  In the evolving landscape of digital commerce, adaptive dynamic pricing
strategies are essential for gaining a competitive edge. This paper introduces
novel {\em doubly nonparametric random utility models} that eschew traditional
parametric assumptions used in estimating consumer demand's mean utility
function and noise distribution. Existing nonparametric methods like
multi-scale {\em Distributional Nearest Neighbors (DNN and TDNN)}, initially
designed for offline regression, face challenges in dynamic online pricing due
to design limitations, such as the indirect observability of utility-related
variables and the absence of uniform convergence guarantees. We address these
challenges with innovative population equations that facilitate nonparametric
estimation within decision-making frameworks and establish new analytical
results on the uniform convergence rates of DNN and TDNN, enhancing their
applicability in dynamic environments.
  Our theoretical analysis confirms that the statistical learning rates for the
mean utility function and noise distribution are minimax optimal. We also
derive a regret bound that illustrates the critical interaction between model
dimensionality and noise distribution smoothness, deepening our understanding
of dynamic pricing under varied market conditions. These contributions offer
substantial theoretical insights and practical tools for implementing
effective, data-driven pricing strategies, advancing the theoretical framework
of pricing models and providing robust methodologies for navigating the
complexities of modern markets.
