Statistical Inference for Welfare under. Complete and Incomplete Information by. Frank Cowell and. Maria-Pia Victoria-Feser. Discussion Paper. Distributional ...
Statistical Inference for Welfare under Complete and Incomplete Information
by
Frank Cowell and
Maria-Pia Victoria-Feser
Discussion Paper No. DARP 47 December 1999
Distributional Analysis Research Programme The Toyota Centre Suntory and Toyota International Centres for Economics and Related Disciplines London School of Economics Houghton Street London WC2A 2AE Tel: 020-7405-7686
ABSTRACT
We show how a collection of results in the literature on the empirical estimation of welfare indicators from sample data can be unified. We also demonstrate how some of these ideas can be extended to empirically important cases where the data have been trimmed or censored.
Keywords: inequality measurement; income distribution; Lorenz curve; influence function, sampling variance; censoring; trimming JEL classification: C13; D63
Correspondence to: Professor F.A. Cowell, STICERD, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK. Maria-Pia Victoria-Feser is at to the Université de Genéve, Faculty of Psychology and Education, University of Geneva, 40, Bd du Pont d'Arve, CH1211 Geneva 4 Tel: + 41 22 705 9104
Partially supported by the Human Capital and Mobility Programme of the EU grant #ERBCHRXCT94067 and ESRC grant reference R000237324: Robust Methods for Comparisons of Income Distribution.
____________________________________________________ © Frank Cowell and Maria-Pia Victoria-Feser. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.
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|h44?} @?_ Ef2fe( f2efb t?} |i ThLTih uLh4*@ Ai B? it|4@|i t f222D `i U@? _h@ |L 4TLh|@?| UL?U*tL?t 6ht| |i B? it|4@|i ? |i L*i t@4T*i t *@h}ih @?_ t}?U@?|*) _gihi?| |@? ? |i |h44i_ t@4T*i At 4}| _i |L |i Thiti?Ui Lu ih) *@h}i ?UL4i @*it Etii Li** @?_ VU|Lh@6itih bbSM 5iUL?_c |i UL?_i?Ui ?|ih@* ? |i |h44i_ t@4T*i t t*}|*) *@h}ih i? L?i tit |i ThLTih uLh4*@ At hi iU|t |i u@U| |@| |i |h44?} ?|hL_Uit @h@M*|) ? |i it|4@|it q r E3k Wu L?i _i?it |i ti| Lu ThLTLh|L?t X ' ^ ' k n ?k G ' c ?k c |i? ED U@? Mi it|4@|i_ M) % & [ e ^ ^ ' ^ %do k%do d ko %d&o ?k &' % & [ d ^ o %d o d ko %do n d ko %d&o ?k &' % & [ [ 2 %do d ko %d&o d ko % n ?k &' ?k &' d&o % & [ ESb %do ^ %do k%do d ko %do ?k &' ff E q I UL?_i?Ui ?|ih@* uLh |i | wLhi?3 Lh_?@|i t |i? # $ t [ %d&o 5q*2 e ^ ^ *?k E k ?k &' Ai U@*U*@|L?t uLh |i hi*@|i wLhi?3 Lh_?@|it @hi LM|@?i_ ? @ t|h@}|uLh @h_ 4@??ih t?} ESb @?_ >k Mi?} it|4@|i_ M) |i LMtihi_ |h44i_ t@4T*i 4i@? L|i |@| L?i U@? @*tL ULLti @ ?4Mih Lu ThLTLh|L?t & ?k Mi|ii? k @?_ k @?_ U@?}i |i ?_Uit @?_ ? ESb |L hitTiU|i*) VE?k c ^ @?_ VE?k c ^ 6hL4 |i 5T@?t _@|@c i UL4T|i |i t|@?_@h_ ihhLht Lu |i U4*@|i ?UL4it uLh |i L*i t@4T*i @?_ @ |h44i_ t@4T*i ihi DI Lu |i _@|@ @i Mii? hi4Li_ uhL4 i@U t_i Lu |i _t|hM|L? 6Lh |i *@||ih i UL4T|i |i t|@?_@h_ ihhLht ML| M) }?Lh?} @?_ M) UL?t_ih?} |i |h44?} Ai hit*|t @hi Thiti?|i_ ? 6}hi 2 ?i ?L|Uit |@| |i t|@?_@h_ ihhLht uLh |i |h44i_ t@4T*i |L| ULhhiU|L? @hi |i *Lit| Lih@** Ai) @hi Uih|@?*) ?L| ULhhiU| MiU@ti |h44?} *i@_t |L @? ?uLh4@|L? *Ltt U tL*_ @TTi@h @t *@h}ih t|@?_@h_ ihhLht |@? ? |i ?|h44i_ U@ti At t @U|@**) |i U@ti i? L?i *LL!t @| |i t|@?_@h_ ihhLht uLh |i |h44i_ t@4T*i | ULhhiU|L?
2
Cumulative income functionals standard errors
15 10 0
5
Standard errors
20
Untrimmed sample Trimmed sample without correction Trimmed sample with correction
0.0
0.2
0.4
0.6
0.8
1.0
Proportions
6}hi 2G 5|@?_@h_ ihhLh UL4T@htL?t uLh UL4T*i|i @?_ |h44i_ t@4T*itc | Lh |L| |h44?} ULhhiU|L?
eS
W?uLh4@|L? L? |i , U*_i_ 5Mt@4T*i
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2 > e/ e^^ n e S^ e S^ / e > ee S^ / e^ > ee S^ / e^ e > e
> e'e Sk n S } r G' rek n r } e / e G' re > e2 / e^ G' e r^ n d^e %^ e S^ o > e %^ e S^
uLh |i +w @?_ |i B? t @? i @4T*ic |@!i |i 5T@?t _@|@ @?_ UL4T|i |i w @?_ +w uLh |i Ui?tLhi_ E+ ' eff @?_ + ' Dff t@4T*i @t i** @t uLh |i u** t@4T*i Ai hi^hi_ t|@|t|Ut @hi @tt4i_ |L Mi !?L? Ai _gihi?| wLhi?3 Uhit @hi Thiti?|i_ ? 6}hi 6Lh |i Ui?tLhi_ U@tic L?i }i|t i @U|*) |i u** t@4T*i U@ti uhL4 U |i Ui?tLhi_ T@h|t @i Mii? 4@t!i_ eS2
hLTLh|L?t i_c ) h@?_L4
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Censored sample
0.8 0.6 0.4 0.0
0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
Proportions
Proportions
Full sample
Full sample
0.8
1.0
0.8
1.0
0
0.8 0.6 0.4 0.2 0.0
200
400
600
Relative Lorenz curve
800
1.0
0.0
Lorenz curve
0.2
Relative Lorenz curve
600 400 200
Lorenz curve
800
1.0
Censored sample
0.0
0.2
0.4
0.6
0.8
1.0
0.0
Proportions
0.2
0.4
0.6
Proportions
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D
L?U*_?} +i4@h!t
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