Principal Component Based Estimation of Finite Population Mean under Multicollinearity

1. Problem Statement

In survey sampling, the use of auxiliary information (variables correlated with the study variable) is a standard technique to improve the precision of population mean estimators. However, when multiple auxiliary variables are used simultaneously, they often exhibit high inter-correlation, a phenomenon known as multicollinearity.

• Consequences: Multicollinearity inflates the variance of conventional estimators (such as ratio, product, and regression estimators), leading to unstable estimates, large standard errors, and poor performance.
• Gap: While Principal Component Analysis (PCA) is a known statistical tool for handling multicollinearity in regression, its application in constructing specific estimators for finite population means under Simple Random Sampling Without Replacement (SRSWOR) requires further development, particularly when auxiliary variables exhibit skewness or extreme fluctuations.

2. Methodology

The authors propose a novel estimation framework that integrates Principal Component Analysis (PCA) with a logarithmic-type transformation to address multicollinearity.

A. Theoretical Framework

• Sampling Design: Simple Random Sampling Without Replacement (SRSWOR).
• Variables:
  • $Y$: Study variable.
  • $X, Z$: Two auxiliary variables (highly correlated).
• PCA Transformation:
  • The correlated auxiliary variables ($X, Z$) are standardized and transformed into orthogonal principal components.
  • The covariance matrix of the auxiliary variables is decomposed into eigenvalues and eigenvectors.
  • The first principal component ($W$), which captures the majority of the variance, is retained to replace the original correlated variables, effectively removing multicollinearity.

B. Proposed Estimators

The study introduces two specific estimators:

1. Log-Type Estimator ($t^*$): A generalized estimator incorporating known parameters of the auxiliary variables ($X, Z$) using logarithmic functions to stabilize ratios and handle skewness.
   $$t^* = y + K \log\left(\frac{X + \psi_1}{x + \psi_1}\right) + (1-K) \log\left(\frac{z + \psi_2}{Z + \psi_2}\right)$$
2. PCA-Based Estimator ($t_{PCA}$): This estimator utilizes the transformed, uncorrelated principal component ($W$) within a similar log-type structure:
   $$t_{PCA} = y + \alpha \log\left(\frac{W + \psi}{w + \psi}\right)$$
   • Here, $\alpha$ is a constant optimized to minimize the Mean Square Error (MSE).

C. Derivation of Properties

• Bias and MSE: The authors derived the Bias and Mean Square Error (MSE) for both proposed estimators up to the first order of approximation using error terms ($e_0, e_1, e_2$).
• Optimization: The constants ($K$ and $\alpha$) are determined by minimizing the MSE expressions, yielding optimal values ($K_{opt}$ and $\alpha_{opt}$) and minimum MSE formulas.

3. Key Contributions

1. Novel Integration: The paper successfully bridges the gap between PCA (a dimensionality reduction technique) and survey sampling estimation, specifically addressing the issue of multicollinearity in finite population mean estimation.
2. Log-Type Transformation: By incorporating logarithmic functions, the proposed estimators are designed to be more robust against skewness and extreme variations in auxiliary data, which often plague conventional ratio estimators.
3. Comprehensive Evaluation: The study provides a rigorous mathematical derivation of bias and MSE, followed by validation through both real-world data analysis and extensive simulation studies.
4. Diagnostic Rigor: The study explicitly quantifies multicollinearity using Variance Inflation Factors (VIF), Condition Indices (CI), and eigenvalues to justify the necessity of the PCA approach.

4. Results and Evaluation

The performance of the proposed estimators ($t^*$ and $t_{PCA}$) was compared against conventional estimators ($t_0$ to $t_5$) using Mean Square Error (MSE) and Percentage Relative Efficiency (PRE).

A. Real Dataset Analysis

• Data Source: Maddala and Lahiri (1959) dataset involving Imports ($Y$), GDP ($X$), and Consumption ($Z$).
• Multicollinearity Check: The correlation between $X$ and $Z$ was extremely high ($\rho_{xz} = 0.9989$), with a VIF of 468.83, indicating severe multicollinearity.
• Performance:
  • The PCA-based estimator ($t_{PCA}$) achieved the lowest MSE (0.1974) and the highest PRE (3310.96%).
  • The proposed log-type estimator ($t^*$) also outperformed all conventional estimators (PRE: 3296.60%).
  • Conventional estimators like the product-cum-product ($t_3$) performed poorly due to the positive correlation structure.

B. Simulation Study

• Setup: A tri-variate normal distribution was simulated ($N=1000$) with varying correlation levels ($\rho_{xz}$ from 0.3 to 0.9) and sample sizes ($n=50, 100, 150$).
• Findings:
  • As $\rho_{xz}$ increased, the VIF and Condition Index increased, confirming worsening multicollinearity.
  • Across all correlation levels and sample sizes, the PCA-based estimator ($t_{PCA}$) consistently yielded the lowest MSE.
  • The efficiency gap widened as multicollinearity became more severe, demonstrating the robustness of the PCA approach.

5. Significance and Conclusion

The study concludes that the presence of multicollinearity significantly degrades the performance of traditional estimators that rely on multiple auxiliary variables. The proposed PCA-based log-type estimator offers a robust solution by:

1. Eliminating Redundancy: Transforming correlated variables into orthogonal components.
2. Stabilizing Estimates: Using log-transformations to handle non-normality and extreme values.
3. Maximizing Efficiency: Consistently outperforming existing methods in terms of MSE and PRE.

Future Directions: The authors suggest extending this methodology to high-dimensional settings, other population parameters (variance, median), alternative sampling designs (stratified, cluster), and scenarios involving non-response or measurement errors.

In summary, this paper provides a statistically sound and practically effective methodology for survey practitioners dealing with highly correlated auxiliary data, ensuring more accurate and stable population mean estimates.