A derivative useful for the linearization functional of inequality ... - arXiv

A functional derivative useful for the linearization of inequality indexes in the design-based framework Lucio Barabesi" , Giancarlo Diana# and Pier Francesco Perri$ "

Department of Economics and Statistics, University of Siena, Piazza San Francesco 7, 53100, Siena (Italy) # Department of Statistical Sciences, University of Padova, Via Cesare Battisti 241, 35121, Padova (Italy) $ Department of Economics, Statistics and Finance, University of Calabria, Via P. Bucci, 87036, Arcavacata di Rende (Italy)

Abstract. Linearization methods are customarily adopted in sampling surveys to obtain approximated variance formulae for estimators of nonlinear functions of finite-population totals such as ratios, correlation coefficients or measures of income inequality - which can be usually rephrased in terms of statistical functionals. In the present paper, by considering the Deville’s (1999) approach stemming on the concept of design-based influence curve, we provide a general result for linearizing large families of inequality indexes. As an example, the achievement is applied to the Gini , the Amato, the Zenga and the Atkinson indexes, respectively. Keywords. Influence function; functional linearization; inequality measure.

1. Introduction. Under the usual design-based approach, let Y be a fixed population of identifiable individuals labeled (at least ideally) by the first R integers, i.e. Y œ Ö"ß á ß R ×, and let C3 be the variable value on the 3-th individual. In this setting, Deville (1999) has considered the discrete measure on ‘ Q œ  $ C3 , 3−Y

where $C represents the Dirac mass at C. Deville (1999) has emphasized that the target population parameter may be generally written as a functional J with respect to Q , namely J ÐQ Ñ. In this case, by supposing that a sample W of size 8 is selected from Y according to a design with firstorder and second-order inclusion probabilities respectively given by 13 and 134 , the empirical measure corresponding to Q is given by s œ  " $ C3 Q 1 3−W 3

s Ñ. If J is homogeneous of degree and the substitution estimator for J ÐQ Ñ may be obtained as J ÐQ α, under broad assumptions Deville (1999) has proven the linearization s Ñ  J ÐQ ÑÑ œ 8R α  IFJ Ð?à Q ÑdÐQ s  Q ÑÐ?Ñ  9: Ð"Ñ , 8R α ÐJ ÐQ

where " ÐJ ÐQ  >$? Ñ  J ÐQ ÑÑ >Ä! >

IFJ Ð?à Q Ñ œ lim

is the influence function in the design-based approach (see also Goga et al., 2009). From a mathematical perspective, IFJ Ð?à Q Ñ is actually the Gâteaux differential of J ÐQ Ñ in the direction

of the Dirac mass at ?. Hence, the role of IFJ Ð?à Q Ñ is central, expecially with the aim of variance s Ñ (Deville, 1999). estimation for the empirical functional J ÐQ In this setting, let us consider the functional which may be expressed as J ÐQ Ñ œ  $? ÑÑ  9ÐPÐQ ÑÑÑ >Ä! > " œ lim Ðf9ÐPÐQ ÑÑT ÐPÐQ  >$? Ñ  PÐQ ÑÑ  9Ð>ÑÑ >Ä! > " œ f9ÐPÐQ ÑÑT lim ÐPÐQ  >$? Ñ  PÐQ ÑÑ œ f9 ÐPÐQ ÑÑT IFP Ð?à Q Ñ >Ä! >

IF9‰P Ð?à Q Ñ œ lim

since P is assumed to be Fréchet differentiable.



Proposition 1. Let J be the functional defined in (1). If P4ßC is Fréchet differentiable for each 4, the influence function of J is given by IFJ Ð?à Q Ñ œ