General asymptotic representations of indexes based on the functional empirical process and the residual functional empirical process and applications

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 $\mathbb{E}h(X)^2<+\infty$]. 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 $$ J_n=\mathbb{E}h(X) + n^{-1/2} \biggr(\mathbb{G}_n(h) + \int_{0}^{1} \mathbb{G}_n(\tilde{f}_s) \ell(s) \ ds + o_{\mathbb{P}}(1)\biggr), \ \ \textit{FGAR} $$