On the differentiability of $φ$-projections in the discrete finite case

TL;DR

This work establishes conditions under which $φ$-projections are continuously differentiable for finite measures on finite spaces and shows that, when the projection set is convex, these conditions follow from easily verifiable criteria. It connects differentiability to the asymptotic theory of minimum $φ$-divergence estimators, enabling explicit influence functions and delta-method–based distributions. The authors provide concrete asymptotic analyses for projections under (i) binomial-vector models, (ii) moment-constrained distributions, and (iii) Fréchet classes of bivariate arrays, including explicit Jacobians and covariance structures. These results enable robust, geometry-aware inference under a broad class of φ-divergences and constraint families, with practical implications for model misspecification, goodness-of-fit testing, and structured probability-vector estimation.

Abstract

In the case of finite measures on finite spaces, we state conditions under which φ- projections are continuously differentiable. When the set on which one wishes to φ- project is convex, we show that the required assumptions are implied by easily verifiable conditions. In particular, for input probability vectors and a rather large class of φ-divergences, we obtain that φ-projections are continuously differentiable when projecting on a set defined by linear equalities. The obtained results are applied to φ- projection estimators (that is, minimum φ-divergence estimators). A first application, rooted in robust statistics, concerns the computation of the influence functions of such estimators. In a second set of applications, we derive their asymptotics when projecting on parametric sets of probability vectors, on sets of probability vectors generated from distributions with certain moments fixed and on Fréchet classes of bivariate probability arrays. The resulting asymptotics hold whether the element to be φ-projected belongs to the set on which one wishes to φ-project or not.

Figures (3)

• Figure 1: Diagram summarizing the various implications of conditions for differentiability when the set $\mathcal{M}$ of interest is convex.
• Figure 2: For Pearson's $\chi^2$ divergence (left), the squared Hellinger divergence (middle) and the Kullback--Leibler divergence (right), graphs of the functions $D_\phi(\mathcal{S}(\cdot) \mid t)$ for 10 vectors $t \in (0,1)^3$ of the form $t = q_0 + (z_1,\dots,z_m)$, where $(z_1,\dots,z_m)$ is drawn from the $m$-dimensional centered normal distribution with covariance matrix $0.01^2 I_m$. In each plot, the vertical dashed line marks the value $\theta_0$ at which the function $D_\phi(\mathcal{S}(\cdot) \mid q_0)$ reaches its minimum and the top insert contains the (strictly positive) value of $J_2[D_\phi(\mathcal{S}(\cdot) \mid q_0)](\theta_0)$.
• Figure 3: Lines representing the linear inequalities appearing in the definition of $\Theta$ in \ref{['eq:Theta:moments']}. The symbol '$\star$' represents the point $(8.896, 24.8704) \in \mathring \Theta$ corresponding to the probability vector of the binomial distribution with parameters $m-1$ and 0.4. For the squared Hellinger divergence, the vector $\theta_0$ at which $D_\phi(\mathcal{S}(\cdot) \mid q_0)$ reaches its minimum is represented by the symbol 'o'. The small oblique cloud of points around 'o' consists of realizations of the value $\theta_n$ at which the function $D_\phi(\mathcal{S}(\cdot) \mid q_n)$ reaches its minimum.

Theorems & Definitions (28)

• Definition 2: $\phi$-di-ver-gen-ce
• Proposition 3
• Definition 4: $\phi$-projection
• Proposition 5: Existence of $\phi$-projection
• Proposition 7: Unicity of $\phi$-projection
• Proposition 10
• Lemma 11
• remark 1
• Proposition 15: Continuity of $\vartheta^*$ at $t_0$
• Corollary 16: Continuity of $\mathcal{S}^*$ at $t_0$