Jackknife-Based Test Statistics

• Jackknife-based test statistics are inferential procedures that use leave-one-out replicates and pseudovalues to estimate bias, variance, and construct test statistics.
• The classical approach computes delete-one replicates to form pseudovalues and yields Wald-type and chi-square calibrated tests for smooth, iid data.
• Modern methods, including jackknife empirical likelihood and infinitesimal jackknife, extend these ideas to handle nonlinear U-statistics, model comparison, and complex data structures.

Jackknife-based test statistics are inferential procedures constructed from leave-one-out replicates, jackknife pseudo-values, or infinitesimal perturbations of data weights. In the classical formulation, the jackknife is used for bias estimation, standard error estimation, confidence limits, and test-statistic construction via pseudo-values; in later developments, jackknife empirical likelihood (JEL) converts nonlinear or (U)-statistic estimating problems into likelihood-ratio statistics with Wilks-type chi-square limits, while infinitesimal-jackknife methods extend the same perturbation logic to covariance estimation, model comparison, and local approximations to cross-validation and bootstrap re-fitting [1606.00497] [1707.04998] [2209.00147] [1907.12116].

1. Classical jackknife construction and the pseudovalue test

The delete-1 jackknife starts from data (X_1,\dots,X_n) and forms the jackknife samples
[
\bold{X}{[i]}={X_1,X_2,\dots,X{i-1},X_{i+1},\dots,X_n}.
]
For an estimator (s(\cdot)), the (i)-th jackknife replicate is
[
\hat{\theta}{(i)}:=s(\bold{X}{[i]}),
\qquad
\hat{\theta}{(\cdot)}=\frac{1}{n}\sum{i=1}^{n}\hat{\theta}_{(i)}.
]
The standard jackknife variance estimator is
[
\widehat{\mathrm{Var}}{jack}(\hat{\theta}) = \frac{n-1}{n}\sum{i=1}^{n}(\hat{\theta}{(i)}-\hat{\theta}{(\cdot)})^2,
]
with
[
SE(\hat{\theta}){jack}=\Big{\frac{n-1}{n}\sum{i=1}^{n}(\hat{\theta}{(i)}-\hat{\theta}{(\cdot)})^2\Big}^{1/2}.
]
The same construction yields the jackknife bias estimate
[
\widehat{\text{bias}}{jack}=(n-1)(\hat{\theta}{(\cdot)}-\hat{\theta}),
]
and the bias-corrected estimator
[
\hat{\theta}{jack}=n\hat{\theta}-(n-1)\hat{\theta}{(\cdot)}.
]

The central inferential object is the pseudovalue
[
ps_i=n\varphi_n(\bold{X})-(n-1)\varphi_{n-1}(\bold{X}{[i]}).
]
The average pseudovalue equals the jackknife estimator, and the pseudovalues can be treated as if they were independent random variables. This leads to the classical asymptotic normal test statistic
[
\frac{\sqrt{n}\big(\frac{1}{n}\sum{i=1}^{n}(n\varphi_n(\bold{X})-(n-1)\varphi_{n-1}(\bold{X}_{[i]}))-\theta\big)}{\hat{S}}
\to N(0,1),
]
where (\hat S) is the usual unbiased sample standard deviation computed from pseudovalues. In this form, jackknife-based testing is a Wald-type procedure centered on the average pseudovalue rather than on a direct plug-in asymptotic variance formula [1606.00497].

For two-sample statistics, the same logic is formalized through asymptotic linearity. If
[
T_n=L_n+R_n,\qquad E(L_n)=0,
]
with
[
L_n=\frac1n\left[\sum_{i=1}^{n_1}h_1(X_i)+\sum_{i=n_1+1}^{n}h_2(X_i)\right],
\qquad
n\,E(R_n^2)=o(1),
]
then the jackknife variance estimator is consistent and asymptotically unbiased for the asymptotic variance
[
\sigma^2(T) = \lambda_1\tau_1^2+\lambda_2\tau_2^2.
]
This framework is used for common mean estimators under ordered variances, where the estimators are piecewise-defined through random weights and the variance analysis is difficult because the estimator changes form depending on whether (S_1^2\le S_2^2) or (S_1^2>S_2^2). Under the stated assumptions, these common-mean estimators satisfy a CLT and admit large-sample Wald intervals based on jackknife variance estimation [1710.01898].

A persistent limitation of the classical jackknife is already explicit in the early resampling literature: it is intended for iid data and smooth or sufficiently linear functionals, is not suitable for correlated data or time series, and may fail for non-smooth estimators such as the median. This limitation became one of the main motivations for pseudo-value likelihood methods and infinitesimal perturbation methods [1606.00497].

2. Jackknife empirical likelihood and Wilks-type ratio statistics

The modern theory of jackknife-based test statistics is dominated by JEL. Across S-Gini indices, probability weighted moments, and Gini correlations, the construction follows the same pattern: start from a (U)-statistic or a ratio of (U)-statistics, rewrite the parameter as the solution of a mean-zero estimating equation, construct jackknife pseudo-values
[
V_i=nT_n-(n-1)T_{n,-i},
]
and apply ordinary empirical likelihood to the linear constraint induced by the pseudo-values. For the relative S-Gini index (R_\nu), for example, the estimating equation is
[
E!\left[h(X_1,\dots,X_\nu;R_\nu)\right]=0,
]
the pseudo-values satisfy
[
\frac1n\sum_{k=1}^n V_k=\hat R_\nu,
]
and the jackknife empirical likelihood is
[
JEL(R_\nu) = \sup \left{ \prod_{k=1}^n p_k: \ p_k\ge 0,\ \sum_{k=1}^n p_k=1,\ \sum_{k=1}^n p_k(V_k-R_\nu)=0 \right}.
]
The maximizer has the usual EL form, and the jackknife empirical log-likelihood ratio
[
J(R_\nu) = 2\sum_{k=1}^n \log(1+\lambda V_k)
]
satisfies
[
J(R_\nu)\Rightarrow \chi^2_1.
]
This yields the test of
[
H_0:R_\nu=R_0
\qquad\text{vs}\qquad
H_1:R_\nu\ne R_0
]
with rejection rule
[
J(R_0)>\chi^2_{1,\,1-\alpha},
]
and the inverted confidence set
[
{R_\nu:\ J(R_\nu)\le \chi^2_{1,\,1-\alpha}}.
]
The same paper emphasizes the computational rationale: the jackknife converts a nonlinear (U)-statistic inference problem into a linear empirical likelihood problem [1707.04998].

For probability weighted moments
[
\beta_r = \mathbb{E}{X F^r(X)},
]
the unbiased estimator is a (U)-statistic based on
[
h(x_1,\dots,x_{r+1})=\max(x_1,\dots,x_{r+1}),
]
and the JEL ratio
[
l(\beta_r) = -\sum_{k=1}^{n}\log!\left[1+\lambda(\widehat V_k-\beta_r)\right]
]
satisfies
[
-2l(\beta_r)\xrightarrow{d}\chi^2_1.
]
The adjusted version AJEL appends the pseudo-value
[
\widehat V_{n+1} = -\frac{a_n}{n}\sum_{k=1}^n \widehat V_k,
\qquad
a_n = \max{1,\log(n/2)},
]
to avoid convex-hull problems; under (a_n=o_p(n^{2/3})),
[
-2l_1(\beta_r)\xrightarrow{d}\chi^2_1.
]
Both JEL and AJEL are then used for confidence intervals and for testing
[
H_0:\beta_r=\beta_r^0
\quad\text{vs}\quad
H_1:\beta_r\ne \beta_r^0
]
by chi-square calibration [1807.04450].

For Gini correlations, the parameter is a ratio of two (U)-statistics,
[
\hat\gamma(X,Y)=\frac{U_1}{U_2},
]
and the estimating kernel is
[
h((x_1,y_1),(x_2,y_2);\gamma)=h_2((x_1,y_1),(x_2,y_2))\gamma-h_1((x_1,y_1),(x_2,y_2)).
]
The resulting single-parameter JEL ratio satisfies
[
-2\log R(\gamma)\xrightarrow{d}\chi^2_1.
]
The same framework also covers testing the equality of the two Gini correlations,
[
\Delta=\gamma(X,Y)-\gamma(Y,X),
\qquad
-2\log R(\Delta)\xrightarrow{d}\chi^2_1,
]
and two-sample comparisons of the difference vector
[
(\delta_1,\delta_2),
\qquad
-2\log R(\delta_1,\delta_2)\xrightarrow{d}\chi^2_2.
]
A recurring theme in these papers is that JEL avoids explicit asymptotic variance estimation while preserving a standard Wilks-type limit under finite-second-moment and non-degeneracy conditions [1806.00792].

Setting	Representative statistic	Null limit
Classical pseudovalue test	(\sqrt{n}(\bar{ps}-\theta)/\hat S)	(N(0,1))
Scalar JEL problems	(J(\theta)), (-2l(\theta)), (-2\log R(\theta))	(\chi^2_1)
Two-sample Gini-correlation difference vector	(-2\log R(\delta_1,\delta_2))	(\chi^2_2)
(K)-sample categorical Gini test	(-2\log R)	(\chi^2_{K-1})
Many-instrument IV trinity	(D,LM,W) or (D^,LM^,W^*)	(\bar\chi^2(\boldsymbol\varphi)) or (\chi^2)

3. Symmetry, goodness-of-fit, and independence tests built from jackknifed (U)-statistics

Several one-sample goodness-of-fit and symmetry problems are now handled by first constructing a scalar departure measure and then applying JEL to the associated pseudo-values. For diagonal symmetry in multivariate data, the null
[
H_0:\; X \overset{d}{=} -X
]
is equivalent to
[
\mathbb{E}\big[|X+X'|-|X-X'|\big]=0.
]
The natural sample statistic is a difference of two (U)-statistics, but under (H_0) it is degenerate, which prevents direct application of standard JEL theory. The paper resolves this by splitting the sample, constructing two pseudo-value arrays (V_i^{(1)}) and (V_j^{(2)}), enforcing a common-mean constraint, and proving the Wilks-type limit
[
l(\theta_0)\xrightarrow{d}\chi^2_1.
]
The resulting test is consistent against any fixed alternative and avoids permutation calibration [1908.06892].

In competing-risks data with two causes of failure, the target null is independence between failure time and cause,
[
H_0:\ T \text{ and } J \text{ are independent}.
]
The paper defines the scalar departure measure
[
A=\int \big(S_1(t)f_2(t)-S_2(t)f_1(t)\big)\,dt
]
and rewrites it as
[
A = P(T_1>T_2,\ J_1=1,\ J_2=2) - P(T_1>T_2,\ J_1=2,\ J_2=1).
]
An order-2 (U)-statistic estimates (A), jackknife pseudo-values
[
V_i = n\hat A_n-(n-1)\hat A_{n-1,i}
]
are formed, and the JEL ratio statistic
[
-2l(A)=2\sum_{i=1}^n \log{1+\lambda(V_i-A)}
]
satisfies
[
-2l(A)\xrightarrow{d}\chi^2_1.
]
In practice the null is (A=0), so independence is tested by (-2l(0)) against a (\chi^2_1) critical value [2110.08747].

For log symmetry on the positive real line, the construction begins from the characterization
[
E(g(X)) - E(g(1/X)) = 0
]
for a positive, real-valued, strictly monotone continuous function (g). Choosing
[
g(x)=xF^\beta(x)
]
produces the PWM-based departure measure
[
\Delta(F) = E!\left(XF^\beta(X)-X^{-1}F^\beta(1/X)\right),
]
which under algebraic rewriting becomes
[
\Delta(F) = \frac{1}{\beta+1} E!\left( \max(X_1,\ldots,X_{\beta+1}) - \frac{1}{\min(X_1,\ldots,X_{\beta+1})} \right).
]
The associated (U)-statistic leads to pseudo-values (\widehat V_k), a JEL ratio
[
R(\Delta) = \max\left{ \prod_{i=1}^{n} np_i: \sum_{i=1}^{n}p_i=1,\ \sum_{i=1}^{n}p_i\nu_i=0 \right},
]
and the limit
[
-2\log R(\Delta)\xrightarrow{d}\chi^2_1.
]
The explicit motivation is that the normal-based test is difficult to implement because the asymptotic variance is difficult to estimate [2410.04082].

The same strategy appears in goodness-of-fit testing for the standard Cauchy law. Using the characterization of Arnold (1979), the paper defines the discrepancy
[
\Delta = E\left(\mathrm{I}\left{\left(\frac{X_1X_2-1}{2X_2}\right)\le X_3\right}\right)-\frac12,
]
estimates it by an unbiased order-3 (U)-statistic, forms jackknife pseudo-values
[
\widehat{J}i = n\widehat{\Delta}^*_n - (n-1)\widehat{\Delta}{n-1}^{*(-i)},
]
and proves
[
-2\log\mathcal{R}(\Delta)\xrightarrow{d}\chi^2_1.
]
The AJEL variant appends
[
\widehat{J}{n+1}=\frac{-k_n}{n}\sum{i=1}^n\widehat{J}_i,
\qquad
k_n=\max{1,\log(n)/2},
]
to enforce feasibility when the convex hull of the jackknife pseudo-values fails to contain zero [2409.05764].

4. Two-sample, (K)-sample, and multivariate jackknife likelihood tests

A major branch of the literature uses jackknife-based test statistics for equality of distributions or equality of functionals across several samples. In the upper-semivariance problem, two independent nonnegative populations with cdfs (F) and (G) are compared under
[
H_0:\ \beta_X(t)=\beta_Y(t),\ \forall t>0,
]
where
[
\beta_X(t)=\int_t^\infty (x-t)^2\,dF(x)=E\big[(X-t)^2\,\mathbf 1(X>t)\big].
]
The paper constructs the scalar departure measure
[
\Delta(F,G)=\int_0^\infty\big(\beta_X(t)-\beta_Y(t)\big)\,(dF(t)+dG(t)),
]
rewrites it as expectations involving pairwise comparisons, and estimates it by a symmetric-kernel (U)-statistic (\widehat\Delta). After pooling the sample and forming pseudo-values
[
\nu_i = nT_1-(n-1)T_{1,i},
]
the JEL ratio
[
R(\Delta)=\max\left{\prod_{i=1}^n np_i:\ \sum_{i=1}^n p_i=1,\ \sum_{i=1}^n p_i\nu_i=0\right}
]
yields the test statistic (-2\log R(\Delta)), with null limit (\chi^2_1). The practical appeal is explicit: JEL avoids the practical difficulty of variance estimation in the normal-based method [2401.09816].

For the (K)-sample homogeneity problem, the paper reformulates
[
H_0: F_1=F_2=\cdots=F_K
]
as testing independence between a continuous random vector (X\in\mathbb R^d) and a categorical variable (Z\in{1,\dots,K}). The categorical Gini correlation
[
\rho_g(X,Z)=\frac{\Delta-\sum_{k=1}^K \alpha_k \Delta_k}{\Delta}
]
characterizes equality of the (K) distributions through (\rho_g(X,Z)=0). The test is built from the Gini contrast
[
U_{n_1,\dots,n_K}=\sum_{k=1}^K \hat\alpha_k (U_n-U_{n_k}),
]
with pooled and groupwise pseudo-values (\hat V_i) and (\hat V_l^k). The resulting JEL log-likelihood ratio
[
-2\log R
]
satisfies
[
-2\log R \xrightarrow{d} \chi^2_{K-1}.
]
A distinctive computational claim is that no permutation procedure is required, unlike many energy-distance or distance-correlation tests [1908.00477].

The multivariate (k)-sample extension treats (U)-statistics based on three or more independent samples. For three samples,
[
U_{n_1,n_2,n_3} = \frac{1}{\binom{n_1}{m_1}\binom{n_2}{m_2}\binom{n_3}{m_3}} \sum\sum\sum h(\cdot),
]
the construction pools all observations into ((\mathbf W_1,\dots,\mathbf W_n)), defines leave-one-out statistics (U_{n-1}^{(-l)}), and forms
[
V_l = nU_n-(n-1)U_{n-1}^{(-l)}.
]
Because the pseudo-values are not identically distributed across the sample blocks, the JEL constraint is written as
[
\sum_{i=1}^n p_i(V_i-EV_i)=0.
]
Under finite-second-moment, non-degeneracy, and sample-balance conditions,
[
-2\log R(\theta)\xrightarrow{d}\chi^2_1.
]
The paper develops this for confidence intervals for differences in VUS measurements and for HUM-type functionals, and repeatedly contrasts JEL with normal approximation and kernel-smoothed bootstrap intervals [2408.14038].

These multi-sample results show a characteristic feature of jackknife-based testing: the underlying estimand can be a high-order, multi-sample, or multivariate (U)-statistic, but the final inferential object is still a low-dimensional empirical-likelihood ratio calibrated by a standard chi-square law.

5. Infinitesimal jackknife, higher-order expansions, and covariance-based tests

The infinitesimal jackknife (IJ) replaces delete-1 recomputation by local differentiation with respect to data weights. For weighted estimating equations
[
\hat\theta(w) := \theta \quad \text{such that}\quad G(\theta, w) := \frac{1}{N}\left(g_0(\theta) + \sum_{n=1}^N w_n g_n(\theta)\right)=0,
]
the higher-order infinitesimal jackknife (HOIJ) is the Taylor expansion of (\hat\theta(w)) in the weights (w) around the all-ones vector (N). The first derivative recovers the ordinary IJ,
[
\dtheta[1][N] = -H^{-1}\Gkw0,
]
and higher-order derivatives are computed recursively from lower-order derivatives and higher-order derivatives of (G). The core recursion uses only one matrix inverse (H^{-1}) for all orders, and the paper proves finite-sample error bounds of the form
[
\sup_{w\in\mathcal W}|\dtheta[k+1]|2 = O(\delta^{k+1}),
\qquad
|\theta{\mathrm{ij}[k]}-\hat\theta(w)|_2 = O(\delta^{k+1}).
]
This work does not develop an explicit hypothesis test statistic or confidence interval, but it makes an important inferential point: for bootstrap weights, the first-order approximation reproduces the usual sandwich covariance estimator, so covariance estimates based on the linear approximation do not improve on asymptotic normal theory; higher-order terms are needed for higher-order bootstrap accuracy [1907.12116].

A direct testing application of the IJ appears in model comparison. Extending the IJ from variance estimation to covariance estimation between two models fitted on the same data, the paper defines directional derivatives (U_i^{(p)}(x)) and estimates the covariance block by
[
\hat{\Sigma}^{(pq)}_{ij} = \frac{1}{n^2}\sum_{k=1}^n U_k^{(p)}(x_i)\,U_k^{(q)}(x_j),
\qquad p,q\in{1,2}.
]
For two predictors evaluated at (x_1,\dots,x_m),
[
F_p = \begin{pmatrix} \hat f_p(x_1) \ \vdots \ \hat f_p(x_m) \end{pmatrix},
]
the covariance of the difference is estimated by
[
\hat\Sigma = \hat\Sigma^{(22)}+\hat\Sigma^{(11)}-\hat\Sigma^{(12)}-\hat\Sigma^{(21)},
]
and the comparison test statistic is
[
(F_1 - F_2)^\top \hat{\Sigma}^{-1} (F_1 - F_2) \sim \chi^2_m.
]
The same machinery is used for testing whether a boosting stage made a statistically significant change, and for uncertainty quantification of sums, differences, and more general linear combinations of models [2209.00147].

Predictive inference provides a different jackknife-based branch. The jackknife+ replaces the classical jackknife interval centered at (\hat\mu(X_{n+1})) by an interval based on the leave-one-out fitted values at the test point,
[
C^{\text{jackknife+}}{n,\alpha}(X{n+1}) =
\left[
q_{n,\alpha}{\hat\mu_{-i}(X_{n+1}) - R_i^{\text{LOO}}},
\;
q_{n,\alpha}{\hat\mu_{-i}(X_{n+1}) + R_i^{\text{LOO}}}
\right].
]
Under exchangeability and permutation invariance of the learning algorithm,
[
\mathbb{P}!\left{Y_{n+1}\in C^{\text{jackknife+}}{n,\alpha}(X{n+1})\right}\ge 1-2\alpha.
]
The same paper is explicit that the original jackknife has no universal guarantee and can have coverage equal to zero. This is not a likelihood-ratio test, but it is an important corrective to the misconception that all leave-one-out jackknife inference is automatically valid in unstable learning problems [1905.02928].

6. High-dimensional IV inference, Bayesian survey pseudo-likelihoods, and scope conditions

In linear IV regression with endogeneity, heteroskedastic disturbances, and many potentially weak instrumental variables, jackknife ideas have been used to construct a full testing trinity. The model is
[
y = X\beta + u,\qquad X = Z\Pi + v,
]
and the paper introduces jackknife objective functions based on quadratic forms and ratio-of-quadratic-forms. For the simple null
[
H_0:\beta=\beta_0,
]
the proposed statistics are
[
D=-r_{\min}\frac{\sigma^2}{k}\left(Q_n(\widehat\beta)-Q_n(\beta_0)\right),
\qquad
LM=r_{\min}\,\tau_n'\Phi_n^{-1}\tau_n,
\qquad
W=r_{\min}\,\psi_n'\Phi_n^{-1}\psi_n.
]
For general linear restrictions
[
H_0:R\beta=r,
]
the paper defines restricted estimators and corresponding statistics (D_a), (LM_a), (W_{1a}), and (W_{2a}). Under the null, the natural asymptotic limit is a weighted sum of chi-squares,
[
T \Rightarrow \bar\chi^2(\boldsymbol\varphi),
]
where (T\in{D,LM,W}). A central contribution is that modified objective functions produce ordinary chi-square limits,
[
T^* \Rightarrow \chi_g^2,
\qquad
T_a^* \Rightarrow \chi_p^2.
]
This is an explicitly jackknife-based testing framework rather than a variance-estimation device, and it is designed for settings where many potentially weak instruments make standard IV inference fragile [2604.15437].

In complex survey sampling, jackknife pseudo-values also support a Bayesian pseudo-likelihood route. For a (U)-statistic parameter (\theta), the pseudo-values are
[
\hat v_i = nT_n - (n-1)T_{n-1}^{(-i)},
]
and the survey-weighted jackknife empirical likelihood imposes
[
\sum_{i\in s} p_i = 1,\qquad \sum_{i\in s} p_i(\hat v_i-\theta)=0,
]
optionally together with the auxiliary-information constraint
[
\sum_{i\in s} p_i \mathbf x_i = \bar{\mathbf X}.
]
Under a non-informative prior (\pi(\theta)\propto 1), the Bayesian jackknife pseudo-empirical likelihood pseudo-posterior is asymptotically normal, with local expansion around the JEL estimator or its weighted analogue. Inference is then obtained through posterior quantiles, and testing proceeds by inversion of credible sets rather than by a frequentist likelihood-ratio cutoff [2309.06781].

Taken together, these developments delineate the scope conditions of jackknife-based testing. Recurrent assumptions are finite second moments of the kernel, non-degenerate first-order projections, and asymptotic balance of sample fractions. Where those conditions fail, several papers introduce explicit repairs: AJEL adds an artificial pseudo-value to fix convex-hull failures, HOIJ uses higher-order derivatives because first-order bootstrap linearization is equivalent to asymptotic normal theory, and jackknife+ modifies the classical jackknife because leave-one-out residual calibration alone does not control instability [1807.04450] [1907.12116] [1905.02928].

The resulting picture is broad but coherent. Jackknife-based test statistics include classical pseudovalue (Z)-statistics, JEL likelihood ratios for scalar and vector (U)-statistic functionals, chi-square and chi-bar-square tests in many-instrument IV models, covariance-aware chi-square tests for model comparison, and Bayesian pseudo-likelihood procedures for complex surveys. What unifies them is not a single formula but a common reduction: a difficult estimator is recast into leave-one-out or infinitesimal perturbation objects that behave like approximately independent estimating values, after which standard normal, empirical-likelihood, or quadratic-form calibration becomes possible.