Caso, Covadonga

Gil, María Angeles

Institute of Mathematics AS CR, v. v. i.

KybernetikaCovadonga Caso; María Angeles GilEstimating income inequality in the stratified sampling from complete data. II. Theasymptotic behaviour and the choice of sample sizeKybernetika, Vol. 25 (1989), No. 4, 312--319Persistent URL: http://dml.cz/dmlcz/125822Terms of use:© Institute of Information Theory and Automation AS CR, 1989Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czK Y B E R N E T I K A - VOLUME 25 (1989), NUMBER 4 ESTIMATING INCOME INEQUALITY IN THE STRATIFIED SAMPLING FROM COMPLETE DATA Part II. The Asymptotic Behaviour and the Choice of Sample Size* COVADONGA CASO, MARIA ANGELES GIL The First Part of this paper deals with the study of the precision of an unbiased estimator of a population income inequality index in the stratified samplings with and without replacement. In this Second Part, we first analyze the asymptotic distribution of the sample income inequality index. Then, we establish different criteria to select the suitable sample size to estimate the popula-tion index with a desired degree of precision. One of these criteria is based on Chebyshev's approach and the other one follows from the preceding asymptotic study. 1. INTRODUCTION When we try to draw statistical conclusions regarding the income inequality in an uncensused population, it would be interesting to know the distribution of the sample index for different sampling methods. Nevertheless, such a distribution cannot be exactly determined, so that the study of the asymptotic behaviour of the sample index, we are going to develop in the next section for the stratified sampling, may be useful. In addition, in most of the problems that try to obtain conclusions about the population income inequality, large samples are available, and the suitability of the preceding study is then justified. Once we have examined the asymptotic distribution of the sample index, we may readily derive some practical results. In this way, we are first going to develop a non-conservative procedure, which allows us to select the sample size needed to estimate the population income ine-quality with a specified degree of precision. This procedure is then compared with the conservative criterion based on Chebyshev's approach. Other statistical procedures, such as confidence intervals and hypotheses testing methods about the population inequality, are finally suggested. * This work was supported in part by the Comision Asesora de Investigation Cientifica y Tecnica (CAICYT) under the contract PB 85-0401. 312 2. ASYMPTOTIC BEHAVIOUR OF THE SAMPLE INCOME INEQUALITY INDEX Consider a finite uncensused population of N income earners which is divided into r non-overlapping strata. Assume that each individual income for a certain period is positive, x*, ..., xM being the possible different income values in the popula-tion (x* > 0). Let Nk be the number of individuals in the fcth stratum (so that, Nj + ... + Nr = N) and let pik and pi denote the probabilities that a randomly selected individual in the /cth stratum and in the whole population, respectively, has an income equal to x* (i = 1, ..., M, k = 1, ..., r) in the considered period of time. Assume that a stratified sample of size n is drawn at random from the population independently in different strata. For the sake of operativeness, we hereafter suppose that the sample is chosen by proportional allocation in each stratum, so that a sample of size nk is drawn at random (with or without replacement) from the /cth stratum, where nkjn = NhJN, k = 1, ..., r. Let fik and fL denote the relative frequencies of individuals in the sample from the /cth stratum and in the sample from the whole population, respectively, with income equal to xf (i = 1, ..., M, k = 1, ..., r) in the considered period of time. According to Definitions 2.1 and 3.1 in the First Part of this paper, [2], I~1(X*) and I~' represent, respectively, the population and sample additively decomposable income inequality index of order - 1 (cf., [ l ] , [3], [4], [5], [13], [14]). Following ideas in [11], [12], and [15], we can now establish Theorem 2.1. The random variable nx,2[l~ l — J - 1 (AT*)] is asymptotically distribut-ed (as nk -> oo, k = 1, ..., r) according to a normal distribution with mean zero and variance equal to ^ = - I £'{[•-(-. 0]2 + [/-(-.*)]2 + 2[/-(**) + 1]/-«(*)} + fc= 1 N + 3[/"2(X*) + r2(X*~l)~] - 4 /~ 1 (**) whenever T2 > 0. Proof. The first order Taylor expansion for I~x(f) =• I~1(f= (flu ...,fMi, ••• . . . , / l r , ...,fMr)) in a neighborhood ofp _ (plt, ..., pM1, ..., plr, ..., pMr) is given by r-(f) = r'(p) + Y t ^ (U - Pik) + Rn i=i fc=i dpik where p* is the (M — 1) x r-dimensional vector (pu, ..., P(M-in> •••> Pin •••> P(\f-i)r)> B(p*) = I~1(p) == I~l(X*) and R„ is the Lagrange remainder term. Thus, M M / * 4 r'(f) = l~\p) + I Vi(fL - Pi) + R„, where vt = J pA^ + % (-.1 i = l \Xi Xj 313 As in the sampling we have considered the random vectors (n1/zfxl, •••, n1/2fM1), ... .... (n1/2flr, ..., n1/2fMr) are independent and each of them has an asymptotic multi­variate normal distribution with mean vectors (n1/2p11, ..., n1/2pM-), ••-, (n1/2plr, ... ..., n1/2pMr), respectively, then the random vector (n1/2f1_, ..., n1/2fM) has an asymptotic multivariate normal distribution with mean vector (nl/2pu, ..., n1/2pM), as a consequence of the reproductivity of the multivariate normal distribution. The remainder term n1/2Rn can be expressed by oo M-l r fii)n(n*\ n1/2R„ = E E I °-f*-l «U2(U - P«)J = j = 2 i = i ft=i dpik 00 M— 1 r = H I ?ffi-3 ^ i , Pik)Y'2 n^ " "P* (fik - ^y-1 As « -> oo, (L-fe — pik)]~x converges in probability to zero (weak laws of large numbers) and [nfik — npik]j[npik(l — pik)]l/2 converges in law to a standard normal random variable (De Moivre's Theorem). Then, according to the well-known properties of the convergence in probability we have that n1/2(fik — pik)' converges in probability to zero as n -> oo, for all / = 2, 3,..., i = 1, ..., M — 1, k = 1, ..., r, so that n1/2Rn converges in probability to zero as n -> oo. Therefore, the random variable n1/2\I~x — I~1(X*)], coinciding with _V / 2t>;. i • (fi. ~ Pi) unless for the remainder term n1/2R„, will be asymptotically distributed according to a normal distribution with mean zero and variance equal to M M r r ^ « n i i vtVj E[(U - Pik) (^ - Pjl)] -i=l j=l k=l 1=1 r N. = - 1 — P + r\k, •) + rH; k)]2 + 3[r2(x*) + /-^X*-1)] + 2r\X*) + 4 = fc = l N - - t § {t'-'(*. ')? + [/-'(•• *)P + 2['~'(**) + 1] • /-W} + fc — 1 J \ + 3[r2(x*) + /^(x*-1)] - 4/-1(X*) whenever T 2 > 0. f_ Remark 2.1. The asymptotic variance T 2 becomes the limit of nVn = Var [n1 / 2 . •(»« ) J a s nk<~*y, (k — 1,..., r), and it is always lower than the asymptotic variance of n1'^!*1 _ /--(„-*)] for the non-stratified random sampling, since ",?, w[ 2 + г"'(*'') + /_1(->*)ľ š - Г t P + t"'(-Д) + Í-Ҷ*. •)]* so that, = - 4 [ / - W + l ] 2 - 4 [ z ' x ( * * ) + 1 ] 2 + 3^-a^) + /-a/^-i)] + 2/-i(AT*) + 4 -- 4 l ( * * ) ] 2 + 3 [ / - 2 ( ^ ) + / - 2 ^ , - ^ - j _ 6 r i { x * ) тz < 314 and this last expression coincides with that for the asymptotic variance of the un-biased estimator in the non-stratified random sampling (r = 1). This last fact corroborates the advisability of stratifying populations in order to estimate income inequality for large samples in each stratum. The following result guarantees that in the asymptotic normal distribution of the statistic n1 / 2[I_ 1 — I~1(X*)~], the asymptotic variance T2 may be replaced by its analogue estimator T2 (where T2 is obtained by replacing the population indices I~\X*), I~2(X*), I~2(X*~X), I~\k, •), J _ 1 ( - , k), and I~l(k) by the corresponding sample indices) without appreciably affecting the accuracy of the approximation. Thus, Theorem 2.2. The random variable n 1 / 2 [ I - 1 — I~1(X*)]jxn is asymptotically distributed according to a standard normal distribution, whenever T2 > 0 and T2 > 0. Proof. On the basis of the considered sampling, we can state that/ i fc converges in probability to pik (i -=- 1, . . . , M, k — 1, ..., r) and, consequently, fL converges in probability to pt. Then, according to the well-known properties of the convergence in probability, we can easily deduce that x2 converges in probability to T2 (and xn converges to T). As T is a positive constant value, and n 1 / 2[I _ 1 — J_ 1(X*)] converges in law to a normal distribution ^ ( 0 , T2), it follows that n 1 / 2[I _ 1 — I~l(X*)~\jx„ converges in law to a standard normal distribution. • 3. THE CHOICE OF SAMPLE SIZE IN ESTIMATING INCOME INEQUALITY FROM COMPLETE DATA Assume that a margin of error s > 0 in the estimated income inequality index has been agreed on, and there is a small risk a > 0 that we are willing to incur that the actual error is larger than e when we use the analogue estimator I~i. According to Theorem 2.1, if the asymptotic variance T2 were a known positive value, we could use the asymptotic distribution of n 1 / 2[I _ 1 — I~1(X*)~\ to look for the suitable sample size to achieve the preceding requirements. Thus, the value n* given by (1) n* = [4V/s2] + 1 (where Xa is the critical point of the standard normal distribution at the significance level \a and [ ] is the greatest integer function), would be adequate for our purposes. Nevertheless, the population value T2 will be unknown in practice and it is in addi-tion impossible to construct a general conservative criterion to choose the sample size similar to that in estimating proportions. In other words, it is not possible to find an upper bound for T2 and replace this unknown value by that bound. In spite of this fact, we could define a non-conservative criterion based on Theorem 315 2.2, so that the value n* given by (2) n* = ft-,/.2] + 1 (n = sample size of a previous sample) would be "approximately" adequate for our purposes. On the other hand, the study of the exact precision of the unbiased estimator (^-1)s, we have developed in the First Part of the present paper (Theorems 3.3 and 3.4), allows us now to state another definitively conservative procedure to select the sample size guaranteeing the specified degree of precision determined by the limit error g and the risk a. In this way, following Chebyshev's approach, we find that such a degree of precision is achieved when we draw at random (and according to a stratified sampling with proportional allocation, independently and with replacement in each stratum) a whole sample of size n*, where n* is the integer such that Var [ ( ^ 4 ) s ] ^ s2oc = s* for all n ^ n*. Obviously, the existence of such an integer n* is confirmed from the fact that Var [ (C ! ) S ] tends to 0 as nk -» GO (k = 1, ..., r), and it maybe obtained in practice by solving an inequation of order s + 2 (where s = number of strata with different sizes). In particular, when the sizes of all strata coincide, n* may be obtained by looking for the minimum integer such that (3) Ae*n3 + BE*n2 + Cn + D > 0 where (if we denote wk = NkJN = nkjn), Ae. = 2e*£w2 j t = i BB. = -[2fi* - 2iw2{[/-1(-,*)]2 + [r\k, -)Y + 2i~\x*). r\k) -& = i - 3[r2(x*) + r2(x*-x)~\ + 4r1(x*) + 2i~1(k)}] c - - P S iw2kwl{[r\k,i)Y + r\k).r\i)} + 2iw2k{-3[r2(.,k) + k=i i = i fc=i + /-*(•, JT1)] + r\x*2) + 2r\x*)} + iiwk{[r\-,k)Y + k=\ + [r\k, -)Y + 2r\x*).r\-,k) - 3[/-2(X*) + r2(x*-iy\ + + 4/~'1(X*) + 2/"1(k)} -> =-l~2i i">k">l{[r1(k,i)Y + r\k).r\i)}-2iwk{-3[r2(;k) + k = 1 I = 1 & = 1 + /-'(-,jr1)] + r\x*2) + 2r\x*)} + 2iw2{2[r\k)Y -k= 1 - 3[r2(k) + r2(x*-x)] + r\k2) + 2rl(k)}~] In the same way, if we take a random sampling without replacement in each stratum, the required objective may be achieved by looking for the minimum integer n* such that Var [(C1)5 0] = fi2« = £* f° r a ^ n ^ n*. Obviously, the existence of such 316 an integer n* is confirmed from the fact that Var [(H~ l)sc] tends to 0 as nk -> Nfe (k — 1, ..., r), and it may be obtained in practice by solving an inequation of order s + 2 (where s = number of strata with different sizes). In particular, when the sizes of all strata coincide, n* may be obtained by looking for the minimum integer such that (4) A£*«3 + Be*n2 + Cn + D > 0 where ...2 -4.-«8*î>Ź-L {2/Vfe(7Nfe-9)[/-Ҷ*)]: Nfc(Nfc-l)ҶNfc-2)(Nfc-3) - ЗNл(5Nfe - 7) [Г2(k) + ГҶ*" 1 ) ] + (2N2 - ЗNk - 1) Г Ҷ *2 ) + + 2(lliV2-15iVfe + 2)ГҶ*)} + £ .t*i (Nfc - 1) (Nг - 1) . {[/-Ҷ*, t)]2 + /-Ҷ*). /-Ҷ/) - 3[ГҶ*, /) + /-Ҷ*" 1, Г 1)] + 1 + ГҶ* 2,/ 2) + 2ГҶ*,/)} + Nfc- 1 {[/-Ҷ*,.)]2 + [rҶ-,*)] 2 + + 2/-ҶX*)./-Ҷ*) - 3[/-Ҷ*, •) + Г 2 ( * - \ •)] + 2ГҶ*) + 2>Vь + 2[ГҶ*,.) + ГҶ.,*)]} + {2ГҶ*).[ГҶ*, •) + (N f c -l)(N f c -2) + /"*(., *)] - [1 + /-*(*)] . [/-*(-, * \ *-2) + /-*(.,*, *2)] + + / _ 1 (*, O + r ^ . , * ) + 2/-1(*)} B£ł= - E H Z ^ , w. --i ЦV f e (N f c - l)2 (N f c -2)(N f e -3) {2N 2(l0N 2-7N f e-9)[/-Ҷ*)]2-- 3N2(7N2 - 5Nfe - 8) [r2(k) + r2(k-')] + (2N3 + 2N2 - HNfe + 1) r-X(lЛ ) + 2(17N3 - 19N2 - 2Nfe - 2) /-Ҷ*)} + X w.(Nfewř + Nřwfe+ 1) . {[Г Ҷ*, /)]2 + /-Ҷ*). /-Ҷ/) + /-Ҷ*2, /2) + + 2 /"Ҷ*, /) - 3[ГҶ*, /) + /-Ҷ*" 1, r1)]} + Nfe + 1 | f ! (Nfe - 1) (N, - 1) + Nfe-1 {2 /-ҷлr*). Г Ҷ*) + [/-Ҷ*, -)]2 + [ Г Ҷ., *)] : - 3[ГҶ*, •) + Г Ҷ * " 1 , •)] + 2[/-Ҷ*) + z-ҷ*, •) + Z-Ҷ-,*)]} + + / 2 V V f e ( ^ + 1 ) , {2Z-Ҷ*). [Z-Ҷ*, •) + ГҶ.,*)] -( N f e - l ) ( N f e - 2 )1 - [i + z-Ҷ*)]. [z-ҷ.,*-\*-2) + z-Ҷ.,*,*2)] + + Z-Ҷ*, .) + Z-Ҷ.,*) + 2Z-Ҷ*)}l 317 c =1 wi {-2Nfc(N2 + 12Nfe-21)([ГҶ*)]2 + L ( N f e - l )2 ( N f e - 2 ) ( N f c - 3 ) + 12Nfc(ЗNfc - 5) [Г2(k) + ГҶ/Г 1)] + (2N3 - 15N2 + 22Nfc - 1) . , .,,,,. . , , ч < / ł Ч ч Л w,(NkN,wk + Nkw, + N,wк) . ГҚk2) - 4(2N3 + ЗN2 - 5Nfe - 1) ľҚk)} - I -& ^ ^ ^ Д ^ *' . {[r Ҷ*, /)]2 + r Ҷ * ) . rҶ7) + ГҚk2, l2) + 2 ГҚk, l) - з [ r Ҷ*, /) + + ГҶ/Г1,/-1)]} - -*« { 2 Г Ҷ * * ) . Г ҶÄ) + [ГҶ*, -)ľ + Џ"Қ%к)Y Nк - 1 - 3[ГҶЛ, •) + Г2(k~\ •)] + 2[Г\k) + ГҚk, •) + rҶ.,/fc)]} -2Nfewfc { 2 f _ 1 ( л ) [ z _ 1 ( ^ ш) + 7 _ 1 ( . 5 ft)- _ p + 7 _ 1 ( i t ) - _ ( N f e - l ) ( N f e - 2 ) . [/-*(., k~\ k~2) + r\-, k, k2)] + 2 /-*(*) + r\k, •) + r*(-, *)}] £ - t - f ^ r [ r ^ r {-2Nfc[/~lW]2 + 3Nfc[r2« + /-»(*-*)] -*_i Nfc - i [_Nfc - i - /-(ft2) - 2 /-(ft)} + t -£&- {[/"(*• ')]2 + r »(*) • r\i) + i = i Nt — 1 + r ^ /2) + 2 /--(*, /) - 3[r *(*, /) + r\k~\ r1)]} The coefficients in (3) and (4) involve unknown population values, that can be approximated by their respective unbiased estimates in the Appendix of the First Part of this paper. It is worth remarking that the application of Chebyshev's ine-quality compensates enough the possible error in approximating the population values by means of their unbiased estimates, so that the criterion to choose the sample size remains usually conservative. Remark 3.1. The integer n* in Inequations (1), (2), (3) and (4), must be chosen so that n*wk, (k — 1, ..., r) are integer numbers, to allow us to obtain in practice integer sample sizes nk. Remark 3.2. Obviously, the procedures to select the sample sizes in (3) and (4) are based on the exact distribution of a random variable, so that they may be applied for small samples and populations (whereas those in (1) and (2) may only be con-sidered for dealing with large samples in each stratum). On the other hand, the criteria based on ( l ) and (2) always determine smaller sample sizes than those derived from (3) and (4). 4. C O N C L U D I N G REMARKS A study similar to that in the Second Part of the present paper has been developed [7] for estimating population entropy and diversity. It would be now interesting 318 to develop another one for estimating the mutual information ([6], [8], [9], [10]) in the stratified sampling. On the other hand, although the research in the First Part of this paper may not easily be accomplished for other income inequality indices, to examine their asymp-totic behaviour becomes simple. In addition, the asymptotic analysis in Section 2 is immediately applicable for defining some useful statistical problems: constructing confidence intervals for the population income inequality, and testing statistical hypotheses concerning this population value, on the basis of the sample income inequality. In virtue of Theorems 2.1 and 2.2, when it is possible to draw large samples in each stratum, these suggested problems may be easily solved. (Received March 29, 1988.) R E F E R E N C E S [1] F. Bourguignon: Decomposable income inequality measures. Econometrica 47 (1979), 901-920. [2] C Caso and M. A. Gil: Estimating income inequality in the stratified sampling from complete data. Part I: The unbiased estimation and applications. Kybernetika 25(1989), 4, 298— 311. [3] F. A. Cowell: On the structure of additive inequality measures. Rev. Econom. Stud. 47 (1980), 521-531. [4] F. A. Cowell and K. Kuga: Additivity and the entropy concept. An axiomatic approach to inequality measurement. J. Econom. Theory 25(1981), 131—143. [5] W. Eichhorn and W. Gehrig: Measurement of inequality in economics. In: Modern Applied Mathematics — Optimization and Operations Research (B. Korte, ed.), North-Holland, Amsterdam 1982. [6] M. A. Gil and C Caso: A note on the estimation of the mutual information in the stratified sampling. Metron 45 (1987), 1-2, 295-301. [7] M. A. Gil and C Caso: The choice of sample size in estimating entropy according to a strati-fied sampling. In: Uncertainty and Intelligent Systems. Proc. IPMU Conf. Urbino (Lecture Notes in Computer Science 313), Springer-Verlag, Berlin 1988. [8] M. A. Gil, M. J. Fernandez and I. Martinez: The choice of the sample size in estimating the mutual information. Appl. Math. Comp. (1988) (accepted for publication). [9] M. A. Gil, R. Perez and P. Gil: The mutual information. Estimation in the sampling without replacement. Kybernetika 23 (1987), 5, 406—419. [10] M. A. Gil, R. Perez and I. Martinez: The mutual information. Estimation in the sampling with replacement. R.A.I.R.O. - Rech. Oper. 20 (1986), 3, 257-268. [11] Z. Lomnicki and S. Zaremba: The asymptotic distributions of estimators of the amount of transmitted information. Inform, and Control 2 (1959), 260—284. [12] T. K. Nayak: On diversity measures based on entropy functions. Commun. Statist. - Theor. Meth. 74(1985), 1, 203-215. [13] A. F. Shorrocks: The class of additively decomposable inequality measures. Econometrica 48 (\9%0), 613-625. [14] D. Zagier: On the decomposability of the Gini coefficient and other indices of inequality. Discussion paper No. 108. Projektgruppe Theoretische Modelle. Universitat Bonn, 1983. [15] J. Zvarova: On asymptotic behaviour of a sample estimator of Renyi's information of order a. In: Trans. Sixth Prague Conf. on Inform. Theory, Statist. Dec. Functions Random Processes. Academia, Prague 1973, pp. 919 — 924. Dr. Maria Angeles Gil, Dr. Covadonga Caso, Departamento de Matemdticas, Universidad de Oviedo, 33071 Oviedo. Spain. 319 

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Estimating income inequality in the stratified sampling from complete data. II. The asymptotic behaviour and the choice of sample size

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Estimating income inequality in the stratified sampling from complete data. II. The asymptotic behaviour and the choice of sample size

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