{"id":41201,"date":"2024-09-26T10:35:44","date_gmt":"2024-09-26T10:35:44","guid":{"rendered":"https:\/\/express24.ir\/d\/product\/%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-%d8%aa%d8%b9%d8%a7%d8%b1%db%8c%d9%81-%d9%85%d8%aa%d9%86%d8%a7%d9%88%d8%a8-%d8%a7%d8%b2-%d8%a7%d9%82%d8%af\/"},"modified":"2024-09-26T10:35:46","modified_gmt":"2024-09-26T10:35:46","slug":"%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-%d8%aa%d8%b9%d8%a7%d8%b1%db%8c%d9%81-%d9%85%d8%aa%d9%86%d8%a7%d9%88%d8%a8-%d8%a7%d8%b2-%d8%a7%d9%82%d8%af","status":"publish","type":"product","link":"https:\/\/express24.ir\/d\/product\/%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-%d8%aa%d8%b9%d8%a7%d8%b1%db%8c%d9%81-%d9%85%d8%aa%d9%86%d8%a7%d9%88%d8%a8-%d8%a7%d8%b2-%d8%a7%d9%82%d8%af\/","title":{"rendered":"\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u062a\u0639\u0627\u0631\u06cc\u0641 \u0645\u062a\u0646\u0627\u0648\u0628 \u0627\u0632 \u0627\u0642\u062f\u0627\u0645\u0627\u062a \u062c\u06cc\u0646\u06cc \u060c \u0647\u0648\u0648\u0631 \u0648 \u0644\u0648\u0631\u0646\u0632 \u0627\u0632 \u0646\u0627\u0628\u0631\u0627\u0628\u0631\u06cc \u0647\u0627 \u0648 \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0645\u0639\u06cc\u0627\u0631 Wasserstein W$_1$"},"content":{"rendered":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc <\/td>\nAlternate definitions of Gini, Hoover and Lorenz measures of inequalities and convergence with respect to the Wasserstein W$_1$ metric<\/td>\n<\/tr>\n
\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc <\/td>\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u062a\u0639\u0627\u0631\u06cc\u0641 \u0645\u062a\u0646\u0627\u0648\u0628 \u0627\u0632 \u0627\u0642\u062f\u0627\u0645\u0627\u062a \u062c\u06cc\u0646\u06cc \u060c \u0647\u0648\u0648\u0631 \u0648 \u0644\u0648\u0631\u0646\u0632 \u0627\u0632 \u0646\u0627\u0628\u0631\u0627\u0628\u0631\u06cc \u0647\u0627 \u0648 \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0645\u0639\u06cc\u0627\u0631 Wasserstein W$_1$<\/td>\n<\/tr>\n
\u0646\u0648\u06cc\u0633\u0646\u062f\u06af\u0627\u0646 <\/td>\nValentin Melot<\/td>\n<\/tr>\n
\u0641\u0631\u0645\u062a \u0645\u0642\u0627\u0644\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc <\/td>\nPDF<\/td>\n<\/tr>\n
\u0632\u0628\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u062a\u062d\u0648\u06cc\u0644\u06cc <\/td>\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc<\/td>\n<\/tr>\n
\u0641\u0631\u0645\u062a \u0645\u0642\u0627\u0644\u0647 \u062a\u0631\u062c\u0645\u0647 \u0634\u062f\u0647 <\/td>\n\u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\n<\/tr>\n
\u0646\u062d\u0648\u0647 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 <\/td>\n\u062f\u0648 \u062a\u0627 \u0633\u0647 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634 (\u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u062f\u0627\u0646\u0644\u0648\u062f\u06cc)<\/td>\n<\/tr>\n
\u062a\u0639\u062f\u0627\u062f \u0635\u0641\u062d\u0627\u062a<\/td>\n50<\/td>\n<\/tr>\n
\u0644\u06cc\u0646\u06a9 \u062f\u0627\u0646\u0644\u0648\u062f \u0631\u0627\u06cc\u06af\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc<\/td>\n\u062f\u0627\u0646\u0644\u0648\u062f \u0645\u0642\u0627\u0644\u0647<\/a><\/td>\n<\/tr>\n
\u062f\u0633\u062a\u0647 \u0628\u0646\u062f\u06cc \u0645\u0648\u0636\u0648\u0639\u0627\u062a <\/td>\nProbability,\u0627\u062d\u062a\u0645\u0627\u0644 ,<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a <\/td>\nSubmitted 19 September, 2024; originally announced September 2024.<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc <\/td>\n\u0627\u0631\u0633\u0627\u0644 \u0634\u062f\u0647 \u062f\u0631 19 \u0633\u067e\u062a\u0627\u0645\u0628\u0631 2024 \u061b\u062f\u0631 \u0627\u0628\u062a\u062f\u0627 \u0633\u067e\u062a\u0627\u0645\u0628\u0631 2024 \u0627\u0639\u0644\u0627\u0645 \u0634\u062f.<\/td>\n<\/tr>\n
\u0627\u0637\u0644\u0627\u0639\u0627\u062a \u0628\u06cc\u0634\u062a\u0631 \u0627\u0632 \u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u062f\u0631 \u067e\u0627\u06cc\u06af\u0627\u0647 \u0647\u0627\u06cc \u0639\u0644\u0645\u06cc <\/td>\n\n INSPIRE HEP<\/a>
\n
\n NASA ADS<\/a>
\n
\n Google Scholar<\/a>
\n
\n Semantic Scholar<\/a>
\n
\n arXiv<\/a><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\r\n\r\n \r\n \r\n \r\n \r\n
\u0641\u0631\u0645\u062a \u0627\u0631\u0627\u0626\u0647 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u062a\u062d\u0648\u06cc\u0644 \u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\r\n <\/tr>\r\n
\u0632\u0645\u0627\u0646 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u0628\u06cc\u0646 2 \u062a\u0627 3 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634<\/td>\r\n <\/tr>\r\n\t
\u06a9\u06cc\u0641\u06cc\u062a \u062a\u0631\u062c\u0645\u0647 <\/td>\r\n \u0628\u0633\u06cc\u0627\u0631 \u0628\u0627\u0644\u0627. \u0645\u0642\u0627\u0644\u0647 \u0641\u0642\u0637 \u062a\u0648\u0633\u0637 \u0645\u062a\u0631\u062c\u0645\u06cc\u0646 \u0628\u0627 \u0645\u062f\u0631\u06a9 \u062f\u0627\u0646\u0634\u06af\u0627\u0647\u06cc \u0645\u062a\u0631\u062c\u0645\u06cc \u062a\u0631\u062c\u0645\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/td>\r\n <\/tr>\r\n\t\t
\u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 <\/td>\r\n \u06a9\u0644\u06cc\u0647 \u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 \u0646\u06cc\u0632 \u062f\u0631 \u0641\u0627\u06cc\u0644 \u062a\u062d\u0648\u06cc\u0644\u06cc \u0648\u0631\u062f \u062f\u0631\u062c \u0645\u06cc\u200c\u0634\u0648\u0646\u062f.<\/td>\r\n <\/tr>\r\n<\/table>\r\n\r\n\n

\u0686\u06a9\u06cc\u062f\u0647<\/h2>\n

This article focuses on some properties of three tools used to measure economic inequalities with respect to a distribution of wealth $\u03bc$: Gini coefficient $G$, Hoover coefficient or Robin Hood coefficient $H$, and the Lorenz concentration curve $L$. To express the distributions of resources, we use the framework of random variables and abstract Borel measures, rather than discrete samples or probability densities. This allows us to consider arbitrary distributions of wealth, e.g. mixtures between discrete and continuous distributions. In the first part (sections~\\ref{section_altdefs_lorenz}–\\ref{section_altdefs_gini_hoover_application}), we discuss alternate definitions of $G$, $H$ and $L$ that can be found in economics literature. The Lorenz curve is defined as the normalized integral of the quantile function (\\cite{gastwirth1971}), which is not the same as saying “$L(p)$ is the share of wealth owned by the $100p$ first centiles of the population” (proposition~\\ref{lorenz_definition_match_condition}). The Gini and Hoover coefficients are introduced in terms of expectation of random variables. In section~\\ref{section_altdefs_gini_hoover}, we interpret Gini and Hoover as geometrical properties of the Lorenz curve (theorem \\ref{theorem_alternate_def_gini} and corollary \\ref{theorem_alternate_def_hoover_max}). In particular, we give a more general and straightforward proof of the main result of \\cite{dorfman1979}. Section~\\ref{section_altdefs_gini_hoover_application} gives two direct applications. We \\emph{en route} prove the (not trivial) fact that the Lorenz curve fully characterizes a distribution, up to a rescaling (proposition~\\ref{bijection_M_Lorspace_R}). The second part of the article (section~\\ref{section_convergence_results}–\\ref{section_weaker_asumptions_cv}) focuses on the consistency of $G(\u03bc)$, $H(\u03bc)$ and $L_\u03bc$ as $\u03bc$ is approximated or perturbated. The relevant tool to use is the Wasserstein metric $\\Wone$, i.e. the $\\Lone$ metric between quantile functions. $\\Wone(\u03bc_n, \u03bc_\\infty) \\to 0$ if and only if underlying random variables converge in distribution and the total amount of wealth converges. In theorem~\\ref{characterization_lorenz_convergence} and proposition~\\ref{convergence_indicators}, we show that if $\\Wone(\u03bc_n, \u03bc_\\infty) \\to 0$, then $G(\u03bc_n) \\to G(\u03bc_\\infty)$, $H(\u03bc_n) \\to H(\u03bc_\\infty)$ and $L_{\u03bc_n} \\to L_{\u03bc_\\infty}$ uniformly. Subsection~\\ref{subsection_topo} discusses topological implications of this fact. Thus, applications \\ref{convergence_corollary_noise}, \\ref{convergence_corollary_sampling}, \\ref{convergence_corollary_quantiles}, \\ref{convergence_corollary_quantiles_and_sampling} and \\ref{convergence_corollary_sampling_kernel} justify that the empirical Gini, Hoover indexes and Lorenz curves computed on a sample or rebuilt with partial information converge to the real Gini, Hoover indexes and Lorenz curve as information increases. Eventually, in section~\\ref{section_weaker_asumptions_cv}, we discuss the situations where the $\\Wone$ convergence is not ensured, but weaker asumptions can be made (convergence in distribution in~\\ref{subsection_weak_convergence}, convergence of means in~\\ref{subsection_convergence_means} or uniform integrability in~\\ref{subsection_ui})<\/p>\n

\u0686\u06a9\u06cc\u062f\u0647 \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc (\u062a\u0631\u062c\u0645\u0647 \u0645\u0627\u0634\u06cc\u0646\u06cc)<\/h2>\n

\u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0628\u0631\u062e\u06cc \u0627\u0632 \u062e\u0648\u0627\u0635 \u0633\u0647 \u0627\u0628\u0632\u0627\u0631 \u0645\u0648\u0631\u062f \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0628\u0631\u0627\u06cc \u0627\u0646\u062f\u0627\u0632\u0647 \u06af\u06cc\u0631\u06cc \u0646\u0627\u0628\u0631\u0627\u0628\u0631\u06cc \u0647\u0627\u06cc \u0627\u0642\u062a\u0635\u0627\u062f\u06cc \u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u062a\u0648\u0632\u06cc\u0639 \u062b\u0631\u0648\u062a $ $ $: \u0636\u0631\u06cc\u0628 \u062c\u06cc\u0646\u06cc $ g $ \u060c \u0636\u0631\u06cc\u0628 \u0647\u0648\u0648\u0631 \u06cc\u0627 \u0636\u0631\u06cc\u0628 \u0631\u0627\u0628\u06cc\u0646 \u0647\u0648\u062f $ H $ \u0648 \u0645\u0646\u062d\u0646\u06cc \u063a\u0644\u0638\u062a \u0644\u0648\u0631\u0646\u0632 $ L $ \u0627\u0633\u062a.\u0628\u0631\u0627\u06cc \u0628\u06cc\u0627\u0646 \u062a\u0648\u0632\u06cc\u0639 \u0645\u0646\u0627\u0628\u0639 \u060c \u0645\u0627 \u0628\u0647 \u062c\u0627\u06cc \u0646\u0645\u0648\u0646\u0647 \u0647\u0627\u06cc \u06af\u0633\u0633\u062a\u0647 \u06cc\u0627 \u062a\u0631\u0627\u06a9\u0645 \u0627\u062d\u062a\u0645\u0627\u0644 \u060c \u0627\u0632 \u0686\u0627\u0631\u0686\u0648\u0628 \u0645\u062a\u063a\u06cc\u0631\u0647\u0627\u06cc \u062a\u0635\u0627\u062f\u0641\u06cc \u0648 \u0627\u0642\u062f\u0627\u0645\u0627\u062a \u0627\u0646\u062a\u0632\u0627\u0639\u06cc Borel \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u0645.\u0627\u06cc\u0646 \u0628\u0647 \u0645\u0627 \u0627\u0645\u06a9\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f \u062a\u0648\u0632\u06cc\u0639 \u062f\u0644\u062e\u0648\u0627\u0647 \u062b\u0631\u0648\u062a \u0631\u0627 \u062f\u0631 \u0646\u0638\u0631 \u0628\u06af\u06cc\u0631\u06cc\u0645 \u060c \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0645\u062b\u0627\u0644\u0645\u062e\u0644\u0648\u0637 \u0628\u06cc\u0646 \u062a\u0648\u0632\u06cc\u0639 \u06af\u0633\u0633\u062a\u0647 \u0648 \u0645\u062f\u0627\u0648\u0645.\u062f\u0631 \u0642\u0633\u0645\u062a \u0627\u0648\u0644 (\u0628\u062e\u0634 \u0647\u0627\u06cc ~ \\ ref {section_altdefs_lorenz}-\\ ref {\u0628\u062e\u0634_ALTDEFS_GINI_HOOVER_APPLICATION}) \u060c \u0645\u0627 \u062f\u0631 \u0645\u0648\u0631\u062f \u062a\u0639\u0627\u0631\u06cc\u0641 \u0645\u062a\u0646\u0627\u0648\u0628 $ g $ \u060c $ h $ \u0648 $ l $ \u0628\u062d\u062b \u0645\u06cc \u06a9\u0646\u06cc\u0645 \u06a9\u0647 \u0645\u06cc \u062a\u0648\u0627\u0646\u062f \u062f\u0631 \u0627\u062f\u0628\u06cc\u0627\u062a \u0627\u0642\u062a\u0635\u0627\u062f \u067e\u06cc\u062f\u0627 \u0634\u0648\u062f.\u0645\u0646\u062d\u0646\u06cc \u0644\u0648\u0631\u0646\u0632 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u06cc\u06a9 \u0627\u0646\u062a\u06af\u0631\u0627\u0644 \u0639\u0627\u062f\u06cc \u0639\u0645\u0644\u06a9\u0631\u062f \u06a9\u0645\u06cc \u062a\u0639\u0631\u06cc\u0641 \u0634\u062f\u0647 \u0627\u0633\u062a (\\ cite {gastwirth1971}) \u060c \u06a9\u0647 \u0628\u0647 \u0647\u0645\u0627\u0646 \u0627\u0646\u062f\u0627\u0632\u0647 \u0646\u06cc\u0633\u062a \u06a9\u0647 \u0645\u06cc \u06af\u0648\u06cc\u062f “$ l (p) $ \u0633\u0647\u0645 \u062b\u0631\u0648\u062a \u0627\u0633\u062a \u06a9\u0647 \u0645\u062a\u0639\u0644\u0642 \u0628\u0647 100 \u062f\u0644\u0627\u0631 $ $ \u0633\u0627\u0646\u062a\u0644\u0647\u0627\u06cc \u0627\u0648\u0644 \u0627\u0633\u062a\u062c\u0645\u0639\u06cc\u062a “(\u06af\u0632\u0627\u0631\u0647 ~ \\ ref {lorenz_definition_match_condition}).\u0636\u0631\u0627\u06cc\u0628 \u062c\u06cc\u0646\u06cc \u0648 \u0647\u0648\u0648\u0631 \u0627\u0632 \u0646\u0638\u0631 \u0627\u0646\u062a\u0638\u0627\u0631 \u0645\u062a\u063a\u06cc\u0631\u0647\u0627\u06cc \u062a\u0635\u0627\u062f\u0641\u06cc \u0645\u0639\u0631\u0641\u06cc \u0645\u06cc \u0634\u0648\u0646\u062f.\u062f\u0631 \u0628\u062e\u0634 ~ \\ ref {sex_altdefs_gini_hoover} \u060c \u0645\u0627 \u062c\u06cc\u0646\u06cc \u0648 \u0647\u0648\u0648\u0631 \u0631\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u0647\u0646\u062f\u0633\u06cc \u0645\u0646\u062d\u0646\u06cc \u0644\u0648\u0631\u0646\u0632 (\u0642\u0636\u06cc\u0647 {refem_alternate_def_gini} \u0648 \u0646\u062a\u06cc\u062c\u0647 \u06af\u06cc\u0631\u06cc \\ refem_alternate_def_hoover_max}) \u062a\u0641\u0633\u06cc\u0631 \u0645\u06cc \u06a9\u0646\u06cc\u0645.\u0628\u0647 \u0637\u0648\u0631 \u062e\u0627\u0635 \u060c \u0645\u0627 \u0627\u062b\u0628\u0627\u062a \u06a9\u0644\u06cc \u0648 \u0633\u0627\u062f\u0647 \u062a\u0631\u06cc \u0627\u0632 \u0646\u062a\u06cc\u062c\u0647 \u0627\u0635\u0644\u06cc \\ cite {dorfman1979} \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u06cc\u0645.\u0628\u062e\u0634 ~ \\ ref {\u0628\u062e\u0634_ALTDEFS_GINI_HOOVER_APPLICATION} \u062f\u0648 \u0628\u0631\u0646\u0627\u0645\u0647 \u0645\u0633\u062a\u0642\u06cc\u0645 \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u062f.\u0645\u0627 \\ empl {\u0645\u0633\u06cc\u0631} \u0648\u0627\u0642\u0639\u06cc\u062a (\u0646\u0647 \u0628\u06cc \u0627\u0647\u0645\u06cc\u062a) \u0631\u0627 \u0627\u062b\u0628\u0627\u062a \u0645\u06cc \u06a9\u0646\u06cc\u0645 \u06a9\u0647 \u0645\u0646\u062d\u0646\u06cc \u0644\u0648\u0631\u0646\u0632 \u0628\u0647 \u0637\u0648\u0631 \u06a9\u0627\u0645\u0644 \u06cc\u06a9 \u062a\u0648\u0632\u06cc\u0639 \u0631\u0627 \u0645\u0634\u062e\u0635 \u0645\u06cc \u06a9\u0646\u062f \u060c \u062a\u0627 \u06cc\u06a9 \u0646\u062c\u0627\u062a (\u06af\u0632\u0627\u0631\u0647 ~ \\ ref {bijection_m_lorspace_r)).\u0642\u0633\u0645\u062a \u062f\u0648\u0645 \u0645\u0642\u0627\u0644\u0647 (\u0628\u062e\u0634 ~ \\ ref {section_convergence_results}-\\ ref {setrace_weaker_asuments_cv}) \u0628\u0631 \u0642\u0648\u0627\u0645 $ g (\u03bc) $ \u060c $ h (\u03bc) $ \u0648 $ l_\u03bc $ \u062a\u0642\u0631\u06cc\u0628 \u0645\u06cc \u06cc\u0627\u0628\u062f.\u06cc\u0627 \u0622\u0634\u0641\u062a\u0647\u0627\u0628\u0632\u0627\u0631 \u0645\u0631\u0628\u0648\u0637\u0647 \u0628\u0631\u0627\u06cc \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u060c \u0645\u062a\u0631\u06cc\u06a9 Wasserstein $ \\ Wone $ \u060c \u06cc\u0639\u0646\u06cc \u0645\u062a\u0631\u06cc\u06a9 $ $ $ \u0628\u06cc\u0646 \u062a\u0648\u0627\u0628\u0639 \u06a9\u0645\u06cc \u0627\u0633\u062a.$ \\ WONE (\u03bc_n \u060c \u03bc_ \\ infty) \\ \u062a\u0627 0 $ \u0627\u06af\u0631 \u0648 \u0641\u0642\u0637 \u062f\u0631 \u0635\u0648\u0631\u062a\u06cc \u06a9\u0647 \u0645\u062a\u063a\u06cc\u0631\u0647\u0627\u06cc \u062a\u0635\u0627\u062f\u0641\u06cc \u062f\u0631 \u062a\u0648\u0632\u06cc\u0639 \u0647\u0645\u06af\u0631\u0627 \u0634\u0648\u0646\u062f \u0648 \u0645\u0642\u062f\u0627\u0631 \u06a9\u0644 \u062b\u0631\u0648\u062a \u0647\u0645\u06af\u0631\u0627 \u0634\u0648\u062f.\u062f\u0631 \u0642\u0636\u06cc\u0647 ~ \\ ref {CENTECTIONISIONIS_LORENZ_CONVERGENCE and \u0648 \u06af\u0632\u0627\u0631\u0647 ~ \\ ref {convergence_indicators} \u060c \u0645\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u06cc\u0645 \u06a9\u0647 \u0627\u06af\u0631 $ \\ wone (\u03bc_n \u060c \u03bc_ \\ infty) \\ to 0 $ \u060c \u0633\u067e\u0633 $ g (\u03bc_n) \\ to g (\u03bc_ \\ \\ infty)$ \u060c $ h (\u03bc_n) \\ to h (\u03bc_ \\ infty) $ \u0648 $ l_ {\u03bc_n} \\ to l_ {\u03bc_ \\ infty} $ \u06cc\u06a9\u0646\u0648\u0627\u062e\u062a.\u0632\u06cc\u0631\u0645\u062c\u0645\u0648\u0639\u0647 ~ \\ ref {subsection_topo} \u062f\u0631 \u0645\u0648\u0631\u062f \u067e\u06cc\u0627\u0645\u062f\u0647\u0627\u06cc \u062a\u0648\u067e\u0648\u0644\u0648\u0698\u06cc\u06a9\u06cc \u0627\u06cc\u0646 \u0648\u0627\u0642\u0639\u06cc\u062a \u0628\u062d\u062b \u0645\u06cc \u06a9\u0646\u062f.\u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u060c \u0628\u0631\u0646\u0627\u0645\u0647 \u0647\u0627\u06cc \u06a9\u0627\u0631\u0628\u0631\u062f\u06cc \\ ref {convergence_corollary_noise} \u060c \\ ref {convergence_corollary_sampling} \u060c \\ ref {convergence_corollary_quantiles} \u060c \\ ref {convergence_corollary_quantiles_quantiles_and_and_sampling {ref \u062c\u06cc\u0646\u06cc \u062a\u062c\u0631\u0628\u06cc \u060c \u0634\u0627\u062e\u0635 \u0647\u0627\u06cc \u0647\u0648\u0648\u0631 \u0648 \u0645\u0646\u062d\u0646\u06cc \u0647\u0627\u06cc \u0644\u0648\u0631\u0646\u0632 \u0645\u062d\u0627\u0633\u0628\u0647 \u0634\u062f\u0647 \u0628\u0631 \u0631\u0648\u06cc \u06cc\u06a9 \u0646\u0645\u0648\u0646\u0647 \u06cc\u0627 \u0628\u0627\u0632\u0633\u0627\u0632\u06cc \u0634\u062f\u0647 \u0628\u0627 \u0622\u0646\u0628\u0627 \u0627\u0641\u0632\u0627\u06cc\u0634 \u0627\u0637\u0644\u0627\u0639\u0627\u062a \u060c \u0627\u0637\u0644\u0627\u0639\u0627\u062a \u062c\u0632\u0626\u06cc \u0628\u0647 \u062c\u06cc\u0646\u06cc \u0648\u0627\u0642\u0639\u06cc \u060c \u0634\u0627\u062e\u0635 \u0647\u0627\u06cc \u0647\u0648\u0648\u0631 \u0648 \u0645\u0646\u062d\u0646\u06cc \u0644\u0648\u0631\u0646\u0632 \u0647\u0645\u06af\u0631\u0627 \u0645\u06cc \u0634\u0648\u0646\u062f.\u0633\u0631\u0627\u0646\u062c\u0627\u0645 \u060c \u062f\u0631 \u0628\u062e\u0634 ~ \\ ref {sex_weaker_asuments_cv} \u060c \u0645\u0627 \u062f\u0631 \u0645\u0648\u0631\u062f \u0645\u0648\u0642\u0639\u06cc\u062a \u0647\u0627\u06cc\u06cc \u06a9\u0647 \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc $ \\ wone $ \u062a\u0636\u0645\u06cc\u0646 \u0646\u0634\u062f\u0647 \u0627\u0633\u062a \u0628\u062d\u062b \u0645\u06cc \u06a9\u0646\u06cc\u0645 \u060c \u0627\u0645\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646 \u0622\u0632\u0645\u0648\u0646 \u0647\u0627\u06cc \u0636\u0639\u06cc\u0641 \u062a\u0631\u06cc \u0627\u0646\u062c\u0627\u0645 \u062f\u0627\u062f (\u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u062f\u0631 \u062a\u0648\u0632\u06cc\u0639 \u062f\u0631 ~ \\ ref {ref_convergence} \u060c \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u062f\u0631 ~ ~ \\ \\ref {subtar_convergence_means} \u06cc\u0627 \u06cc\u06a9\u067e\u0627\u0631\u0686\u0647 \u0633\u0627\u0632\u06cc \u06cc\u06a9\u0646\u0648\u0627\u062e\u062a \u062f\u0631 ~ \\ ref {subsection_ui})<\/p>\n\r\n\r\n \r\n \r\n \r\n \r\n
\u0641\u0631\u0645\u062a \u0627\u0631\u0627\u0626\u0647 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u062a\u062d\u0648\u06cc\u0644 \u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\r\n <\/tr>\r\n
\u0632\u0645\u0627\u0646 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u0628\u06cc\u0646 2 \u062a\u0627 3 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634<\/td>\r\n <\/tr>\r\n\t
\u06a9\u06cc\u0641\u06cc\u062a \u062a\u0631\u062c\u0645\u0647 <\/td>\r\n \u0628\u0633\u06cc\u0627\u0631 \u0628\u0627\u0644\u0627. \u0645\u0642\u0627\u0644\u0647 \u0641\u0642\u0637 \u062a\u0648\u0633\u0637 \u0645\u062a\u0631\u062c\u0645\u06cc\u0646 \u0628\u0627 \u0645\u062f\u0631\u06a9 \u062f\u0627\u0646\u0634\u06af\u0627\u0647\u06cc \u0645\u062a\u0631\u062c\u0645\u06cc \u062a\u0631\u062c\u0645\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/td>\r\n <\/tr>\r\n\t\t
\u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 <\/td>\r\n \u06a9\u0644\u06cc\u0647 \u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 \u0646\u06cc\u0632 \u062f\u0631 \u0641\u0627\u06cc\u0644 \u062a\u062d\u0648\u06cc\u0644\u06cc \u0648\u0631\u062f \u062f\u0631\u062c \u0645\u06cc\u200c\u0634\u0648\u0646\u062f.<\/td>\r\n <\/tr>\r\n<\/table>\r\n\r\n\n","protected":false},"excerpt":{"rendered":"

\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc Alternate definitions of Gini, Hoover and Lorenz measures of inequalities and convergence with respect to the […]<\/p>\n","protected":false},"featured_media":27,"comment_status":"open","ping_status":"open","template":"","meta":{"pmpro_default_level":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}}},"product_cat":[21],"product_tag":[],"class_list":{"0":"post-41201","1":"product","2":"type-product","3":"status-publish","4":"has-post-thumbnail","6":"product_cat-21","7":"pmpro-has-access","8":"desktop-align-left","9":"tablet-align-left","10":"mobile-align-left","12":"first","13":"instock","14":"downloadable","15":"shipping-taxable","16":"purchasable","17":"product-type-simple"},"yoast_head":"\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u062a\u0639\u0627\u0631\u06cc\u0641 \u0645\u062a\u0646\u0627\u0648\u0628 \u0627\u0632 \u0627\u0642\u062f\u0627\u0645\u0627\u062a \u062c\u06cc\u0646\u06cc \u060c \u0647\u0648\u0648\u0631 \u0648 \u0644\u0648\u0631\u0646\u0632 \u0627\u0632 \u0646\u0627\u0628\u0631\u0627\u0628\u0631\u06cc \u0647\u0627 \u0648 \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0645\u0639\u06cc\u0627\u0631 Wasserstein W$_1$ - \u0641\u0631\u0648\u0634\u06af\u0627\u0647 \u0627\u06a9\u0633\u067e\u0631\u0633<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/express24.ir\/d\/product\/\u062a\u0631\u062c\u0645\u0647-\u0641\u0627\u0631\u0633\u06cc-\u0645\u0642\u0627\u0644\u0647-\u062a\u0639\u0627\u0631\u06cc\u0641-\u0645\u062a\u0646\u0627\u0648\u0628-\u0627\u0632-\u0627\u0642\u062f\/\" \/>\n<meta property=\"og:locale\" content=\"fa_IR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u062a\u0639\u0627\u0631\u06cc\u0641 \u0645\u062a\u0646\u0627\u0648\u0628 \u0627\u0632 \u0627\u0642\u062f\u0627\u0645\u0627\u062a \u062c\u06cc\u0646\u06cc \u060c \u0647\u0648\u0648\u0631 \u0648 \u0644\u0648\u0631\u0646\u0632 \u0627\u0632 \u0646\u0627\u0628\u0631\u0627\u0628\u0631\u06cc \u0647\u0627 \u0648 \u0647\u0645\u06af\u0631\u0627\u06cc\u06cc \u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0645\u0639\u06cc\u0627\u0631 Wasserstein W$_1$ - 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