CC BY 4.0

Original authors: Rajesh Singh, Shobh Nath Tiwari, Yashveer Singh

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 weight of every apple in a vast orchard without weighing them all. To make a better guess, they look for a helpful clue, or "auxiliary variable," that is already known for the entire orchard, such as the average number of leaves on the trees. If the number of leaves is closely related to the weight of the fruit, using that information can sharpen the estimate. This is the foundation of ratio estimation, a method that has been refined for decades to improve accuracy. However, this traditional approach hits a wall when the data being counted—like the number of hospital visits or traffic accidents—does not follow a smooth, predictable curve. Count data often behaves differently, with many zeros and occasional spikes that break the assumptions of standard linear models. When these standard tools are forced onto such data, they can produce impossible results, like predicting negative numbers of accidents, or simply fail to capture the true pattern.

To solve this, a team of researchers from Banaras Hindu University in India has developed a new way to estimate population averages that is specifically designed for count data. Instead of forcing the data into a straight line, they utilized a Poisson regression model, a statistical tool built to handle counts that naturally vary over time or space. The researchers proposed a new mathematical formula that combines this specialized regression approach with the concept of using auxiliary information. They tested this new estimator against several existing methods using real-world data from two very different fields: air pollution and traffic safety. In one case, they looked at weekly hospital admissions for respiratory issues alongside air pollution levels; in the other, they examined traffic fatalities against the number of motor vehicles in various provinces. The results were striking. In the traffic safety dataset, the new method reduced the error of the estimate by a massive margin, performing thousands of times better than the standard baseline in some comparisons.

The researchers did not stop at real-world examples; they also ran thousands of computer simulations to see how the method would hold up under different conditions. They generated artificial populations where the relationship between the count data and the auxiliary clue varied in strength, and they tested sample sizes ranging from small to moderately large. Across every scenario, the new estimator consistently outperformed the older, traditional methods. It produced more accurate averages with less error, regardless of whether the sample size was small or large, or how strongly the variables were linked. The study suggests that for anyone working with count data—such as epidemiologists tracking disease cases or urban planners monitoring accidents—this new approach offers a significantly more reliable way to understand the true average of a population.

The core of the discovery lies in how the new formula handles the relationship between the known information and the unknown count. Traditional methods often treat the connection between variables as a simple, straight line, which can lead to errors when the data is skewed or contains extreme values. The new approach, however, uses a specific type of curve that respects the nature of count data, ensuring that predictions remain realistic and positive. By fine-tuning the constants within this formula, the researchers found a "sweet spot" that minimizes the difference between the estimated average and the true average. In their tests with real data, the new method achieved a level of precision that the older ratio estimators simply could not match. For instance, when analyzing traffic accident deaths, the new estimator was nearly twenty-three times more efficient than the standard method used as a baseline, meaning it required far less data to achieve the same level of certainty.

While the study focused on specific datasets and simulated scenarios, the implications for statistical practice are clear. The authors demonstrate that sticking to traditional linear methods for count data can lead to suboptimal results, and that switching to a regression model designed for counts can dramatically improve accuracy. They acknowledge that their work is a step forward, but not the final word. The researchers point out that future work could address even more complex situations, such as data with an excessive number of zeros or missing values, by adapting their method to more specialized models. For now, the study provides a robust, tested tool for improving the accuracy of population estimates in fields where counting events is essential, offering a clearer picture of reality than the methods currently in widespread use.

Technical Summary: An Efficient Estimator of Population Mean Using a Poisson Regression-Based Approach

Problem Statement
In sample survey theory, estimating the population mean is a primary objective, often enhanced by utilizing auxiliary information. While ratio and regression estimators are standard when a positive relationship exists between the study and auxiliary variables, classical linear regression models are frequently inappropriate for count data. Linear models may produce negative predictions and violate the assumption of constant variance, which is critical for hypothesis testing. Although Poisson regression is the standard for modeling count responses, there is a need for more efficient estimators of the population mean that specifically leverage the Poisson regression framework under Simple Random Sampling (SRS), particularly when competing estimators (such as those by Kadilar and Cingi or Koç) may not fully exploit the distributional properties of count data.

Methodology
The study proposes a novel estimator for the population mean, denoted as $Z^*$ , designed for use under SRS with count data. The methodology proceeds as follows:

Theoretical Framework: The study utilizes a Poisson regression model where the study variable $Y_i$ represents event counts following a Poisson distribution. The slope coefficient ( $b_P$ ) is derived from this Poisson regression model rather than traditional least squares.
Proposed Estimator Construction: Inspired by Raghav et al. and Koç, the authors define the new estimator $Z^*$ as a combination of a modified ratio term and a regression term, incorporating an exponential function of the ratio of the difference and sum of the auxiliary variable means:
$Z^* = \left[ \alpha \bar{y} \left\{ \frac{1}{2} \left( \frac{\bar{X}}{\bar{x}} + \frac{\bar{x}}{\bar{X}} \right) \right\} + b_P(\bar{X} - \bar{x}) \right] \exp\left( \frac{\bar{X} - \bar{x}}{\bar{X} + \bar{x}} \right)$
Here, $\alpha$ is a scalar constant determined by minimizing the Mean Square Error (MSE).
Derivation of Properties: Using first-order approximation and error terms ( $\epsilon_0, \epsilon_1$ ), the authors derive the expressions for the Bias and the MSE of $Z^*$ . The MSE is expressed in the form $MSE(Z^*) = \alpha^2 A + \alpha B + C$ . The optimal value of $\alpha$ is found by minimizing this expression ( $\alpha = -B/2A$ ), yielding the minimum MSE: $MSE_{min} = C - B^2/4A$ .
Comparative Analysis: The theoretical efficiency of $Z^*$ $Z^{*}$ is compared against:
- Traditional ratio estimators by Kadilar and Cingi ( $\bar{y}_{KC1}$ to $\bar{y}_{KC5}$ ).
- Poisson regression-based ratio estimators by Koç ( $\bar{y}_1$ to $\bar{y}_5$ ).
- A Poisson regression-based mean estimator by Shahzad et al. ( $\bar{y}_U$ ).
- Poisson regression-based ratio-cum-regression estimators by Jana et al. ( $Z_1$ to $Z_5$ ).

Key Contributions and Results
The paper validates the proposed estimator through both empirical studies using real-world datasets and extensive simulation studies.

Empirical Study: Two real populations were analyzed:
1. Population I: Air pollution data (Afyon, Turkey), linking weekly hospital admissions ( $Y$ ) to PM10 levels ( $X$ ).
2. Population II: Traffic accident data (Turkey, 2019), linking traffic deaths ( $Y$ ) to the number of motor vehicles ( $X$ ).
  In both cases, the proposed estimator $Z^*$ demonstrated the lowest Mean Square Error (MSE) and the highest Percentage Relative Efficiency (PRE). For Population II, $Z^*$ achieved a PRE of 2323.13% relative to the conventional estimator $\bar{y}_1$ , significantly outperforming all competing estimators.
Simulation Study: A simulation was conducted using R software with a finite population ( $N=1000$ ) where $X$ followed a uniform distribution and $Y$ followed a Poisson distribution with $\lambda = \exp(\beta_0 + \beta_1 X)$ . The study varied sample sizes ( $n=20, 30, 40$ ) and regression coefficients ( $\beta_1 = 0.75, 0.85, 0.95$ ).
- Across all scenarios, $Z^*$ consistently yielded the minimum MSE and maximum PRE compared to the estimators of Kadilar and Cingi, Koç, and Jana et al.
- The results confirmed that as sample size increased, the MSE decreased for all estimators, but the proposed estimator maintained superior efficiency.

Significance and Claims
The authors claim that the proposed estimator $Z^*$ offers a statistically significant improvement over existing methods for estimating the population mean of count data. The study asserts that:

The theoretical derivation of bias and MSE provides a robust foundation for the estimator's performance.
The empirical and simulation results validate that $Z^*$ is more efficient than traditional ratio estimators and other Poisson regression-based approaches currently in the literature.
The use of the exponential function and the specific combination of ratio and regression terms within the Poisson framework effectively captures the relationship between the study and auxiliary variables for count data.

The paper concludes that $Z^*$ is recommended for analyzing count datasets where auxiliary information is available. The authors modestly suggest that future work could extend this methodology to handle over-dispersion, excess zeros (via zero-inflated models), missing data, and other sampling designs like stratified or ranked set sampling, but they do not claim these extensions have been achieved in the current study.

Technical Summary: An Efficient Estimator of Population Mean Using a Poisson Regression-Based Approach

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