Maximum likelihood estimation of the Weibull distribution with reduced bias

Technical Summary: Maximum Likelihood Estimation of the Weibull Distribution with Reduced Bias

Problem Statement
The Weibull distribution is a standard model for survival data, characterized by a shape parameter ($k$) and a scale parameter ($\lambda$). While the maximum likelihood (ML) estimate for the scale parameter $\lambda$ exhibits negligible bias even in small samples, the ML estimate for the shape parameter $k$ is known to be strongly biased upward, particularly when sample sizes are small or when data is subject to Type I censoring. Existing bias-correction methods, such as those by Ross, Yang and Xie, and Shen and Yang, often rely on complex simulation-based procedures (e.g., bootstrap), non-linear function fitting, or modified profile likelihoods that require numerical optimization beyond standard ML routines.

Methodology
The authors derive new, first-order bias-adjusted maximum likelihood estimates (MMLE) for the shape parameter $k$ under two scenarios: complete data and Type I censored data. The methodology relies on the finite-sample bias approximation formula developed by Cox and Snell.

1. Theoretical Derivation:

   • The authors apply the Cordeiro and Klein approach to the Weibull distribution to derive the expected Fisher information matrix ($K$) and the matrix of expected third-order derivatives of the log-likelihood ($A$).
   • For complete data, they derive an explicit analytical approximation for the bias of the ML estimate of $k$. The bias is shown to be approximately $k \times (1.3795/n)$.
   • For Type I censored data, they derive a more complex bias formula dependent on the proportion of uncensored observations ($p$). This involves the incomplete gamma function and its derivatives. The resulting bias adjustment factor, $f(p)/n$, approaches the complete data constant as $p \to 1$ and diverges as $p \to 0$.
   • The reduced-bias estimate is calculated simply as $\tilde{k}_{ML} = \hat{k}_{ML} - \text{Bias}(\hat{k}_{ML})$, where the bias term is evaluated at the standard ML estimate.

2. Computational Advantage:

   • Unlike competing methods (e.g., Shen and Yang's profile ML or bootstrap-based corrections), the proposed MMLE does not require numerical simulation or iterative re-estimation. It can be computed trivially using any software that already implements standard Weibull ML estimation by applying the derived correction factor post-hoc.

Key Results
The paper validates the proposed estimator through extensive simulations ($10^5$ runs) and real-data applications.

• Simulation (Complete Data): Across sample sizes $n \in \{10, 20, 50\}$ and shape parameters $k \in \{0.5, 1, 5, 10\}$, the MMLE significantly reduces bias compared to standard ML. It also demonstrates lower Mean Squared Error (MSE) and Kullback–Leibler (KL) divergence compared to the conditional ML (MLC) of Yang and Xie. While the profile ML (MLP) of Shen and Yang shows slightly lower bias in some cases, the MMLE achieves virtually identical MSE and KL divergence with far less computational complexity.
• Simulation (Type I Censored Data): The MMLE outperforms standard ML, MLC, and MLP in terms of bias, MSE, and KL divergence, particularly for small sample sizes ($n=20$) and high censoring proportions ($p \leq 0.5$). Notably, the MLC method can fail (yielding negative estimates) when the number of uncensored observations is low ($d < 2$), whereas the MMLE remains stable.
• Real Data Applications:
  • Cable Insulation Failure: Applying the method to failure voltage data ($n=20$), the standard ML estimates for the shape parameter were 9.38 and 9.14. The bias-adjusted MMLE estimates were 8.74 and 8.51, respectively, correcting the known upward bias.
  • Criminal Recidivism: In a subsample of 20 observations with high censoring, the standard ML estimate for $k$ was 1.72, while the MMLE was 1.39. The latter was much closer to the estimate derived from the full dataset (1.37), demonstrating the robustness of the bias correction in small, heavily censored samples.

Significance and Claims
The authors position this work as a practical improvement to existing statistical tools. The primary contribution is the derivation of explicit, closed-form bias correction formulas for the Weibull shape parameter that are applicable to both complete and Type I censored data.

The paper claims that the proposed MMLE is:

1. Significantly less biased and more efficient than the standard ML estimate and the conditional ML estimate (MLC).
2. Computationally simple, as it requires no additional numerical optimization or bootstrap simulation, allowing for easy integration into existing software packages that already perform Weibull ML fitting.
3. Robust in scenarios where other methods (like MLC) may fail due to insufficient uncensored observations.

The authors also note a secondary contribution: the derivation of a novel formula for the Kullback–Leibler divergence between two Weibull distributions under Type I censoring, a result not widely known in the literature. While the paper suggests the methodology could theoretically extend to more complex censoring schemes (like progressive type I interval censoring), it acknowledges that analytic expressions for these cases are cumbersome and currently best handled via numerical implementation in existing software (e.g., the R package bccp).