Estimation of Neutrosophic Population Mean Incorporating Auxiliary Information under Simple Random Sampling

Technical Summary: Estimation of Neutrosophic Population Mean Incorporating Auxiliary Information under Simple Random Sampling

Problem Statement
Traditional statistical methods for estimating population means rely on exact, crisp data. However, real-world data often contain uncertainty, vagueness, ambiguity, or indeterminacy (e.g., in weather forecasts, economic growth monitoring, or temperature measurements). While fuzzy logic addresses imprecision, it does not explicitly model indeterminacy. Neutrosophic statistics, an extension of classical statistics introduced by Smarandache (2014), addresses this gap by allowing observations to contain both determinate and indeterminate components. Despite the potential of neutrosophic statistics, its application in sampling theory for estimating population parameters remains under-explored compared to classical methods. Existing estimators often suffer from bias or suboptimal efficiency when applied to indeterminate data. This research addresses the need for an estimator that incorporates auxiliary information to estimate the neutrosophic population mean with reduced bias and Mean Squared Error (MSE).

Methodology
The study operates under the Simple Random Sampling Without Replacement (SRSWOR) framework within a neutrosophic environment. The authors define neutrosophic variables (study variable $Y_N$ and auxiliary variable $X_N$) as intervals $[L, U]$ to represent indeterminacy.

1. Proposed Estimator Formulation:
   The authors propose a new class of "almost unbiased" neutrosophic estimators ($t_N^*$). This estimator is constructed as a linear combination of three components:

   • The standard neutrosophic sample mean ($\bar{y}_N$).
   • A modified ratio-type exponential estimator involving auxiliary parameters ($a_N, b_N$) and constants ($K_{1N}, K_{2N}$).
   • A logarithmic ratio-type estimator involving a constant ($K_{3N}$).

   The general form is:
   $$t_N^* = l_{0N}\bar{y}_N + l_{1N}\bar{y}_N \left(\frac{a_N\bar{X}_N + b_N}{a_N\bar{x}_N + b_N}\right)^{K_{1N}} \exp\left(K_{2N}\left[\frac{(a_N\bar{X}_N + b_N) - (a_N\bar{x}_N + b_N)}{(a_N\bar{X}_N + b_N) + (a_N\bar{x}_N + b_N)}\right]\right) + l_{2N}\bar{y}_N\left(1 + \log\left(\frac{\bar{x}_N}{\bar{X}_N}\right)^{K_{3N}}\right)$$
   where $l_{0N}, l_{1N}, l_{2N}$ are statistical constants determined to minimize bias and MSE.

2. Theoretical Derivation:
   Using first-order approximations, the authors derive the expressions for the Bias and MSE of the proposed estimator.

   • Bias: The estimator is designed to be "almost unbiased," meaning its first-order bias is extremely small. The authors apply a linear constraint ($\sum l_{iN} \text{Bias}(t_{iN}^*) = 0$) to eliminate the first-order bias term.
   • MSE: The MSE is derived and minimized with respect to the optimal value of the combined constant $l_N$. The minimum MSE is shown to be equivalent to that of the traditional linear regression estimator in the neutrosophic context: $MSE_{min}(t_N^*) = f_N \bar{Y}_N^2 C_{yN}^2 (1 - \rho_{yxN}^2)$.

3. Variants:
   The paper demonstrates that the proposed estimator encompasses various existing neutrosophic estimators (ratio, product, exponential ratio, exponential product, and those using coefficients of variation and kurtosis) as special cases by adjusting the constants $a_N, b_N, K_{1N}, K_{2N}, K_{3N}$.

Key Contributions

• Novel Estimator: Development of a generalized, almost unbiased neutrosophic estimator that integrates auxiliary information (means, coefficients of variation, correlation, and kurtosis) to estimate the population mean under uncertainty.
• Theoretical Framework: Mathematical derivation of the bias and MSE for the proposed estimator under first-order approximation, proving its efficiency relative to existing methods.
• Comprehensive Evaluation: The study validates the theoretical findings through two distinct approaches:
  • Empirical Study: Analysis of four real-world datasets with indeterminacy: NVIDIA stock prices, ServiceNow stock prices, seasonal temperature records, and natural growth rates.
  • Simulation Study: A robust simulation involving 10,000 replications across Normal, Gamma, and Lognormal distributions with varying sample sizes ($n=100, 200, 300$) to test performance under symmetric and skewed population structures.

Results
The performance of the proposed estimator ($t_N^*$) was compared against 13 existing neutrosophic estimators ($t_{1N}$ through $t_{13N}$) using Mean Squared Error (MSE) and Percentage Relative Efficiency (PRE).

• Empirical Findings: Across all four real-world datasets, the proposed estimator consistently yielded the lowest MSE and the highest PRE.
  • Dataset 1 (NVDA): MSE range $[94.39, 105.88]$ vs. existing ranges up to $[1710, 1808]$; PRE range $[547, 558]$.
  • Dataset 4 (Growth Rate): MSE range $[0.06, 0.22]$ vs. existing ranges up to $[2.70, 8.15]$; PRE range $[1230, 1462]$.
• Simulation Findings: Under Normal, Gamma, and Lognormal distributions, the proposed estimator maintained superior efficiency.
  • In the Lognormal population (highly skewed), the proposed estimator achieved PRE values up to $248.86$, significantly outperforming competitors.
  • The estimator demonstrated robustness across different sample sizes and distributional shapes, consistently minimizing the neutrosophic MSE interval.

Significance and Claims
The paper claims that the proposed estimator effectively utilizes auxiliary information to estimate the finite population mean even when data is unclear, vague, ambiguous, or indeterminate. The authors assert that:

1. The estimator is almost unbiased with a first-order bias that is negligible.
2. It provides superior performance over traditional and existing neutrosophic estimators in terms of both MSE and PRE.
3. The neutrosophic framework is particularly advantageous for fields involving uncertainty, such as environmental monitoring, forecasting, and complex system evaluation.

The study concludes that the proposed method offers a more flexible and accurate approach for statistical analysis in the presence of indeterminacy, filling a gap in the current literature on neutrosophic sampling theory.