Lepage Test: Dual Perspectives

• Lepage test refers to two distinct criteria: one for verifying local variationality in geometric ODE systems via Lepage forms and another for combining rank-based location-scale tests in nonparametric statistics.
• In the geometric context, the test employs Lepage forms and Helmholtz conditions to determine if a system of differential equations originates from a Lagrangian, ensuring variational consistency.
• In statistics, the test consolidates location and scale components to detect differences between samples, with recent weak-null modifications enhancing its robustness and accuracy.

Searching arXiv for papers on “Lepage test” and closely related “Lepage equivalents” to ground the article in published work. tool call: arxiv_search tool call: arxiv_search {"^^^^^^^^4query4 test\"^^^^^^^^4 OR abs:\4^^^^^^^^"Lepage test\"^^^^^^^^4 OR ti:\4^^^^^^^^"Lepage equivalents\"","max_results":^^^^4all:\4query4^^^^,"sort_by":"relevance"} In current technical usage, the term Lepage test refers to two unrelated constructions. In the geometric calculus of variations, it denotes a criterion formulated through Lepage forms that detects whether a system of differential equations is variational, or whether a Lagrangian is null. In nonparametric statistics, it denotes a two-sample location-scale test obtained by combining a location component and a scale component. The two usages share a name but arise in different mathematical domains (&&&^^^^4query4^^^^&&&, &&&^^^^4all:\4^^^^&&&).

^^^^4all:\4^^^^. Distinct technical meanings

The current literature attaches the label Lepage test to several criteria built around different objects: source forms for ordinary differential equations, Lepage equivalents of Lagrangians on jet bundles, and combined rank statistics for two-sample inference. The following summary isolates the main settings.

Usage	Setting	Criterion
Variational ODEs	Source form PRESERVED_PLACEHOLDER_^^^^4query4^^^^ on PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^	PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^ is locally variational iff the associated Lepage PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^-form is closed
Field-theoretic variational calculus	Lagrangian $\lambda$ on jet bundles	A suitably constructed Lepage equivalent has a closure property equivalent to vanishing Euler–Lagrange expressions
Nonparametric statistics	Two independent samples	A combined location-scale statistic is asymptotically $\chi^2_2$ under the null hypothesis

This terminological split is important because the geometric and statistical tests are conceptually unrelated. In the geometric literature, Krupková, Urban–Volná, Rossi, Saunders, and Voicu–Garoiu–Vasian study closure and exactness properties of Lepage forms and Lepage equivalents (&&&^^^^4query4^^^^&&&, &&&^^^^4 OR ti:\4^^^^&&&, Voicu et al., 2021). In statistics, Fligner–Policello, Fong–Huang, and the recent weak-null modifications concern rank-based inference for simultaneous location and scale changes (&&&^^^^4all:\4^^^^&&&).

^^^^4 OR abs:\4^^^^. The geometric Lepage test for second-order ordinary differential equations

For second-order autonomous ODEs on a smooth manifold $M$, one works with the product fibration

$Y=\mathbb{R}\times M\to \mathbb{R},$

with $J^1Y\simeq \mathbb{R}\times TM$ and $J^2Y\simeq \mathbb{R}\times T^2M$. A general second-order ODE system is encoded by a source form

PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^

where the PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^ are the basic contact PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^-forms. The system is locally variational if locally there exists a Lagrangian PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^ whose Euler–Lagrange expressions

PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^

reproduce PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5 (&&&^^^^4query4^^^^&&&).

Krupková’s construction associates to PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6 a Lepage PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7-form PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8 on PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9. In coordinates, Urban–Volná give the explicit formula

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^

The Lepage test is the equivalence

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^

In coordinates, this is exactly the classical Helmholtz conditions:

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR ti:\4^^^^

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\44^^^^

An equivalent formulation is that local variationality forces each PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^5 to be affine in PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^6,

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^7

with symmetry and differential conditions on PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^8 and PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^9 reproducing PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4query4^^^^. In this form, the test is an inverse-problem criterion: it decides whether a given second-order system arises as Euler–Lagrange equations.

^^^^4 OR ti:\4^^^^. Exactness, global variationality, and homogeneous systems

Closedness of the Lepage PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4all:\4^^^^-form gives only local variationality. Global variationality is stronger: one asks for a globally defined horizontal PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR abs:\4^^^^-form PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR ti:\4^^^^ such that PRESERVED_PLACEHOLDER_^^^^4 OR ti:\44^^^^. In the Lepage-equivalent formalism,

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^5

where PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^6 is the Cartan form of PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^7. Hence the existence of a global Lagrangian is equivalent to solvability of the global exactness equation

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^8

on PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^9; the horizontal part $\lambda$^^^^4query4^^^^ then yields a global $\lambda$^^^^4all:\4^^^^ (&&&^^^^4query4^^^^&&&).

Urban–Volná analyze this problem by decomposing

$\lambda$^^^^4 OR abs:\4^^^^

where $\lambda$^^^^4 OR ti:\4^^^^ and $\lambda$4 are closed $\lambda$5-forms on $\lambda$6. They construct a homotopy operator on $\lambda$7 showing that $\lambda$8 is always exact, and reduce global exactness of $\lambda$9 to exactness of a certain closed $\chi^2_2$^^^^4query4^^^^-form $\chi^2_2$^^^^4all:\4^^^^ on $\chi^2_2$^^^^4 OR abs:\4^^^^. Two consequences are singled out. First, if the leading pure-$\chi^2_2$^^^^4 OR ti:\4^^^^ component $\chi^2_2$4 vanishes, then $\chi^2_2$5, so $\chi^2_2$6 is globally exact and $\chi^2_2$7 is globally variational. Second, in general, solvability of $\chi^2_2$8 is governed by $\chi^2_2$9; if $M$^^^^4query4^^^^ one may solve it topologically, but no universal constructive formula is known for arbitrary $M$^^^^4all:\4^^^^.

A particularly important case is homogeneity. Rossi and Urban–Volná show that every locally variational second-order ODE with $M$^^^^4 OR abs:\4^^^^ homogeneous of degree $M$^^^^4 OR ti:\4^^^^ or $M$4 is automatically globally variational. Writing

$M$5

with $M$6 and $M$7 homogeneous of degree $M$8 and $M$9 respectively, one constructs two global $Y=\mathbb{R}\times M\to \mathbb{R},$^^^^4query4^^^^-forms $Y=\mathbb{R}\times M\to \mathbb{R},$^^^^4all:\4^^^^ and $Y=\mathbb{R}\times M\to \mathbb{R},$^^^^4 OR abs:\4^^^^ on $Y=\mathbb{R}\times M\to \mathbb{R},$^^^^4 OR ti:\4^^^^ and sets

$Y=\mathbb{R}\times M\to \mathbb{R},$4

Its differential reproduces $Y=\mathbb{R}\times M\to \mathbb{R},$5, hence $Y=\mathbb{R}\times M\to \mathbb{R},$6 on all of $Y=\mathbb{R}\times M\to \mathbb{R},$7.

The standard examples are geodesic equations. For a Riemannian metric $Y=\mathbb{R}\times M\to \mathbb{R},$8, the energy Lagrangian

$Y=\mathbb{R}\times M\to \mathbb{R},$9

is homogeneous of degree $J^1Y\simeq \mathbb{R}\times TM$^^^^4query4^^^^ in $J^1Y\simeq \mathbb{R}\times TM$^^^^4all:\4^^^^, and its Euler–Lagrange equations are homogeneous of degree $J^1Y\simeq \mathbb{R}\times TM$^^^^4 OR abs:\4^^^^ in $J^1Y\simeq \mathbb{R}\times TM$^^^^4 OR ti:\4^^^^. For Finsler geometry, if $J^1Y\simeq \mathbb{R}\times TM$4 is positive-homogeneous of degree $J^1Y\simeq \mathbb{R}\times TM$5, then the energy $J^1Y\simeq \mathbb{R}\times TM$6 is degree-$J^1Y\simeq \mathbb{R}\times TM$7 homogeneous and yields the spray equations

$J^1Y\simeq \mathbb{R}\times TM$8

which are homogeneous of degree $J^1Y\simeq \mathbb{R}\times TM$9 in $J^2Y\simeq \mathbb{R}\times T^2M$^^^^4query4^^^^. In both cases the homogeneity hypothesis removes the topological obstruction, and one recovers globally defined Lagrangians.

4. Field-theoretic Lepage equivalents and closure tests for null Lagrangians

In higher-order field theory, a Lagrangian is a horizontal $J^2Y\simeq \mathbb{R}\times T^2M$^^^^4all:\4^^^^-form

$J^2Y\simeq \mathbb{R}\times T^2M$^^^^4 OR abs:\4^^^^

on a jet bundle $J^2Y\simeq \mathbb{R}\times T^2M$^^^^4 OR ti:\4^^^^ of a fibered manifold $J^2Y\simeq \mathbb{R}\times T^2M$4, with $J^2Y\simeq \mathbb{R}\times T^2M$5. A Lepage equivalent $J^2Y\simeq \mathbb{R}\times T^2M$6 is an $J^2Y\simeq \mathbb{R}\times T^2M$7-form whose horizontal part is $J^2Y\simeq \mathbb{R}\times T^2M$8 and whose first contact component of $J^2Y\simeq \mathbb{R}\times T^2M$9 is the Euler–Lagrange source form. In the formulation of Vitolo–Palese–Rossi, the operational Lepage test for an arbitrary PRESERVED_PLACEHOLDER_^^^^4all:\4query4query4^^^^-form PRESERVED_PLACEHOLDER_^^^^4all:\4query4all:\4^^^^ is: compute its horizontal part PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR abs:\4^^^^, compute PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR ti:\4^^^^, and check whether the PRESERVED_PLACEHOLDER_^^^^4all:\4query44^^^^-contact component vanishes,

PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^5

Then PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^6 is a Lepage equivalent of PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^7 if and only if PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^8 and PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^9. Equivalently,

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4query4^^^^

with PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4all:\4^^^^ the Euler–Lagrange form and no higher contact terms (Palese et al., 2020).

Saunders reformulates the closure property in the variational bicomplex. Given a homotopy operator PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR abs:\4^^^^ for the horizontal differential PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR ti:\4^^^^, one defines the fundamental Lepage form

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\44^^^^

This is again a Lepage equivalent, and it satisfies the closure criterion

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^5

so the vanishing of its horizontal differential is exactly equivalent to PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^6 being a null Lagrangian. Saunders also gives a global version PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^7 by choosing a symmetric linear connection PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^8 on the manifold of independent variables and using Anderson’s global homotopy operator PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^9 (&&&^^^^4 OR ti:\4^^^^&&&).

Voicu, Garoiu, and Vasian prove an analogous closure property for general Lagrangians of any order using a canonical construction. Starting from the Vainberg–Tonti Lagrangian PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4query4^^^^ with the same Euler–Lagrange form as PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4all:\4^^^^, and a local decomposition PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4 OR abs:\4^^^^, they define a Lepage equivalent

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4 OR ti:\4^^^^

and show

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\44^^^^

For reducible second-order Lagrangians PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^5, they obtain the reduced first-order form

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^6

again with

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^7

Although the construction is local in general, the paper states that in many cases of physical interest it becomes global, notably for vector or tensor bundles and Lagrangians of order at most PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^8, including the Einstein–Hilbert Lagrangian, Lovelock generalizations, and Horndeski-type theories (Voicu et al., 2021).

A common source of confusion is that the ODE and field-theoretic versions do not test the same property. For ODE source forms, the test detects local variationality; for field-theoretic Lagrangians, the recent closure-property constructions detect nullity or vanishing Euler–Lagrange expressions.

5. The classical Lepage test in nonparametric statistics

In statistics, the classical Lepage test addresses the two-sample independent location-scale problem. Let

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^9

be independent samples. The strong null hypothesis is

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4query4^^^^

equivalently no location shift PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4all:\4^^^^ and no scale change PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4 OR abs:\4^^^^. The alternatives include location-only,

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4 OR ti:\4^^^^

scale-only,

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\44^^^^

and the full location-scale alternative

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^5

with PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^6 or PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^7 (&&&^^^^4all:\4^^^^&&&).

The location component is the Wilcoxon–Mann–Whitney statistic

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^8

Under PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^9,

PRESERVED_PLACEHOLDER_^^^^4all:\44query4^^^^

The scale component is the Ansari–Bradley statistic PRESERVED_PLACEHOLDER_^^^^4all:\44all:\4^^^^: order the pooled sample PRESERVED_PLACEHOLDER_^^^^4all:\44 OR abs:\4^^^^, assign symmetric ranks PRESERVED_PLACEHOLDER_^^^^4all:\44 OR ti:\4^^^^, and sum the assigned ranks over the PRESERVED_PLACEHOLDER_^^^^4all:\444^^^^ observations. Under PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^5,

PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^6

and

PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^7

PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^8

Lepage’s combined statistic is

PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^9

Under PRESERVED_PLACEHOLDER_^^^^4all:\4max_results4query4^^^^, both standardized components are asymptotically PRESERVED_PLACEHOLDER_^^^^4all:\4max_results4all:\4^^^^ and independent, hence

PRESERVED_PLACEHOLDER_^^^^4all:\4max_results4 OR abs:\4^^^^

The main limitation of the classical construction lies in the assumptions required by its components. The Wilcoxon–Mann–Whitney test assumes equal population variances under the null of no location shift, and the Ansari–Bradley test requires equal population medians for its null distribution to be distribution-free. These assumptions motivate the recent weak-null modifications.

6. Weak-null modifications and recent extensions

The recent weak-null framework modifies both components of the classical Lepage test. For location, Fligner and Policello proposed

PRESERVED_PLACEHOLDER_^^^^4all:\4max_results4 OR ti:\4^^^^

where PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^54 is a data-based consistent estimator under the weak null PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^55. Fong and Huang identified a small bias in PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^56 and proposed

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^57

with PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^58 exactly equal to PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^59 under PRESERVED_PLACEHOLDER_^^^^4all:\4sort_by4query4^^^^ even when PRESERVED_PLACEHOLDER_^^^^4all:\4sort_by4all:\4^^^^. For scale, a new estimator of the Ansari–Bradley variance is introduced:

PRESERVED_PLACEHOLDER_^^^^4all:\4sort_by4 OR abs:\4^^^^

leading to

PRESERVED_PLACEHOLDER_^^^^4all:\4sort_by4 OR ti:\4^^^^

These replacements yield five modified Lepage-type statistics under the weak null (&&&^^^^4all:\4^^^^&&&).

Variant	Location term	Scale term
PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^64	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^65	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^66
PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^67	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^68	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^69
PRESERVED_PLACEHOLDER_^^^^4all:\4relevance4query4^^^^	PRESERVED_PLACEHOLDER_^^^^4all:\4relevance4all:\4^^^^	PRESERVED_PLACEHOLDER_^^^^4all:\4relevance4 OR abs:\4^^^^
PRESERVED_PLACEHOLDER_^^^^4all:\4relevance4 OR ti:\4^^^^	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^74	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^75
PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^76	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^77	PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^78

Under the weak null hypothesis, each standardized component is asymptotically PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^79 and uncorrelated, so each PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8^^^^4query4^^^^ is asymptotically PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8^^^^4all:\4 For small samples, the paper recommends Monte Carlo or permutation of group labels; critical values for PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8^^^^4 OR abs:\4^^^^ were tabulated for sample sizes as small as PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8^^^^4 OR ti:\4^^^^.

The simulation study examines five distributions—Exponential, PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^84, Gamma, Beta, and Uniform—in two sample-size regimes, PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^85 and PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^86. All modified tests maintain near nominal PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^87 type I error in small samples. In large samples, PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^88 tends to be slightly conservative; PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^89 are slightly liberal; and PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9^^^^4query4^^^^ is very close. Across non-Normal, skewed, or heavy-tailed cases, the modified tests except PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9^^^^4all:\4^^^^ uniformly outperform PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9^^^^4 OR abs:\4^^^^. For example, for ExponentialPRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9^^^^4 OR ti:\4^^^^ versus ExpPRESERVED_PLACEHOLDER_^^^^4all:\4^^^^94 with PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^95, the reported powers are approximately PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^96 for PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^97, PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^98 for PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^99, PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4query4^^^^ for PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4all:\4^^^^, PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4 OR abs:\4^^^^ for PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4 OR ti:\4^^^^, and PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query44^^^^ for PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^5; at PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^6, the corresponding powers are about PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^7, PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^8, PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^9, PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4query4^^^^, and PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4all:\4^^^^.

The biomedical illustration uses newborn platelet counts from infants of mothers treated with prednisone PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4 OR abs:\4^^^^ versus untreated controls PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4 OR ti:\4^^^^. Visual inspection suggests both higher median and higher variability in the treated group. The one-tailed right PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\44^^^^-values at PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^5 are reported as PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^6 for PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^7, less than PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^8 for PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^9, less than PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4query4^^^^ for PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4all:\4^^^^, PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4 OR abs:\4^^^^ for PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4 OR ti:\4^^^^, and less than PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\44^^^^ for PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^5 and PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^6. The paper’s practical recommendation is to prefer PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^7 for general two-sample location-scale problems where variances or medians may differ, and to use permutation or Monte Carlo critical values for very small samples.

The statistical literature therefore uses the Lepage name for a combined rank procedure whose modern form is no longer tied to the strong null PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^8. By contrast, the geometric literature uses the name for closure or exactness criteria in the inverse problem of the calculus of variations. The coexistence of these meanings is now a stable feature of the literature rather than a notational anomaly.