Expectation-Over-Transformation Objective

• The expectation-over-transformation objective is a framework that defines target statistics via a bijective transformation applied to both observations and predictions.
• It constructs strictly consistent losses by transforming inputs with functions like power or logarithmic functions, generalizing traditional mean estimation.
• The methodology unifies empirical loss strategies and theoretical principles, ensuring elicitable functionals and enhancing predictive modeling across applications.

An expectation-over-transformation objective, also termed a “$g$-transformed expectation” functional, refers to a class of target statistics and associated strictly consistent loss functions derived by applying a bijective transformation to both the realization and prediction variables of a strictly consistent loss function. This framework generalizes the elicitation of the mean to a broader array of functionals, enabling the systematic construction and analysis of loss functions relevant for diverse statistical and machine learning tasks (Tyralis et al., 23 Feb 2025).

1. Definition of the g-Transformed Expectation Functional

Let $Y$ be a real-valued random variable with probability law $P$ defined on $D \subseteq \mathbb{R}$, and let $g : D \to \mathbb{R}$ be a bijection with inverse $g^{-1}$. Provided $P$ has finite $g$-moment, i.e., $\mathbb{E}[|g(Y)|] < \infty$, the $g$-transformed expectation $T_g$ is defined as: $T_g(P) := g^{-1}\big(\mathbb{E}[g(Y)]\big)$ Thus, the functional first computes the expectation of $g(Y)$ and then inverts the transformation to deliver a statistic on the original scale. The construction covers a wide range of functionals, with special cases subsuming familiar quantities such as the arithmetic mean, power means, and the geometric mean.

2. Strictly Consistent Losses for Expectation-over-Transformation Functionals

Corresponding to $T_g$, a strictly consistent loss (scoring function) can be built by transforming both the realization and prediction using $g$ and applying a standard strictly consistent loss for the mean on the $g$-scale. The most elementary example is the squared error: $L_g(y, z) = \big(g(z) - g(y)\big)^2$ More generally, for a strictly convex, differentiable function $\phi : \mathbb{R} \to \mathbb{R}$ (a "potential"), the generalized $g$–Bregman divergence is used: $$L_{\phi,g}(y,z) = \phi\big(g(z)\big) - \phi\big(g(y)\big) - \phi'\big(g(z)\big)\big(g(z) - g(y)\big)$$ When $\phi(t) = t^2$, $L_{\phi,g}$ reduces to $L_g(y,z)$. Theorem (strict consistency for $T_g$) establishes that if $\mathbb{E}[|\phi(g(Y))|] < \infty$, then for all $P$, the risk $R(z) = \mathbb{E}[L_{\phi,g}(Y, z)]$ is uniquely minimized at $z^* = T_g(P) = g^{-1}(\mathbb{E}[g(Y)])$, so $L_{\phi,g}$ is strictly consistent, and $T_g$ is elicitable (Tyralis et al., 23 Feb 2025).

3. Identification Functions and Necessity

Associated with these losses is the identification function: $V_g(y, z) = g(z) - g(y)$ Strict consistency is characterized by the oriented identification function: $\mathbb{E}[V_g(Y, z)] = 0$ if and only if $z = T_g(P)$. Osband’s principle asserts that the existence of such an identification function is necessary and sufficient for strict consistency of the loss (Tyralis et al., 23 Feb 2025).

4. Special Cases and Illustrative Examples

The expectation-over-transformation framework encompasses several important functional forms. Notable examples include:

Transformation $g$	$T_g(P)$ (Functional)	Strictly Consistent Loss $L_g(y, z)$
Identity $g(t)=t$	$\mathbb{E}[Y]$	$(z-y)^2$
Power $g(t)=t^a$ ($t\geq0,a\neq0$)	$\left(\mathbb{E}[Y^a]\right)^{1/a}$	$(z^a-y^a)^2$
Geometric $g(t)=\log t$ ($t>0$)	$\exp(\mathbb{E}[\log Y])$	$(\log z - \log y)^2$
Entropic $g(t)=\exp(a t)$ ($a\neq0$)	$\frac{1}{a}\log\mathbb{E}[e^{a Y}]$	$(e^{a z}-e^{a y})^2$
Box–Cox $g(t)=(t^a-1)/a$ ($a\neq0$)	$$\left(a\,\mathbb{E}\left[\frac{Y^a-1}{a}\right]+1\right)^{1/a}$$	$\left(\frac{z^a-1}{a}-\frac{y^a-1}{a}\right)^2$

This systematic approach extends to composite quantities, such as the mean and variance of $g(Y)$, and to "g-transformed expectiles" by transforming the arguments of the expectile loss function.

5. Relation to Elicitability and Consistency Theory

Expectation-over-transformation objectives generalize the well-studied case of strictly consistent losses for elicitable functionals, such as the mean, via variable transformation. Given a strictly convex loss eliciting $\mathbb{E}[g(Y)]$, any bijection $g$ allows this to be "pulled back" to a strictly consistent loss for $T_g$, generalizing Osband’s revelation principle from transformations of the prediction variable to joint transformations of both variables (Tyralis et al., 23 Feb 2025). Existence of a strictly consistent loss for $T_g$ guarantees the functional’s elicitability.

6. Applied Implications and Empirical Links

The framework provides theoretical justification for empirical approaches that optimize loss functions of the form $(y^a - z^a)^2$ or other transformed losses. For example, calibration of hydrologic models with $(y^a-z^a)^2$ as loss has been observed to enhance high-flow prediction when $a$ increases; theoretically, this corresponds to the model targeting $T_g$ for power transformation $g$ (Tyralis et al., 23 Feb 2025). Under log-normal $P = \text{Lognormal}(\mu, \sigma^2)$, with $g(t)=t^a$, $T_g(P) = \exp(\mu + a \sigma^2/2)$, showing explicit dependence on the transformation parameter $a$.

Extensions such as g-transformed expectiles arise by analogous transformation of asymmetric quadratic expectile loss, eliciting $z=g^{-1}${$\tau$-expectile} of $g(Y)$. Skill-score variants, multi-dimensional extensions, and proper scoring-rule analogues generalize directly by these principles.

7. Generalization and Theoretical Significance

The expectation-over-transformation objective synthesizes insights from consistent loss function theory, identification functions, and transformation-based elicitation. It unifies empirical strategies and theoretical methods for constructing loss functions tailored to non-standard objectives, advancing principled methodologies for predictive modeling across domains (Tyralis et al., 23 Feb 2025).