{"ID":2886023,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.04905","arxiv_id":"2508.04905","title":"General asymptotic representations of indexes based on the functional empirical process and the residual functional empirical process and applications","abstract":"The objective of this paper is to establish a general asymptotic representation (\\textit{GAR}) for a wide range of statistics, employing two fundamental processes: the functional empirical process (\\textit{fep}) and the residual functional empirical process introduced by Lo and Sall (2010a, 2010b), denoted as \\textit{lrfep}. The functional empirical process (\\textit{fep}) is defined as follows: $$ \\mathbb{G}_n(h)=\\frac{1}{\\sqrt{n}} \\sum_{j=1}^{n} \\{h(X_j)-\\mathbb{E}h(X_j)\\}, $$ \\Bin [where $X$, $X_1$, $\\cdots$, $X_n$ is a sample from a random $d$-vectors $X$ of size $(n+1)$ with $n\\geq 1$ and $h$ is a measurable function defined on $\\mathbb{R}^d$ such that $\\mathbb{E}h(X)^2\u003c+\\infty$]. It is a powerful tool for deriving asymptotic laws. An earlier and simpler version of this paper focused on the application of the (\\textit{fep}) to statistics $J_n$ that can be turned into an asymptotic algebraic expression of empirical functions of the form $$ J_n=\\mathbb{E}h(X) + n^{-1/2} \\mathbb{G}_n(h) + o_{\\mathbb{P}}(n^{-1/2}). \\ \\ \\ \\textit{SGAR} $$ \\Bin However, not all statistics, in particular welfare indexes, conform to this form. In many scenarios, functions of the order statistics $X_{1,n}\\leq$, $\\cdots$, $\\leq X_{n,n}$ are involved, resulting in $L$-statistics. In such cases, the (\\textit{fep}) can still be utilized, but in combination with the related residual functional empirical process introduced by Lo and Sall (2010a, 2010b). This combination leads to general asymptotic representations (GAR) for a wide range of statistical indexes $$ J_n=\\mathbb{E}h(X) + n^{-1/2} \\biggr(\\mathbb{G}_n(h) + \\int_{0}^{1} \\mathbb{G}_n(\\tilde{f}_s) \\ell(s) \\ ds + o_{\\mathbb{P}}(1)\\biggr), \\ \\ \\textit{FGAR} $$","short_abstract":"The objective of this paper is to establish a general asymptotic representation (\\textit{GAR}) for a wide range of statistics, employing two fundamental processes: the functional empirical process (\\textit{fep}) and the residual functional empirical process introduced by Lo and Sall (2010a, 2010b), denoted as \\textit{l...","url_abs":"https://arxiv.org/abs/2508.04905","url_pdf":"https://arxiv.org/pdf/2508.04905v1","authors":"[\"Gane Samb Lo\",\"Tchilabalo Abozou Kpanzou\",\"Gandasor Bonyiri Onesiphore Da\"]","published":"2025-08-06T22:03:53Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\"]","methods":"[]","has_code":false}