Fuzzy Confidence Intervals

• Fuzzy confidence intervals are statistical intervals assigning fractional coverage probabilities, ensuring exact frequentist coverage without overconservatism.
• They are constructed via the Neyman–Pearson lemma to minimize expected interval length while meeting global coverage constraints.
• The method extends to classical models like Binomial, Poisson, and Normal, and informs uncertainty quantification in fuzzy logic systems and regression discontinuity designs.

Fuzzy confidence intervals generalize classical and randomized confidence sets by permitting the assignment of fractional coverage probabilities across the parameter space. This approach facilitates exact frequentist coverage without overconservatism in discrete models and enables the construction of optimal intervals—minimizing expected length—by leveraging hypothesis-testing duality through the Neyman–Pearson lemma. The fuzzy confidence interval framework has further inspired robust quantification of uncertainty in other areas, such as fuzzy logic systems for prediction intervals, and specialized applications to scenarios with irregular design, such as fuzzy regression discontinuity setups.

1. Formal Definition and Theoretical Framework

Let $(\Omega, \mathcal{A}, \mu(\cdot|\theta))$ denote a statistical experiment with parameter $\theta \in \Theta$ and observed data $\omega \in \Omega$. For a fixed confidence level $\gamma \in (0,1)$, a measurable function

$\psi: \Omega \times \Theta \longrightarrow [0,1]$

is a $100\gamma$% fuzzy confidence interval if, for every $\tau \in \Theta$,

$$\int_\Omega \psi(\omega|\tau) \, d\mu(\omega|\tau) \geq \gamma.$$

For a given realization $\omega$, $\psi(\omega|\cdot)$ is the membership function of a fuzzy set on $\Theta$. If $\psi$ is $\{0,1\}$-valued, this coincides with a classical confidence set. Fractional values for $\psi$ over subregions of $\Theta$ yield a genuinely fuzzy or randomized interval.

In contrast to classical and boundary-randomized procedures, the fuzzy approach enables exact finite-sample size control in discrete settings, eliminating systematic overcoverage (Felix et al., 29 Dec 2025).

2. Neyman–Pearson Construction and Optimality

The construction of optimal fuzzy intervals is based on the duality between acceptance rules in hypothesis testing and confidence set inversion. For a “reference” parameter value $o \in \Theta$, the fuzzy acceptance rule $\psi_o$ is constructed to minimize the expected length at $\theta = o$, subject to the global coverage constraint.

The key steps are:

• Consider the test of $H_0: \theta = o$ versus $H_1: \theta = \tau$.
• Let $\nu = \mu(\cdot|o)$ and $\mu = \mu(\cdot|\tau)$, and define the Radon–Nikodym derivative (likelihood ratio) $Y(\omega) = d\nu/d\mu(\omega)$.
• For any $\gamma \in (0,1)$, form the acceptance function

$$\psi^*(\omega) = \begin{cases} 1, & Y(\omega) < Q(\gamma),\ 0, & Y(\omega) > Q(\gamma),\ (\gamma - \mu(A)) / \mu(C), & Y(\omega) = Q(\gamma), \end{cases}$$

where $Q(\gamma)$ is the $\gamma$-quantile of $Y$ under $\mu$, and $A, B, C$ are the sets where $Y(\omega)$ is $<$, $>$, or $=$ to this quantile. Assembling $\psi^*_o( \cdot | \tau )$ for all $\tau$ yields the membership function $\psi_o$ of the fuzzy interval.

Theorem 2.1 in (Felix et al., 29 Dec 2025) establishes that, among all rules with coverage at least $\gamma$, this $\psi^*$ minimizes the expected length for $\theta=o$.

3. Explicit Forms in Classical Models

The methodology specializes to standard statistical families:

Binomial$(n,\theta)$: $\Omega = \{0,1,\ldots,n\}$ with

$$Y(\omega) = \left( \frac{o(1-\tau)}{\tau(1-o)} \right)^\omega \left( \frac{1-o}{1-\tau} \right)^n.$$

The cutoff index $i$ solves

$$\sum_{j=0}^{i-1} \binom{n}{j}\tau^j(1-\tau)^{n-j} < \gamma \leq \sum_{j=0}^{i} \binom{n}{j}\tau^j(1-\tau)^{n-j}.$$

Equation (5.5) in (Felix et al., 29 Dec 2025) gives

$$\psi_o(\omega|\tau) = \begin{cases} 1, & \tau > I^{-1}(1-\gamma, i+1, n-i),\ 0, & \tau \leq I^{-1}(1-\gamma, i, n-i+1),\ \text{fractional}, & \text{otherwise}, \end{cases}$$

where $I^{-1}$ is the inverse regularized beta function.

Poisson$(\theta)$: With $Y(\omega) = \exp( -(o-\tau) ) (o/\tau)^\omega$, and similar partitioning, the acceptance function takes the form (5.10): $$\psi_o(\omega|\tau) = \begin{cases} 1, & \tau > \chi^2_{2\omega+2,1-\gamma},\ 0, & \tau \leq \chi^2_{2\omega,1-\gamma},\ \frac{\gamma - \sum_{j<\omega}e^{-\tau}\tau^j/j!}{e^{-\tau} \tau^\omega / \omega!}, & \text{otherwise}. \end{cases}$$

Normal$(\theta, \sigma^2/n)$: For $X \sim N(\theta, \sigma^2/n)$,

$$Y(x) = \exp \left\{ \frac{n}{\sigma^2} (o-\tau)x - \frac{n}{2\sigma^2}(o^2-\tau^2) \right\}.$$

For the continuous case, fractional allocations are not needed and the one-sided region exactly reproduces the classical confidence interval with truncation to $[a,b]$ as needed.

4. Properties: Coverage, Minimaxity, and Uniqueness

• Exact Coverage: By construction, $\psi_o$ achieves $$\int_\Omega \psi_o(\omega|\tau) d\mu(\omega|\tau) \geq \gamma$$ for all $\tau$, with fractional allocation on the equality set ensuring exactness.
• Minimized Expected Fuzzy Length: Letting $\nu$ be a measure on $\Theta$, the expected length

$$EL(\theta, \psi) = E_\theta \left[ \int_\Theta \psi(\omega|\tau)\, d\nu(\tau) \right]$$

is minimized by $\psi_\theta$ among all $\psi \in \mathcal{F}_\gamma$ (Felix et al., 29 Dec 2025).

• Bernoulli/Universal Bound: For the binomial case, the minimal maximal expected length is achieved by the fuzzy interval (Theorem 2.3).

5. Comparative Performance and Examples

Relative performances are quantified in (Felix et al., 29 Dec 2025) via expected lengths at fixed $\theta$. For example, for binomial $n=10$, $\gamma=0.95$ at $\theta=0.5$:

Method	Expected Length
Lower bound	0.40
$\psi_o$ $(o=0.5)$	0.42
Geyer–Meeden	0.48
Agresti–Coull	0.51
Asymptotic (Wald)	0.45

At $\theta=5$ for Poisson:

Method	Expected Length
Lower bound	2.10
$\psi_o$ $(o=5)$	2.12
Geyer–Meeden	2.40
Score	2.35

In high-variance or small-sample scenarios, the fuzzy method offers intervals that are tangent to the theoretical lower bound at the reference point and outperform standard approaches over a region around this point.

6. Implementation Details and Practical Guidance

Evaluation requires computation of either beta-quantiles or tail probabilities for the binomial, Poisson tail sums or $\chi^2$ quantiles for the Poisson, and standard normal quantiles for the normal model. The R package FRCI implements these constructions for the major parametric families.

The choice of reference value $o$ can be informed by prior knowledge, an empirical Bayes approach (e.g., $o = \operatorname{MLE}$), or profile-likelihood considerations. Regardless of this choice, the procedure retains exact frequentist coverage. If the parameter space is bounded, the membership function is truncated outside the domain.

7. Connections, Generalizations, and Alternative Fuzzy Intervals

Fuzzy confidence intervals underpin more elaborate uncertainty quantification frameworks:

• Type-2 Fuzzy Logic Systems (FLSs) produce high-quality prediction intervals by generalizing Zadeh's type-2 fuzzy sets using $\alpha$-plane representations. In these systems, the prediction interval derives directly from the type-reduced set at the lowest $\alpha$-plane, optimizing both empirical coverage and sharpness. Models such as Z-GT2-FLS achieve both accurate point prediction (low RMSE) and tight envelope coverage (high PICP with low PINAW) using deep learning optimizers and novel decoupling of primary and secondary membership functions (Guven et al., 2024).
• Fuzzy Regression Discontinuity Designs employ confidence sets that adapt to the uncertainty induced by local structure, weak identification, or discrete running variables. Anderson–Rubin-type tests and interval inversion furnish bias-aware confidence sets, controlling type I error under a wide spectrum of empirical scenarios (Noack et al., 2019).

A plausible implication is that the fuzzy confidence interval principle, founded on test inversion and exact coverage, provides a unifying paradigm for uncertainty quantification across discrete, nonregular, and fuzzy-logic-driven predictive modeling.

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References:

• "A Fuzzy Approach for Randomized Confidence Intervals" (Felix et al., 29 Dec 2025)
• "Bias-Aware Inference in Fuzzy Regression Discontinuity Designs" (Noack et al., 2019)
• "Zadeh's Type-2 Fuzzy Logic Systems: Precision and High-Quality Prediction Intervals" (Guven et al., 2024)