We develop a unified framework for flexible distributional learning based on expectile regression with adaptive basis functions, allowing one to capture heterogeneous covariate effects across different regions of the outcome distribution. Building on this foundation, we introduce a series of methodological contributions that extend expectile regression to increasingly complex data settings.

First, we propose a flexible nonparametric framework for expectile regression using reproducing kernel Hilbert spaces (RKHS), motivated by longitudinal studies in human biology in which aspects of the distribution of offspring anthropometry covary with parental characteristics. We develop a computationally efficient algorithm based on over-relaxed alternating direction method of multipliers (ADMM) to estimate expectiles across multiple distributional levels, and establish valid joint inference procedures for a collection of expectiles using both cross-fitting and robust analytic approaches.

Second, we extend expectile regression to event time data subject to right censoring and left truncation, motivated by biomedical and public health studies where outcomes are incompletely observed and covariate effects may vary across the lifespan. Our motivating application is to understand how lifespans in different demographic groups correspond to neighborhood deprivation, allowing for different effects on early and late mortality. To capture such patterns, we estimate conditional expectiles of patient lifespans using weighting to account for censoring and truncation. We then derive asymptotic linear expansions of the estimators and construct robust sandwich variance estimators, enabling valid inference for distributional contrasts, including comparisons across demographic groups and difference-in-difference analyses across expectile levels.

Third, we develop a unified framework for multivariate generalized expectile regression to analyze multi-output longitudinal data, motivated by applications in which multiple related outcomes are measured repeatedly over time and exhibit complex dependence. Examples include biomedical studies where multiple health indicators are tracked for each patient, or demographic data where event counts in geographic strata evolve jointly over time. Such data may exhibit heterogeneous covariate effects that predict different features of the response distribution. We begin by extending expectile regression to have a link function for each response, enabling the specification of models with additive and multiplicative structures. We formulate the problem as a stacked estimating equation system capturing dependence across outcomes, across time, and across distributional levels without requiring specification of a working correlation structure. We develop cluster-robust sandwich covariance estimators that support valid inference for joint hypotheses, enabling simultaneous assessment of distributional effects across outcomes and expectile levels.

Finally, we introduce a new class of interpretable distributional summaries based on expectile L-moments (EL-moments), motivated by the need for robust and informative measures of distributional shape that can be modeled in relation to covariates. Classical measures such as skewness and kurtosis are often sensitive to extreme observations and are not readily adapted to regression settings, while quantile-based summaries lack smoothness and can be difficult to integrate into unified modeling frameworks. By projecting the expectile function onto a shifted Legendre polynomial basis, we obtain EL-moments that provide interpretable summaries of location, scale, asymmetry, and tail behavior. We further extend these summaries to conditional settings via expectile regression, enabling covariate-dependent characterization of distributional features. We develop an influence-function-based framework for inference, yielding consistent covariance estimators for both the EL-moment coefficients and their derived ratios.