On the differentiability of $ϕ$-projections in the discrete finite case
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915248184754176 |
|---|---|
| author | Geenens, Gery Kojadinovic, Ivan Martini, Tommaso |
| author_facet | Geenens, Gery Kojadinovic, Ivan Martini, Tommaso |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the differentiability of $ϕ$-projections in the discrete finite case Geenens, Gery Kojadinovic, Ivan Martini, Tommaso Statistics Theory 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. |
| title | On the differentiability of $ϕ$-projections in the discrete finite case |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2407.05997 |