Generalized Orlicz Premia

• Generalized Orlicz premia are multiplicative premium functionals based on non-convex Orlicz-type loss functions, preserving monotonicity, positive homogeneity, and law invariance.
• The framework unifies diverse risk measures including geometric means, expectiles, and Lᵖ-quantiles, offering a common approach to risk assessment and robust statistics.
• Key structural properties such as cash-additivity, dual representations, and geometric convexity provide practical insights for linking arithmetic and multiplicative risk measures.

Generalized Orlicz premia are multiplicative premium functionals defined from an Orlicz-type loss function $\Phi$ that need not be convex. In the formulation introduced in "Generalized Orlicz premia" (Aygün et al., 12 Jul 2025), the premium of a bounded nonnegative random variable $X$ is

$$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$

with the convention that if $\mathbb{P}(X=0)>0$ and $\Phi(0)=-\infty$, then $H_\Phi(X)=0$. The framework extends the classical convex Orlicz premium by relaxing convexity while preserving monotonicity, positive homogeneity, normalization, and law-invariance, and it is broad enough to include the geometric mean, quantiles, expectiles, and $L^p$-quantiles (Aygün et al., 12 Jul 2025).

1. Definition and analytic setting

The construction is developed on a nonatomic probability space $(\Omega,\mathcal{F},\mathbb{P})$, with equalities and inequalities between random variables understood $\mathbb{P}$-almost surely. In the classical Orlicz setting, for a convex Young function $\Psi:[0,\infty)\to[0,\infty]$ with $X$0, the Orlicz space is

$X$1

and the associated Luxemburg norm is

$X$2

When the loss function is not convex, the corresponding admissible set is generally not a linear space. For that reason, the generalized theory takes as natural domain

$X$3

and, for geometric constructions,

$X$4

A generalized Orlicz function $X$5 is required to satisfy

$X$6

$X$7

This normalization enforces $X$8.

If $X$9 is finite, strictly increasing and continuous, then

$$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$0

This identifies $$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$1 as the unique scaling factor that normalizes expected loss to one. The paper presents this as the multiplicative counterpart of certainty equivalents and connects it to M-estimation-type functionals through root-finding for the estimating equation $$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$2 (Aygün et al., 12 Jul 2025).

The generalized definition should be distinguished from the classical convex Orlicz premium. In the convex case, if $$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$3 is a convex Young function with $$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$4 and $$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$5, then

$$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$6

and $$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$7 is a norm on $$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$8 with the usual duality based on the conjugate

$$H_\Phi(X) := \inf \left\{k>0\,\middle|\, \mathbb{E}\big[\Phi(X/k)\big]\le 1\right\},$$9

2. Structural properties

For $\mathbb{P}(X=0)>0$0 satisfying the generalized assumptions above, the premium $\mathbb{P}(X=0)>0$1 on $\mathbb{P}(X=0)>0$2 retains several structural properties that are standard in the convex theory. Specifically,

$\mathbb{P}(X=0)>0$3

Thus the premium is monotone, positively homogeneous, and normalized (Aygün et al., 12 Jul 2025).

The paper also gives deterministic bounds. Writing

$\mathbb{P}(X=0)>0$4

one has

$\mathbb{P}(X=0)>0$5

These inequalities locate the premium between two scale parameters determined by the essential supremum of $\mathbb{P}(X=0)>0$6 and the effective support of $\mathbb{P}(X=0)>0$7.

Left-continuity of $\mathbb{P}(X=0)>0$8 is used to obtain min-attainment and a threshold criterion. If $\mathbb{P}(X=0)>0$9, then

$\Phi(0)=-\infty$0

and

$\Phi(0)=-\infty$1

The exposition emphasizes that without left-continuity, min-attainment and this threshold equivalence may fail.

Convexity of the premium is equivalent, up to mild auxiliary conditions, to convexity of the loss function. If $\Phi(0)=-\infty$2 is convex, then $\Phi(0)=-\infty$3 is convex. Conversely, under mild additional conditions, convexity of $\Phi(0)=-\infty$4 implies convexity of $\Phi(0)=-\infty$5. The premium is also law-invariant: $\Phi(0)=-\infty$6

A useful interpretive point is that generalized Orlicz premia are presented as return risk measures rather than monetary risk measures. In this terminology, return risk measures are normalized, monotone, positively homogeneous maps on $\Phi(0)=-\infty$7, whereas cash-additivity for premia,

$\Phi(0)=-\infty$8

is distinct from translation invariance for monetary risk measures, which takes the form $\Phi(0)=-\infty$9 (Aygün et al., 12 Jul 2025).

3. Canonical examples and the scope of the framework

A central contribution of the generalized formulation is that it accommodates economically and statistically important functionals that fall outside the convex Orlicz setting.

The geometric mean is obtained by taking

$H_\Phi(X)=0$0

which is concave and satisfies $H_\Phi(X)=0$1. In that case

$H_\Phi(X)=0$2

The relevant domain restriction is $H_\Phi(X)=0$3 almost surely, or $H_\Phi(X)=0$4, so that $H_\Phi(X)=0$5 is finite. The paper identifies this as a prototypical geometrically convex example (Aygün et al., 12 Jul 2025).

Expectiles are also recovered exactly. For $H_\Phi(X)=0$6, define

$H_\Phi(X)=0$7

Then

$H_\Phi(X)=0$8

the $H_\Phi(X)=0$9-expectile of $L^p$0. Equivalently, $L^p$1 is the unique solution of

$L^p$2

or the minimizer of

$L^p$3

The first-order condition matches the balancing equation.

The paper further records several additional examples. With a suitable step function $L^p$4, $L^p$5 yields the left $L^p$6-quantile. For $L^p$7, the choice $L^p$8 yields

$L^p$9

More generally,

$(\Omega,\mathcal{F},\mathbb{P})$0

gives an $(\Omega,\mathcal{F},\mathbb{P})$1-quantile: $(\Omega,\mathcal{F},\mathbb{P})$2 solves

$(\Omega,\mathcal{F},\mathbb{P})$3

These examples clarify the breadth of the generalized framework. The geometric mean is multiplicative and non-convex, expectiles are elicitable and cash-additive, and quantile-type constructions emerge from step or power losses. This suggests that generalized Orlicz premia function as a unifying device for functionals that are usually treated in separate literatures on risk measurement, robust statistics, and forecast evaluation (Aygün et al., 12 Jul 2025).

4. Cash-additivity and collapse to $(\Omega,\mathcal{F},\mathbb{P})$4-quantiles

One of the main structural theorems concerns the consequences of imposing cash-additivity,

$(\Omega,\mathcal{F},\mathbb{P})$5

Under the assumptions that $(\Omega,\mathcal{F},\mathbb{P})$6 is finite, continuous and strictly increasing, the paper proves that if $(\Omega,\mathcal{F},\mathbb{P})$7 is cash-additive, then there exist $(\Omega,\mathcal{F},\mathbb{P})$8 and $(\Omega,\mathcal{F},\mathbb{P})$9 such that

$\mathbb{P}$0

If, in addition, $\mathbb{P}$1 is convex, or alternatively concave, then necessarily $\mathbb{P}$2 and

$\mathbb{P}$3

so that $\mathbb{P}$4 is an expectile (Aygün et al., 12 Jul 2025).

This yields a generalized collapse theorem. In the classical convex theory, cash-additivity is associated with a “collapse to the mean.” The new result extends that statement by showing that, without assuming convexity of $\mathbb{P}$5, cash-additivity forces the premium into the family of $\mathbb{P}$6-quantiles. Convexity or concavity then refines the conclusion to the expectile case.

The relevant notion of $\mathbb{P}$7-quantile, for $\mathbb{P}$8, is any minimizer

$\mathbb{P}$9

The paper notes that existence follows under mild integrability, and that uniqueness holds for $\Psi:[0,\infty)\to[0,\infty]$0 under strictly convex loss. Special cases include $\Psi:[0,\infty)\to[0,\infty]$1, which yields the mean, and $\Psi:[0,\infty)\to[0,\infty]$2, which yields the median(s).

The theory also records one-sided variants. For

$\Psi:[0,\infty)\to[0,\infty]$3

one obtains cash-subadditivity if $\Psi:[0,\infty)\to[0,\infty]$4 and cash-superadditivity if $\Psi:[0,\infty)\to[0,\infty]$5, whereas exact cash-additivity forces $\Psi:[0,\infty)\to[0,\infty]$6.

The broader implication is that additive consistency in premium units is highly restrictive in the multiplicative Orlicz setting. A plausible implication is that non-convexity does not generate arbitrary new cash-additive premia; rather, it enlarges the family from classical linear expectile-type objects to the full $\Psi:[0,\infty)\to[0,\infty]$7-quantile class (Aygün et al., 12 Jul 2025).

5. Geometric convexity and dual representation

A substantial part of the theory replaces ordinary convexity by geometric convexity. For functions $\Psi:[0,\infty)\to[0,\infty]$8, GG-convexity means

$\Psi:[0,\infty)\to[0,\infty]$9

and GA-convexity means

$X$00

The paper proves the equivalences

$X$01

together with GG-convex $X$02 GA-convex, and the fact that for monotone convex $X$03, GA-convexity holds automatically (Aygün et al., 12 Jul 2025).

For functionals $X$04, GG-convexity is defined by

$X$05

while GA-convexity uses the corresponding arithmetic combination on the right-hand side. Within this framework, if $X$06 is GA-convex, then $X$07 is GG-convex; conversely, under mild conditions, GG-convexity of $X$08 implies GA-convexity of $X$09.

The dual representation in the convex arithmetic case takes the form

$X$10

with multiplicative penalty

$X$11

For convex Orlicz premia $X$12, the penalty is explicitly

$X$13

where $X$14 is the convex conjugate of $X$15.

In the geometrically convex case, the dual form becomes multiplicative: $X$16 with

$X$17

For GA-convex generalized Orlicz premia $X$18,

$X$19

The exposition describes this as structurally analogous to the convex case but with the geometric mean as the basic block.

The connection between arithmetic and geometric duality is made explicit by relative entropy. If a convex return risk measure admits

$X$20

then it also admits

$X$21

where

$X$22

and

$X$23

is the relative entropy. This identifies a multiplicative transformation linking the arithmetic and geometric dual forms (Aygün et al., 12 Jul 2025).

6. Elicitability, stability, and relation to Haezendonck–Goovaerts premia

The paper’s final major characterization concerns elicitability. A functional $X$24 is elicitable if there exists a strictly proper scoring function $X$25 such that, for all integrable $X$26,

$X$27

with unique minimizer. Equivalently, there exists an identification function $X$28 with $X$29 and a strict identification property.

A necessary condition for elicitability is the CxLS property, meaning convex level sets under mixtures on distributions. The paper shows that if $X$30 is law-invariant, GG-convex, and has CxLS, then $X$31 for some GA-convex Orlicz function $X$32. If $X$33 is law-invariant, convex, and has CxLS, then $X$34 for some convex Orlicz function $X$35. Generalized Orlicz premia are therefore characterized as the law-invariant return risk measures that are GG-convex or convex and elicitable via CxLS (Aygün et al., 12 Jul 2025).

Two examples are emphasized. The geometric mean satisfies

$X$36

with strictly consistent score $X$37 and identification $X$38. Expectiles satisfy

$X$39

and admit the identification

$X$40

This generalized theory is naturally compared with the classical Haezendonck–Goovaerts premium principles, which are Orlicz-based premia on the natural domain $X$41. For a finite-valued Young function $X$42 with $X$43 and $X$44, the Haezendonck–Goovaerts premium is

$X$45

where

$X$46

These premia are sublinear, monotone, translation invariant, law-invariant, and Lipschitz in the Orlicz norm. They always satisfy the Fatou property, and they satisfy the Lebesgue property if and only if $X$47 fulfills the $X$48 condition (Gao et al., 2019).

The dual representation of the Haezendonck–Goovaerts premium is

$X$49

where

$X$50

This classical result provides the convex Orlicz benchmark against which generalized Orlicz premia are formulated. The generalized theory departs from the linear-space setting of $X$51, accommodates non-convex $X$52, and replaces arithmetic dual blocks by multiplicative geometric ones when GG-convexity is imposed. A plausible implication is that generalized Orlicz premia extend rather than supplant the Haezendonck–Goovaerts framework: the latter remains the natural convex, translation-invariant model on Orlicz spaces, while the former captures multiplicative and elicitable functionals that classical convexity excludes (Gao et al., 2019).