of inequality indexes in the design-based framework

Transcription

1 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 Siena (Italy) # Department of Statistical Sciences University of Padova Via esare Battisti Padova (Italy) $ Department of Economics Statistics and Finance University of alabria Via P. Bucci 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 Öß á ß R and let 3 be the variable value on the 3-th individual. In this setting Deville (1999) has considered the discrete measure on Qœ$ 3 3 Y where $ represents the Dirac mass at. Deville (1999) has emphasized that the target population parameter may be generally written as a functional J with respect to Q namely JÐQÑ. In this case by supposing that a sample W of size 8 is selected from Y according to a design with first- order and second-order inclusion probabilities respectively given by 13 and 134 the empirical measure corresponding to Q is given by Qœ s 1 $ 3 3 W 3 and the substitution estimator for JÐQÑmay be obtained as JÐQÑ s. If J α under broad assumptions Deville (1999) has proven the linearization is homogeneous of degree α 8R ÐJ ÐQÑ s J ÐQÑÑ œ α 8R IF J Ð?à QÑd ÐQs QÑÐ?Ñ 9: ÐÑ where Ð?à QÑ œ ÐJ ÐQ > Ñ J ÐQÑÑ > IF J lim $? >Ä! is the influence function in the design-based approach (see also Goga et al. 2009). From a mathematical perspective IF J Ð?à QÑ is actually the Gâteaux differential of J ÐQÑ in the direction

2 of the Dirac mass at?. Hence the role of IF J Ð?àQÑ is central expecially with the aim of variance estimation for the empirical functional JÐQÑ s (Deville 1999). In this setting let us consider the functional which may be expressed as JÐQÑ œ ÐP ÐQÑÑ QÐÑ < d (1) where P ÐQÑ œ ÐPßÐQÑßáßP5ßÐQÑÑ is a vector of functionals (eventually) indexed by and 5 < À È is a function family assumed to be differentiable and regularly indexed by. he inequality measures commonly considered in practice are members of the functional family J or may be expressed at most as : ÐJ ÐQÑÑ œ Ð : J ÑÐQÑ where : À È is a smooth function - in the next Section some illustrative examples are providedþ In order to obtain the linearization of the functional J defined in expression (1) the following results are useful. Lemma 1 is introduced for the sake of completeness since its proof (and the connected assumptions) are not usually reported in statistical literature in its precise version. For more details on Gâteaux and Fréchet differentiability see e.g. Behmardi and Nayeri (2008) and the references therein. Lemma 1. If PÐQÑ œ ÐP ÐQÑßáßP5ÐQÑÑ is a vector of functionals and 9 À È is a differentiable function let us consider the functional 9ÐPÐQÑÑ œ Ð 9 PÑÐQÑ. By assuming that P4 is Fréchet differentiable for each 4 the influence function of Ð 9 PÑis given by 9 P P IF Ð?àQÑ œ f9ðpðqññ IF Ð?àQÑ where IFPÐ?àQÑœÐIFP Ð?àQÑßáßIFP Ð?àQÑÑ. 5 Proof. By Peano's form of aylor's formula it holds 9 ÐPÐQ > $? ÑÑ œ 9ÐPÐQÑÑ f9 ÐPÐQÑÑ ÐPÐQ > $? Ñ PÐQÑÑ 9ÐmPÐQ > $ Ñ PÐQÑmÑ? 5 and since 9ÐmPÐQ > $? Ñ PÐQÑmÑ œ 9Ð>Ñ by definition it follows that IF9 PÐ?à QÑ œ lim Ð9ÐPÐQ > $? ÑÑ 9ÐPÐQÑÑÑ œ lim Ðf9 ÐPÐQÑÑ ÐPÐQ > $? Ñ PÐQÑÑ 9Ð>ÑÑ œ f9 ÐPÐQÑÑ lim ÐPÐQ > $? Ñ PÐQÑÑ œ f9ðpðqññ IFPÐ? àqñ since P is assumed to be Fréchet differentiable. Proposition 1. Let J be the functional defined in (1). If P4ß is Fréchet differentiable for each 4 the influence function of J is given by IFJ Ð?à QÑ œ <? ÐP? ÐQÑÑ f< ÐPÐQÑÑ IFP Ð?à QÑ dqðñ where IFP Ð?àQÑœÐIFP Ð?àQÑßáßIF P Ð?àQÑÑ. ß 5ß Proof. By definition it holds

3 IF Ð?à QÑ œ lim ÐJ ÐQ > $ Ñ J ÐQÑÑ œ lim Ð < ÐPÐQ > $? ÑÑ dðq > $? ÑÐÑ < ÐPÐQÑÑ dqðññ œ lim Ð< ÐPÐQ > $? ÑÑ < ÐPÐQÑÑÑ dqðñ lim <? ÐP? ÐQ > $? ÑÑ. >Ä! J? For a fixed? we have lim < ÐP ÐQ > $ ÑÑ œ < ÐP ÐQÑÑ >Ä!????? and since < is continuously indexed by it reads IF Ð?à QÑ œ lim Ð< ÐPÐQ > $ ÑÑ < ÐP ÐQÑÑÑ dqðñ < ÐP? ÐQÑÑ œ f< ÐP ÐQÑÑ IFP Ð?à QÑ d QÐÑ <? ÐP? ÐQÑÑ J?? on the basis of the differentiability of < and Lemma 1. Hence Proposition 1 provides a simple rule for obtaining the influence function corresponding to the functional (1). Finally it should be remarked that by means of Lemma 1 the influence function of Ð : JÑ also follows i.e. IF Ð?à QÑ œ : ÐJ ÑIF Ð?à QÑ : J by assuming that : be differentiable. 2. Application to some inequality indexes. We show the usefulness of Proposition 1 for the linearization of some inequality measures commonly adopted in practice (for a general treatment of inequality indexes see the classical monograph by owell 2011). As a first example the celebrated concentration index is considered. In this setting Langel and illé (2012a) have provided the expression of the influence function by developing an ad hoc differential rule. he same result may be simply achieved by using Proposition 1. Indeed the finite-population version of the concentration index (see e.g. Berger 2008) may be expressed as the functional where #LÐQÑ KÐQÑ œ d QÐÑ RÐQÑXÐQÑ w J RÐQÑ œ dqðbñ is actually the population size R rephrased as a functional and XÐQÑ œ BdQÐBÑ is the population total. In addition by assuming that MF is the usual indicator function of a set F

4 LÐ QÑ œ MÒBß ÒÐÑdQÐBÑ actually represents the number of individuals whose variable value is less than or equal to a given. In the following we also assume that OÐ QÑ œ BMÒß ÒÐBÑd QÐBÑ which obviously turns out to be the total of the variable values greater than or equal to a given. Hence in this case we have PÐQÑ œ ÐLÐ QÑßRÐQÑßXÐQÑÑ while < ÐP ÐQÑÑ #LÐQÑ œ RÐQÑXÐQÑ and :ÐJ Ñ œ J Þ hus with a slight abuse in notation i.e. by suppressing the argument of the functionals for the sake of simplicity it holds that # L L f< ÐP ÐQÑÑ œ ß ß RX R X and IF P Ð?à QÑ œ ÐMÒ?ß ÒÐÑßß?Ñ. Hence by applying Proposition 1 after some algebra it follows that IF K Ð?à QÑ œ # Ð? L Ñ ÐK Ñ?? O? RX R X which coincides with the expression given by Langel and illé (2012a). As a second example we consider the Amato index which has recently received renewed interest for its properties (Arnold 2012). he influence function for the Amato index is not available in literature. o this aim on the basis of the continuous-population expression of the Amato index (Arnold 2012) the finite-population counterpart of this inequality measure may be given as the functional # EÐQÑ œ QÐÑ RÐQÑ # XÐQÑ # d. Hence in this case PÐQÑ œ ÐRÐQÑßXÐQÑÑ while < ÐP ÐQÑÑ œ # RÐQÑ XÐQÑ # # and trivially :ÐJ Ñ œ J. By adopting the same notational simplification as above and by assuming that. œxîris the population mean it holds that # f < X ÐPÐQÑÑ œ ß. # # R $ X $ and IF P Ð?à QÑ œ Ðß?Ñ. Hence by applying Proposition 1 it turns out that

5 IF E Ð?à QÑ œ.?.#?# dqðñ d QÐÑ. X R#. # # X#. # # As a third example we consider an inequality measure recently proposed by Zenga (2007) - the so-called Zenga new index - which has received considerable attention (see e.g. Langel and illé 2012b). Langel and illé (2012b) have introduced the finite-population version of the Zenga new index based on smoothed quantiles and have provided the linearization of the corresponding functional. However by considering the continuous-population expression as proposed by Zenga (2007 expression (5.6) in his paper) the natural finite-population counterpart of the Zenga new index may be given as the functional ^ÐQÑ œ ÐRÐQÑ LÐ QÑÑÐX ÐQÑ OÐ QÑÑ d QÐÑ. RÐQÑLÐQÑOÐQÑ Hence in this case PÐQÑ œ ÐLÐQÑß OÐQÑß RÐQÑßXÐQÑÑ while < ÐP ÐQÑÑ ÐRÐQÑ LÐ QÑÑÐX ÐQÑ OÐ QÑÑ œ RÐQÑLÐQÑOÐQÑ and :ÐJ Ñ œ J Þ By adopting in turn the same notational simplification for the argument of the functionals it holds that XO. ÐRLÑ X O RL f< ÐPÐQÑÑ œ ß ß ß LO # LO # R# O RLO and IF P Ð?à QÑ œ ÐM ÐÑß? M ÐÑßß?Ñ from which it follows that IF Ò?ß Ò Ò?ß Ò ÐR L? ÑÐX O? Ñ ÐX Ñ Ð?à QÑ œ M Ò?ß Ò Ð Ñ O dqðñ RLO?? LO MÒ?ß ÒÐÑÐRLÑ XO.? dqðñ dqðñ LO R # # O? RL d QÐÑ. R LO ^ # For the final example the Atkinson index is assumed. he finite-population counterpart of this inequality measure may be expressed as the functional # EÐQÑœ % % RÐQÑ XÐQÑ % % ÎÐ% Ñ d QÐÑ where % Ò!ß Ò. Hence in this case P ÐQÑ œ ÐRÐQÑß X ÐQÑÑ > while < ÐP ÐQÑÑ œ % RÐQÑ XÐQÑ % % and :ÐJ Ñ œ J ÎÐ% Ñ. Hence with the usual notational simplification for the argument of the functionals it holds that

6 % % % f< ÐPÐQÑÑ œ ß RX % % R X and IF P Ð?à QÑ œ Ðß?Ñ. By applying Proposition 1 and after some algebra it follows % E%?? % IF E% Ð?à QÑ œ R Ð%. Ñ. % % where. < œr < d QÐÑ represents the < -th population moment. Bibliography Arnold B.. (2012) On the Amato inequality index Statistics & Probability Letters Behmardi D. and Nayeri E. (2008) Introduction of Fréchet and Gâteaux derivative Applied Mathematical Sciences Berger Y.G. (2008) A note on the asymptotic equivalence of jackknife and linearization variance estimation for the Gini coefficient Journal of Official Statistics owell F.A. (2011) Measuring inequality 3rd edition Oxford University Press New York. Deville J.. (1999) Variance estimation for complex statistics and estimators: linearization and residual techniques Survey Methodology Goga. Deville J.. and Ruiz-Gazen A. (2009) Use of functionals in linearization and composite estimation with application to two-sample survey data Biometrika Langel M. and illé Y. (2012a) Variance estimation of the Gini index: revisiting a result several times published Journal of the Royal Statistical Society Series A Langel M. and illé Y. (2012b) Inference by linearization for Zenga s new inequality index: a comparison with the Gini index Metrika Zenga M. (2007) Inequality curve and inequality index based on the ratios between lower and upper arithmetic means Statistica e Applicazioni