arXiv subjects

Manuela-Simona Cojocea

Publications and source records attributed to Manuela-Simona Cojocea.

3 recordsLinked to original sources

arXiv2026

Statistical Inference for Probability Barycenters and Kolmogorov Moments

A probability coordinate chart is a continuous strictly increasing bijection that transports observations to the open unit interval, where averaging is always well defined. For barycentric inference, the first coordinate moment is returned through the inverse chart. Higher initial coordinate moments similarly generate initial Kolmogorov moments by pullback, while centred coordinate moments remain on the probability scale. We develop inference for these quantities under fixed, intrinsic, and estimated charts. For a fixed benchmark chart, the coordinate mean is the primary inferential object. Confidence intervals are constructed on the probability scale and then transported through the inverse chart, preserving the geometry and allowing asymmetry on the observation scale. Hoeffding's inequality also provides finite-sample distribution-free intervals. The intrinsic case requires different treatment. The empirical cumulative distribution function is not an admissible chart, and its natural generalised plug-in construction collapses exactly to a middle order statistic. Feasible intrinsic inference therefore reduces to median inference. This self-induced functional is distinguished from the frozen oracle construction, which has a different asymptotic variance. For estimated location-scale charts, the asymptotic expansion contains a calibration correction for the first-order effect of learning the chart from the same sample. This yields an influence-function variance estimator and a bootstrap procedure that recalibrates the chart in every resample. Joint covariance theory is developed for vectors of initial and centred coordinate moments, with initial Kolmogorov moments obtained by inverse-chart pullback. The framework separates variation on the probability scale, amplification by the inverse chart, and uncertainty from chart calibration.

stat.ME↗

arXiv2026

G-Exponential Families through Probability Coordinates

We develop a probability geometric construction of generalized exponential families based on probability coordinate charts. A classical exponential family on the unit interval is transported to value space through a chart, producing what we call a G-exponential family. The construction changes the geometry in which exponential structure is represented while leaving the exponential function itself unchanged. This distinguishes it from the algebraic deformations used in the Tsallis and Kaniadakis frameworks. The transported families inherit the structure of the probability coordinate framework. The probability coordinate of each family member follows exactly the original exponential family on the unit interval. Initial Kolmogorov moments are therefore obtained by pulling classical coordinate moments back to value space. For the canonical family, the probability barycenter is the pullback of the mean parameter, while the Fisher information coincides with the Kolmogorov variance. Tail behaviour is inherited from the chart. Every family member is tail equivalent to the chart density, so heavy tails arise from geometry and remain unchanged in order under probability coordinate tilting. This is the opposite of classical exponential tilting. The sufficient statistic is bounded by construction, maximum likelihood reduces to moment matching in probability coordinates, and the maximum likelihood estimator has a bounded influence function. Thus robustness arises from geometric compactification. We also characterise G-exponential families as minimisers of relative entropy to the chart law under coordinate moment constraints and prove that transport preserves the information geometry of the family. A Cauchy chart example and a Monte Carlo study confirm efficient, calibrated, and bounded influence estimation even though every family member has an infinite mean.

math.ST↗

arXiv2026

Chart-Generated Probability Geometry and Kolmogorov Expectations

This paper develops a geometric reinterpretation of probability in which a cumulative distribution function is not used merely as a passive coordinate label, but as a generator of metric geometry. An admissible probability chart $G:I\to(0,1)$ pulls the Euclidean distance of the probability interval back to value space. This creates a decisive distinction from ordinary coordinate invariance: if $G$ is replaced by another chart $H$ while the observations and their law are kept fixed, the induced metric-measure structure changes. Probability charts preserve ordinal structure while altering metric notions such as distance, dispersion, boundary proximity, and barycentric centrality. Averaging linearly in the chart coordinate and pulling the result back defines the Fr\'echet barycenter of $X$ under $d_G$. These functionals coincide with Kolmogorov-Nagumo means, but the chart-generated viewpoint explains their chart dependence geometrically: changing the chart changes the loss being minimized, rather than merely rewriting one fixed centre in new coordinates. A rigidity result shows that, within normalized probability charts, preserving the barycenter for every law forces the chart itself to remain unchanged. Under the intrinsic chart the coordinates are uniform and the barycenter is the median; under an external benchmark chart, tail mismatch is represented by excess boundary occupation. Laws of large numbers, central limit theorems, and a non-asymptotic concentration inequality are established for the transformed functionals. Probability-coordinate moments exist for every law supported on the chart domain and determine the law uniquely, in contrast with classical moment non-existence and indeterminacy on unbounded value spaces.

math.PR↗