Estimation Method under Three-Parameter Generalized Exponential Model: Consistency, Uniqueness and its Applications

1. Problem Statement

The paper addresses the statistical challenges associated with estimating parameters for the three-parameter Generalized Exponential (GE) distribution, denoted as $GE(\alpha, \beta, \gamma)$, where $\alpha$ is the scale, $\beta$ is the shape, and $\gamma$ is the location parameter.

• Non-Regular Family: The GE distribution belongs to a "non-regular" family of distributions because the support of its probability density function (PDF) depends on the unknown location parameter $\gamma$.
• Failure of Standard MLE: In such non-regular cases, the standard Maximum Likelihood Estimation (MLE) often fails. Specifically:
  • The likelihood function can become unbounded when the shape parameter $\beta < 1$ and the location parameter $\gamma$ approaches the smallest observation in the sample.
  • Regularity conditions required for asymptotic normality and the existence of the Fisher Information matrix are violated.
  • Consequently, standard MLEs may not exist, or if they do, their asymptotic properties are undefined.
• Existing Alternatives: Previous methods, such as the Location-Scale-Parameter-Free (LSPF) and Location-Parameter-Free (LPF) approaches, have been proposed to handle these issues. However, the theoretical properties (specifically uniqueness and consistency) of LPF estimators for the GE distribution had not been rigorously proven, and the LSPF method suffers from high computational complexity and potential bias accumulation due to sequential estimation steps.

2. Methodology

The authors propose a refined application and theoretical analysis of the Location-Parameter-Free (LPF) estimation method.

• Data Transformation: The method utilizes order statistics $X_{(1)} < X_{(2)} < \dots < X_{(n)}$. It defines a location-invariant statistic:
  $$V_{(i)} = X_{(i)} - X_{(1)}, \quad i = 1, \dots, n$$
  Note that $V_{(1)} = 0$. The joint distribution of $V_{(2)}, \dots, V_{(n)}$ depends only on the scale ($\alpha$) and shape ($\beta$) parameters, effectively removing the location parameter $\gamma$ from the likelihood function.
• Likelihood Construction:
  • A bounded likelihood function $\ell_v(\alpha, \beta)$ is constructed based on the transformed data $v_2, \dots, v_n$.
  • The authors prove that this likelihood function is bounded and differentiable with respect to $\alpha$ and $\beta$, overcoming the unboundedness issue of the standard MLE.
• Estimation Procedure:
  1. Step 1: Maximize $\ell_v(\alpha, \beta)$ to obtain estimators $\hat{\alpha}_v$ and $\hat{\beta}_v$.
  2. Step 2: Estimate the location parameter $\gamma$ using the first order statistic $X_{(1)}$. To correct for bias, a bias-corrected estimator is proposed:
     $$\hat{\gamma}_v = X_{(1)} - \hat{\alpha}_v \int_0^\infty (1 - (1 - e^{-y})^{\hat{\beta}_v})^n dy$$
  3. Quantile Estimation: Quantiles are estimated using the plug-in principle with the derived estimators.

3. Key Contributions

The paper makes several theoretical and practical contributions:

• Theoretical Proofs:
  • Uniqueness: The authors prove that the likelihood function $\ell_v(\alpha, \beta)$ is unimodal with respect to one parameter when the other is fixed. While global unimodality of the bivariate function is complex, they establish conditions ensuring a unique maximum exists for specific cases.
  • Consistency: They provide a rigorous proof that the LPF estimators for $\alpha$, $\beta$, and the bias-corrected $\gamma$ are consistent (converge in probability to the true parameters as $n \to \infty$).
• Computational Efficiency:
  • The study demonstrates that the LPF method is significantly faster than the LSPF method.
  • Reason: The LPF method reduces the dimensionality of the integration required for the likelihood function compared to LSPF.
  • Result: In simulations, LPF reduced computation time from hours (LSPF) to minutes for moderate sample sizes (e.g., $n=100$).
• Bias Reduction: The LPF method avoids the sequential accumulation of bias inherent in the LSPF method, which estimates parameters step-by-step using previous estimates.

4. Results

The performance of the LPF method was evaluated via a Monte Carlo simulation study (10,000 replications) across various sample sizes ($n=20, 50, 100, 200$) and shape parameters ($\beta = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0$). It was compared against MLE (where valid), MMLE I, MMLE III, and LSPF.

• Parameter Estimation:
  • Shape ($\beta$) & Scale ($\alpha$): LPF and LSPF performed similarly and generally outperformed other methods.
    • For $\beta > 1$, LPF was recommended for scale estimation due to lower bias and RMSE.
    • For $\beta \le 1$, MMLE III often performed best, though LPF was competitive for larger sample sizes ($n \ge 50$).
  • Location ($\gamma$): LPF consistently outperformed all other methods (including LSPF) in terms of both bias and RMSE, particularly when $\beta \ge 1$.
• Quantile Estimation:
  • LPF provided the lowest bias for quantile estimates when $\beta \ge 1$.
  • For small $\beta$ ($\le 1$), MMLE III showed superior performance, but LPF remained robust for $n > 20$.
• Rejection Rates:
  • LPF had a higher rejection rate (due to numerical instability) compared to MMLE I, but significantly lower computational cost than LSPF.
  • MMLE I had the lowest rejection rate (near zero) across all scenarios.
• Real Data Application:
  • The method was applied to an electrical lifetime dataset (40 observations).
  • Goodness-of-fit tests (Kolmogorov-Smirnov and Cramér-von Mises) indicated that LPF, LSPF, and MMLE III provided excellent fits, with LPF performing slightly better.
  • Bootstrap confidence intervals were constructed, showing stable estimates.

5. Significance

This paper is significant for the following reasons:

1. Theoretical Rigor: It fills a gap in the literature by providing the first analytical proofs of uniqueness and consistency for LPF estimators in the context of the three-parameter GE distribution.
2. Practical Utility: It offers a computationally efficient alternative to the LSPF method. The drastic reduction in simulation time (from hours to minutes) makes LPF viable for large-scale simulation studies and real-time engineering applications.
3. Robustness: By avoiding the sequential bias accumulation of LSPF and the unbounded likelihood issues of standard MLE, the LPF method provides a robust framework for analyzing skewed lifetime data, particularly in reliability engineering where location parameters and high quantiles are critical.
4. Guidance for Practitioners: The study provides clear guidelines on which estimation method to choose based on the shape parameter ($\beta$) and sample size ($n$), helping researchers select the optimal estimator for their specific data characteristics.

In conclusion, the authors successfully establish the LPF method as a theoretically sound and computationally superior approach for estimating parameters in the three-parameter Generalized Exponential distribution, particularly when dealing with non-regular conditions where standard MLE fails.