Jackknife Empirical Likelihood (JEL) Method

• Jackknife Empirical Likelihood (JEL) is a nonparametric method that uses jackknife pseudo-values to transform nonlinear constraints into a linearized inference problem.
• The AJEL extension augments the pseudo-values with an artificial value to guarantee a well-defined empirical likelihood ratio for all parameter values.
• JEL enables robust confidence interval construction and improved coverage accuracy while addressing computational challenges in U-statistic-based estimations.

The Jackknife Empirical Likelihood (JEL) method is an extension of empirical likelihood (EL) that incorporates jackknife pseudo-values to enable nonparametric likelihood-based inference for estimators—most notably those based on nonlinear or U-statistic estimating equations. JEL overcomes key computational and theoretical limitations of the standard EL approach, particularly in the presence of nonlinear constraints, while preserving Wilks' theorem and the associated asymptotic chi-square calibration. The Adjusted Jackknife Empirical Likelihood (AJEL) modification further guarantees well-definedness of the empirical likelihood ratio for all parameter values, removing the convex hull constraint that may otherwise impede existence of solutions.

1. Foundational Concepts and Motivation

Traditional EL constructs nonparametric likelihood ratios under constraints imposed by estimating equations. For regular mean-type constraints, this results in tractable optimization and the Wilks phenomenon: $-2\log R(\theta)$ converges in distribution to a chi-square under mild conditions. However, if the constraint is nonlinear (e.g., a nonlinear function of a U-statistic), the maximization becomes computationally infeasible, and the zero vector must belong to the interior of the convex hull of the estimating equations—otherwise the statistic is undefined.

The JEL framework resolves these challenges by replacing nonlinear constraints with jackknife pseudo-values, turning the estimation into a linearized sample mean problem. For a U-statistic $U_n$ of degree $m$ with symmetric kernel $h$, and parameter of interest $\theta = \mathbb{E}[h(Z_1, ..., Z_m)]$ from a sample $\{Z_i\}_{i=1}^n$, the jackknife pseudo-values are defined by: $V_i = n U_n - (n-1)U_{n-1}^{(-i)}$ where $U_{n-1}^{(-i)}$ is the U-statistic computed after removing $Z_i$. These pseudo-values satisfy $\mathbb{E}(V_i) = \theta$ and have mild dependence.

The AJEL method augments the set of pseudo-values with an artificial pseudo-value: $V_{n+1} = -a_n U_n,$ where $a_n > 0$ (typically $a_n = (\log n)/2$ or any $a_n = o_p(n^{2/3})$), ensuring that $0$ lies in the convex hull. Thus, AJEL overcomes the domain restriction (convex hull constraint) inherent in EL and JEL by construction.

2. Mathematical Formulation

Log-Likelihood Ratio Construction

Given the set $\{V_1, ..., V_n, V_{n+1}\}$, define the empirical likelihood profile for parameter value $\theta$ as: $$L(\theta) = \sup \left\{ \prod_{i=1}^{n+1} (n+1)p_i : \sum_{i=1}^{n+1} p_i = 1,\, \sum_{i=1}^{n+1} p_i(V_i - \theta) = 0 \right\}.$$ The AJEL log-likelihood ratio is

$$\ell_n(\theta) = -\sum_{i=1}^{n+1} \log\left\{1 + \lambda(V_i - \theta)\right\},$$

where $\lambda$ is a Lagrange multiplier solving

$$\sum_{i=1}^{n+1} \frac{(V_i - \theta)}{1 + \lambda (V_i - \theta)} = 0.$$

In both the one-sample and two-sample U-statistic cases, the pseudo-values are constructed by leave-one-out deletions from the sample(s), and the constraint is imposed on their (weighted) mean.

Asymptotic Distribution

Under regularity conditions, for both JEL and AJEL,

$-2\ell_n(\theta_0) \xrightarrow{d} \chi^2_1$

as $n \to \infty$, where $\theta_0$ is the true parameter value. This is a direct analogue of Wilks’ theorem, meaning AJEL-based confidence sets and tests inherit the asymptotic chi-square calibration of the original EL approach.

3. Properties and Theoretical Guarantees

• Existence and Well-Definedness: The augmentation in AJEL guarantees that for all $\theta$, the required constraint can be satisfied and the empirical likelihood is defined, eschewing the need to assign $-\infty$ to the log-likelihood at “problematic” values (as in Owen, 2001).
• Coverage Properties: Simulations show markedly improved coverage accuracy for the AJEL method, particularly in small samples, relative to standard JEL. Confidence intervals tend to be slightly longer, but the trade-off is favorable as improved coverage probabilities are achieved.
• Flexibility: AJEL is widely applicable for statistical procedures based on U-statistics, including but not limited to ROC curve estimation, mean residual life differences, and comparison of indices.
• Robustness: The pseudo-value construction simplifies dependence and nonlinearity issues, allowing for reliable inference even with nonlinear estimating equations.

4. Practical Implementation and Applications

Step-by-Step Workflow

1. Calculate U-statistic: Compute $U_n$ for the parameter of interest.
2. Calculate Pseudo-values: For $i=1, ..., n$, compute $U_{n-1}^{(-i)}$ and form $V_i = nU_n - (n-1)U_{n-1}^{(-i)}$.
3. Augment with Artificial Pseudo-value: Compute $V_{n+1} = -a_n U_n$ for suitably chosen $a_n$.
4. Set Up Empirical Likelihood: Solve for $\lambda$ in the log-likelihood ratio expression as above, using the $n+1$ pseudo-values.
5. Inference: For a confidence interval at level $1-\alpha$ for $\theta$, solve $\{-2\ell_n(\theta) \leq \chi^2_{1,1-\alpha}\}$, where $\chi^2_{1,1-\alpha}$ is the $1-\alpha$ quantile of the $\chi^2_1$ distribution.

Example Applications

• Probability Weighted Moments (PWM): AJEL yields approximately 2.5% higher coverage probabilities over JEL for $n=20$ or $n=30$ in simulation for the estimation of $E[XF(X)]$ with kernel $h(x, y) = \max\{x, y\}/2$.
• ROC Curve Area (AUC): AJEL improves coverage by about 1% over JEL for certain sample sizes in estimation of $P(Y > X)$.
• Real Medical Data (DMD dataset): Construction of confidence intervals for probabilities and diagnostic accuracy using AJEL produced competitive or marginally better performance compared to JEL.

5. Limitations and Implementation Considerations

• Choice of $a_n$: The selection of the adjustment parameter $a_n$ is not unique—any sequence with $a_n = o_p(n^{2/3})$ suffices, but practical recommendations (e.g., $a_n = (\log n)/2$) are adopted for stability.
• Computational Complexity: The most intensive step is the repeated computation of U-statistics under leave-one-out sample reductions and adjustment, but the pseudo-value linearization makes maximization tractable compared to nonlinear EL maximization.
• Interval Length: There is a modest increase in average interval length, but this is offset by superior coverage, especially in small $n$ contexts.
• Generalizability: AJEL is applicable in any context in which U-statistics form the basis of the estimator, as these admit pseudo-value representations.

6. Significance and Further Directions

The introduction and theoretical analysis of AJEL address one of the central challenges in empirical likelihood for U-statistics—ensuring the existence and reliability of the likelihood ratio under arbitrary parameter values without imposing additional sample-size regime restrictions or requiring ad hoc assignment of infinite log-likelihoods. The method provides an effective tool for practitioners confronting small sample inference, complex nonlinear estimation equations, and the need for robust nonparametric confidence intervals and tests.

Possible extensions include further empirical exploration of the choice of $a_n$, systematic comparison across a broader range of U-statistics–based procedures, and adaptation to stratified, high-dimensional, or dependent data structures. The AJEL framework, by combining computational tractability and rigorous theoretical guarantees, has established itself as a central method within modern nonparametric inference for functionals estimated via U-statistics.

Table: Key Steps in AJEL Inference

Step	Description
1. U-statistic	Compute $U_n$ from data
2. Pseudo-values	For $i=1$ to $n$, compute $V_i = nU_n - (n-1)U_{n-1}^{(-i)}$
3. Adjustment	Add $V_{n+1} = -a_n U_n$ (typically $a_n = (\log n) / 2$)
4. Likelihood	Set up $L(\theta)$ as above; maximize over probability weights $p_i$
5. Lagrange	Solve for $\lambda$ in the constraint equation
6. Inference	Use $-2\ell_n(\theta) \leq \chi^2_{1, 1-\alpha}$ for interval or hypothesis test

The AJEL method generalizes the key strengths of jackknife and empirical likelihood—bias correction, variance estimation, and nonparametric likelihood inference—to the general class of U-statistics, ensuring inferential reliability with minimal additional computational or conceptual complexity.