Nonparametric and Gaussian bivariate transvariation theory: its applications to economics
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Nonparametric and Gaussian bivariate transvariation theory: its applications to economics Camilo Dagum1
Received: 1 December 2016 / Accepted: 15 March 2017 / Published online: 29 August 2017 © Sapienza Università di Roma 2017
Abstract This paper is concerned with the bivariate transvariation theory. It presents an historical account of the subject and a development of the theory. It deals with the probability of transvariation and its related concepts, namely, transvariability and its maximum; the r th moment of transvariation, its maximum and the r th intensity of transvariation; area of transvariation and discriminative value. These concepts are developed without parametric constraint and under the assumption of Gaussian distribution. The sample variances and covariances of the transvariation parameters estimators are herein deduced. The applications, performed on economic variables, underline the fruitfulness of transvariation theory as a quantitative method to deal with comparative statics analysis. Keywords Transvariation theory · Bivariate transvariation · Applications to economics
1 Introduction The theory of transvariation, formulated by Corrado Gini, was first introduced into mathematical statistics in Gini’s “II Concetto di Transvariazione e le sue Prime Applicazioni”, published in 1916 [1]. Editors’ Note: This version of the paper has been edited by Francesco Dotto, research fellow at the Department of Statistical Sciences, University of Rome “La Sapienza”, email:[email protected]. A longer version of this paper was released as research Memorandum No 99 of the Econometric Research Program at Princeton University in June 1968. We would like to warmly thank Estela B. Dagum and Bo E. Honoré, Director of the Gregory C. Chow Econometric Research Program at Princeton University, for their authorization to publish the paper in this special issue. Original Note: The subject of this paper was discussed in the Econometric Research Program Seminar under the direction of Professor Oskar Morgenstern and in the Department of Statistics Seminar under the direction of Professor John W. Tukey. The author is very much indebted to them for their interest in the subject and stimulating comments. The author expresses his appreciation to Mr. Peter Kaminsky, a Princeton University student, for his diligence and collaboration in preparing the final English version.
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Camilo Dagum [email protected] Econometric Research Program, Princeton University, Princeton, USA
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Gini’s formulation of this theory grew out of his attempt to develop a proper statistical method for the solution of the following problem: given the sign of the difference between the means or medians, of two populations, find a probabilistic statement regarding the sign or the intensity of the difference (the common area, etc. . .) between two random observations corresponding to each of the populations. The probabilistic statement of each characteristic defines a parameter of transvariation, i.e., the probability of tra
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