Fuzzy Logistic Liu-type Parameter Estimators

• FLLTPE are advanced fuzzy logistic estimators that use Liu-type shrinkage to mitigate instability and variance inflation in high-collinearity settings.
• They generalize ridge- and Liu-type methods by employing IRLS optimization to balance bias and variance for improved model performance.
• Empirical studies demonstrate that FLLTPE achieve lower mean square error and superior goodness-of-fit compared to traditional fuzzy maximum likelihood and ridge methods.

Fuzzy Logistic Liu-type Parameter Estimators (FLLTPE) are advanced shrinkage-based estimators formulated for fuzzy logistic regression models where both the regression parameters and the response variable are modeled as triangular fuzzy numbers. FLLTPE specifically target instability and variance inflation in parameter estimates due to multicollinearity among predictors. The approach generalizes ridge- and Liu-type penalization to the fuzzy logistic setting, yielding estimators that demonstrate improved mean square error (MSE) and predictive goodness of fit compared to traditional fuzzy maximum likelihood and ridge methods, as validated by simulation and real-data studies involving high-collinearity structures (Shemail et al., 27 Dec 2025).

1. Framework for Fuzzy Logistic Regression

In the fuzzy logistic regression framework, each response $\tilde{y}_i$ is represented as a triangular fuzzy number $\tilde{y}_i = (y_{iL}, y_{iM}, y_{iU})$, and each regression parameter $\tilde{\beta}_j$ likewise is a triangular fuzzy number $(\beta_{jL}, \beta_{jM}, \beta_{jU})$. The membership function for such numbers is defined piecewise, and componentwise operations (addition, scalar multiplication) are strictly observed. The model—given $n$ observations with crisp predictors $x_i\in\mathbb{R}^p$—specifies the fuzzy probability as

$$\tilde{\pi}_i = \frac{ \exp( \tilde{\beta}_0 + \tilde{\beta}_1 x_{i1} + \cdots + \tilde{\beta}_p x_{ip} ) } { 1 + \exp( \tilde{\beta}_0 + \tilde{\beta}_1 x_{i1} + \cdots + \tilde{\beta}_p x_{ip} ) },$$

where calculation uses fuzzy arithmetic rules (Shemail et al., 27 Dec 2025).

For computational feasibility and algorithmic iteration (e.g., IRLS), the mid-point vectors $\beta$ and $y_M$ are frequently used, and inference is based on penalized log-likelihood of these midpoints.

2. Liu-type Regularization and FLLTPE Construction

The genesis of FLLTPE lies in the generalization of the Liu estimator to the context of fuzzy logistic regression. The classic ridge estimator introduces a penalty $k$ on the $\ell_2$ norm of the parameter vector, while the Liu-type variants operationalize an additional shrinkage governed by a parameter $d$. For the fuzzy case, the Liu-type estimators minimize

$$\ell_{\text{pen}}(\beta) = \ell(\beta) - \tfrac{1}{2}k\beta^\top\beta,$$

but with $k$ itself selected via an optimality criterion connected to $d$, following the formulation of Asar & Genç (2016). The Liu-type estimator interpolates between unpenalized and ridge forms, inducing beneficial bias-variance trade-offs under fuzzy MSE.

The optimization can be implemented using an IRLS scheme, with each update solving a system

$(X^\top W X + k I)\beta = X^\top W z,$

where $W=\text{diag}(\pi_i(1-\pi_i))$ at the midpoints and $z$ encodes the working response for the current $\beta^{(t)}$. FLLTPE is designed as an extension optimizing both $k$ and $d$ for minimal fuzzy MSE.

3. Selection of Liu-type Parameters

FLLTPE requires careful determination of the Liu-shrinkage parameter $d$ and consequently the penalty $k$. The parameter $d$ is chosen from $(0,1)$, typically via robust estimates (median, min, or max over the set $\{ \lambda_j a_j^2/(1+\lambda_j) \}$, where $\lambda_j$ are eigenvalues of $X^\top W X$ and $a_j$ are the response components in the eigenbasis). With $d$ fixed, $k$ is set according to

$$k = \frac{ \sum_{j=1}^p \lambda_j a_j^2 (1-d) - d \sum_{j=1}^p a_j^2 }{ \sum_{j=1}^p a_j^2 },$$

which aligns the estimator to the optimal bias–variance configuration with respect to the fuzzy MSE criterion (Shemail et al., 27 Dec 2025).

4. Comparative Theoretical Properties

Under the adopted IRLS approximation, the FLLTPE and related Liu-type estimators yield closed-form expressions for bias, variance, and MSE:

• Bias:

$$\text{Bias}(\hat{\beta}_{\text{FLRE}}) = -(X^\top W X + kI)^{-1}k\beta_0$$

• Variance:

$$\text{Var}(\hat{\beta}_{\text{FLRE}}) = (X^\top W X + kI)^{-1} (X^\top W X) (X^\top W X + kI)^{-1}$$

• MSE:

$$\text{MSE}(\hat{\beta}_{\text{FLRE}}) = (X^\top W X + kI)^{-1} (X^\top W X) (X^\top W X + kI)^{-1}$$

The result is that nonzero $k$ (and, by construction, appropriately chosen $d$) lowers estimator variance while maintaining a manageable increase in bias, minimizing overall estimation error in scenarios of strong multicollinearity (Shemail et al., 27 Dec 2025).

5. Simulation-Based Performance Assessment

Monte Carlo studies on fuzzy logistic models with $p\in\{3,5,7\}$ predictors, $n\in\{25,50,100,150\}$, and correlation $\rho\in\{0.70, 0.90, 0.99\}$ demonstrate that FLLTPE consistently achieves the lowest MSE and superior goodness-of-fit metrics relative to FMLE, FLRE, and FLLE. For example, in settings with $p=7$ and $n=150$, the trend in MSE and goodness-of-fit for FLRE confirms sensitivity to increased predictor collinearity, but FLLTPE achieves superior error reduction, outperforming all compared estimators in these metrics (Shemail et al., 27 Dec 2025).

6. Application to Real Fuzzy Data

A case study on kidney failure data with $n=35$ patients and predictors (blood sugar, hemoglobin, age—all highly collinear) provides empirical validation. Fuzzy ratings for the binary outcome were encoded as triangular numbers reflecting Low, Medium, High categories. Table 9 in the referenced work records that FLLTPE delivered the smallest MSE (≈0.5342) and best GOF among all estimators, improving substantially over both the fuzzy MLE (MSE=1.8073) and ridge (MSE=1.4152), and surpassing the original Liu estimators (FLLE ≈ 1.2882, FLLTE ≈ 0.5672) (Shemail et al., 27 Dec 2025). This suggests FLLTPE's strong practical value in domains characterized by observation ambiguity and multicollinear predictors.

7. Conclusion and Outlook

FLLTPE, as introduced and analyzed in the cited work, generalizes Liu-type penalization methods to the fuzzy logistic regression context, addressing the twin challenges of fuzziness in data/parameters and severe linear dependency among predictors. The approach yields estimators that are computationally tractable via IRLS, statistically robust against multicollinearity, and empirically superior in mean square error and goodness of fit to previously established techniques. Ongoing research may further refine fuzzy parameter selection heuristics and extend the framework to alternative membership function shapes, but current evidence establishes FLLTPE as the preferred estimator in high-collinearity, fuzzy-valued logistic modeling environments (Shemail et al., 27 Dec 2025).