Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality
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Russell Davidson, JeanYves Duclos
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Title:
Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality
Author:
Russell Davidson, JeanYves Duclos
Year:
2000
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Econometrica
Volume:
68
Issue:
6
Start Page:
1435
End Page:
1464
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English
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Updated: November 18th, 2012
Abstract:
Ecoi~or?retr.icn.Vol. 68. No. 6 (November. 2000). 1435 1464
STATISTICAL INFERENCE FOR STOCHASTIC
DOMINANCE AND FOR THE MEASUREMENT
OF POVERTY AND INEQUALITY
BY RUSSELLDAVIDSONAND JEANYVESDUCLOS'
We derive the asymptotic sampling distribution of various estimators frequently used to order distributions in terms of poverty. ~velfare. and inequality. This includes e5timators of most of the poverty indices currently in use. as ~vell as estimators of the curves used to infer stochastic dominance of any order. These curves can he used to deterlnine whether poverty, inequality, or social ~velfare is greater in one distribution than in another for general classes of indices and for ranges of possible poverty lines. We also tierive the sampling distribution of the lnasilnal poverty lines up to which Lve may confidelltly asert that poverty is greater in one distribution than in another. The sampling distribution of convenient dual estimators for the measurement of poverty is also establishetl. The statistical results are established for deterministic or stochastic poverty lines ai well as for paired or independent samples of incomes. Our results are briefly illustrated using data for four countries drawn froin the Luxembourg Income Study data bases.
KEYWORDS:
Stochastic dominance. povertp, inequality. distributionfrec statistical infer ence. orderrestricted inference.
SINCETHE INFLUENTIAL WORK of Atkinson (19701, considerable effort has been devoted to making comparisons of welfare distributions more ethically robust, by making judgements only when all members of a wide class of inequality indices or social welfare functions lead to the same conclusion, rather than concentrat ing on some particular index. More recently, pleas have been made for similar robustness in poverty measurement, following up on the criticism in Sen (1976) of the headcount ratio and the poverty gap as not taking into account the intensity and the depth of poverty respectively. Such pleas are found, for instance, in Atkinson (1987), Foster and Shorrocks (1988a,b), and Howes (1993). Robustness is also needed to guard against the uncertainty and the frequent lack of agreement regarding the choice of a precise poverty line.
1The paper is part of the research program of the TMR network "Living Stantlards. Inequality and Taxation" [Contract No. ERBFMRXCT 9802481 of the European Communities, \vhose financial support is gralef~~llp acknowledged. This research was also supported by grants from the Social Sciences and Humanities Research Council of Canatla, from the Fonds FCAR of the Province of QuChec, and from the MIMAP program of the International Dc\elopment Research Centre. We are grateful to Timothy Smeeding. Koen Vleminckx. and StCphan Wagner for assistance in accessing and processing the data, to Paul Maktlissi for helpful comments. and to Nicolas Beaulieu and Nicole Charest for research and secretarial assistance. We are grateful to numerous seminar participants for comsnents on earlier versions of this paper, and to two anonymous referees and a coeditor for many valuable suggestions that have much improved the paper. Duclos also acknowledges the hospitality of the School of Ecollornics at the University of New South Wales. \\#here he Lvas a Lisiting fellow during the revision of the paper.
In this paper, we study estimation and inference in the context of inequality, welfare, and poverty orderings. Our main objective is to show how to estimate orderings that are robust over classes of indices and ranges of poverty lines, and how to perform statistical inference on them.'
In the next section, we review the definitions of the various indices in which we are interested for the distributions of entire populations, and we note some of the relationships among them."n Section 3, we study estimators of these indices, based on samples drawn from the populations, and we derive their asynlptotic distributions. In particular, we discuss the statistical consequences of
sing estimated poverty lines. We also provide estimates of the thresholds up to which one population stochastically dominates another at a given order, and of cumulati\re poverty gap (CPG) curves. Our results apply equally to the case of observations drawn from independent distributions and to the case in which dependent observatiolls are drawn fro~n a joint distribution, as for instance when, with panel data, there are several observations of the same individual. We obtain as a corollary the distributions of the two most popular classes of poverty indices, both for deterministic and for sampledependent poverty lines. The first is the class of additive poverty indices, which include the Foster, Greer, and Thorbecke (1984) indices, which themselves include the headcount and average poverty gap measures, the Clark, Hamming, and Ulph (1981), Chakravarty (1983), and Watts (1968) indices. The second is the class of linear poverty indices, which can be expressed as weighted areas underneath CPG curves. Members of that linear class include the poverty indices of Sen (1 976): Takayama (1979); Thon (1979), Kakwani (1980) Hagenaars (1987): Shorrocks (1995): and Chakravarty (1997).
Statistical inference for inequality or poverty indices could be performed without recourse to the asymptotic theory of this paper by use of the bootstrap, with resampling of the observed data serving to provide estimates of the needed variances and covariances. However, it is well known that the bootstrap yields better results when applied to asymptotic pivots, and it is therefore a better idea to use our results in order to construct such asymptotic pivots before using the bootstrapsee Horowitz (1997) for an account of the relevant issues.
Finally, in Section 4, we provide a brief illustration of our techniques using crosscountry data from the Luxembourg Income Study data bases. Most of the proofs are relegated to the Appendix.
'work on these line5 can be found in. for illstance. Beach and Davidson, (19831, Beach and Kichmond (19851, Bishop, Chakraborti. and Thistle (19891, Howes (19931, Anderion (19961, David son and Duclos (1997).
'~oster (19841, Chalir;ivarty (1990), Foster and Sen (19971, and Zheng (l997a1, among others, can also he consulted for a revlew of different aspects of the social welfare, poverty, and inequality literatures.
STOCHASTIC DOMINAUCE 1437
2. STOCHASTIC DOMINANCE AND POVERTY INDICES
Consider two distributions of incomes, characterized by the cumulative distri bution functions (CDFs) FAand FB,with support contained in the nonnegative real line. We use term "income" throughout the paper to signify a measure of individual welfare, which need not be money income. Let Di(x) = F,(s) and
for any integer s 2 2, and let Di(x) be defined analogously. It is easy to check inductively that we can express D".x) for any order .r as
Distribution B is said to dominate distribution A stochastically at order s if D;(x) 2 Dij(x) for all x E R. For strict4 dominance, the inequality must hold strictly over some interval of positive measure. Suppose that a poverty line is established at an income level z > 0. Then we will say that B (stochastically) dominates A at order s up to the poverty line if D$x) 2Di(x) for all xIZ.
Firstorder stochastic dominance of A by B up to a poverty line z implies that F,(x) 2FB(x) for all x Iz. This is equivalent to the statement that the proportion of individuals below the poverty line (the headcount ratio) is always (weakly) greater in A than in B, for any poverty line not exceeding z.
Secondorder dominance of A by B up to a poverty line z implies thatD:(s)
2D;(.X), that is, that
for all x 5 z. When the poverty line is z, the por,ert)!gap for an individual with income y is defined as
The notation x+ will be used throughout the paper to signify max(x, 0). In addition, censored irzconze y is defined for a given poverty line z as min(y, z). We can see from (3) that stochastic dominance at order 2 up to z implies that, for all poverty lines x Iz, the average poverty gap in A, D~(x),is greater than that in B, Di(x). The approach is easily generalized to ally desired order s.
Ravallion (1994) and others have called the graph of D1(x) a poverty incidence curve, that of D~(.x) a poverty deficit curve (see also Atkinson (198711, and that of Di(x) a poverty severity curve. D'(x) is shown in Figure 1 for two distributions A and B. Distribution B dominates A for all common poverty
"ince the main focus of this paper is siatistical, we Lvill not distinguish strict and nonstrict dominance. since near the margin no statistical test call do so.
R. DAVIDSON AND J.Y. DUCLOS
FIGCRE1.Poverty incidence curves for two distributions A and B.
lines below z. The area underneath D1(x) for x between 0 and z equals the average poverty gap D~(z), which is clearly greater for A than for B.
Following Atkinson (19871, we consider the class of poverty indices, defined over poverty gaps, that take the form
These can be regarded as absolute indices since equal additions to both z and y do not affect them. In Atkinson (1987), Foster and Shorrocks (1988a), and McFadden (1989), it is shown, in the context of risk aversion, that, for all indices
(5) for which T is differentiable and increasing with ~(0) = 0, lTA(x) &(x) for all x z if and only if B stocl~astically dominates A up to z at first order. This class of indices, along with the headcount ratio, for which it is easy to see that the result holds as well, will be denoted P'. Similarly, the class P2 is defined by convex increasing functions T with ~(0) = 0. The use of indices in p2 is analogous to using social evaluation functions that obey the Dalton principle of transfers (see the discussion of this in Atkinson (1987)). It is easy to show that all indices in P2are greater for A than for B for all x Izif and only if B stochastically dominates A up to z at second order. In general, for any desired order s, we can define the class Pqo contain those indices (5) for which T(')(x)2 0 for x> 0, T(~')(O) 2 0, and T(')(O) = 0 for i = 0,. . . ,s 2. Then it is easy to show that lTA(x)2 &(x) for all x5z for all 17~Pf and only if B dominates A up to z at order s. The classes Pbf poverty indices can be interpreted using the generalized transfer principles of Kolm (19761, Fishburn and Willig (19841, and Shorrocks (1987).' Note that, for class P2, we need not
"01. s = 1, 2, Foster and Shorrocks (1988b) show how some of these dominance relationships can he extended to poverty indices (or censored social welfare functions) that map be nonadditive.
STOCHAS TIC DOMIULWCF 1439
require that .rr '(0) = 0,but, for s > 2, all the derivatives of .rr LIPto order s 2 must vanish at 0."
A useful concept for the analysis of poverty is the mzximum coinmon poverty line z, up to which B stochastically dominates A at order s. All indices in P' will then indicate greater poverty in A than in B for any poverty line no greater than Z,. If B stochastically dominates A (at first order) for low thresholds z, then either B dominates A everyvhere (in which case we have firstorder welfare dominance in the sense of Foster and Shorrocks (1988b)), or else there is a reversal at the value z, defined by
z, is illustrated in Figure 1.If z, is below the maximum possihle income, we can repeat the exercise at order 2. Either B dominates A at second order every where,' or there exists z, defined by
This procedure call be continued either until stochastic dominance at some order s is achiehed evewhere. or until z, has become greater than what is seen as a reasonable maximum possible value for the poverty line (or welfare censoring threshold) z. It is shown in l,einma 1 in the Appendix that stochastic dominance of A and B up to any finite z ivzll be achieved Lor some suitably large kalue of s. This result confirms the illterpretation of stochastic dominance for general r given by Fishburn and Willig (1984) in terms of principles that give increasing weights to transfers occurring at the bottom of the distribution. The limit as r + has a Rawlsian flavor, since. in that limit. only the very bottom of the two distributions determines which dominates the other for large A.
In comparing poverty across time, societies, or eco~lomic environments, it can be desirable to use different poverty lines for different income distributions." This is particularly common in studies ol poverty in developed economies where a proportion of median or average incomes is often used as a "poverty line" to make crosscountry comparisons.~e may continue to use the classev PA as above, but now. in order to compare A and B, we use two different poverty lines z, and z,. Then it is easily shown that there is at least as much poverty in A as in B, according to all indices in P', if and only if D;(z, x) DA(z, r)
or a rs, the FGT indices J"(z) defined below obey this continuity condition and tlir~s belong to the classcs P'. Besides. if an additive index of the type /l(z) = i;G(j,, z)ilF(y) defiiicd belo\?. helor~gsto P'. then, for y > I, all additive indiceh of tile form (8(y. z)IY will belong to I" ' '.
'This is ecluivalent to Generalized Lorenz dominance of the distribution of incomes in R over that in A. and to secondorder w~elfare dominance.
"ee. for instance, Grcer and Thorhccke (1986) aliil Ravallion and Bidani (1994). where povelty lines are cstirnated for diffeient socioeconomic groups, and Sen (1981; p.21) on the issue of comparing poverty of t\vo societies with either common 01different "standards of minimum necessities."
%n this, see, for instance. Srneeding, Rainwrater, and O'Higgins (1990). Van dell Bosch et 211. (1993), Gustafsson and Nivorozhkina (1996), or Atkinson (1995).
2 0 for all x 2 0.'' This just involves checking wllether B dominates A for all pairs of poverty lines of the form (z, x, z, x) with x 2 0. As x varies, the absolute difference between the two poverty lines remains constant. Of course, this relation no longer constitutes stochastic dominance at order .r.
The popular FGT (see Foster Greer, and Thorbecke (1984)) class of additive poverty indices is defined by"
These indices are clearly related to the criteria for stochastic dominance, as was noted by Foster and Shorrocks (1988a, b). I11 fact, if a is an integer, it follows from (2) that Aa(x) = (a l)!Da(x).
For any one member of the FGT class of indices, there may be a range of common poverty lines for which poverty in A is greater than in B. For any such line z, the index AQl~ows more poverty in A than in B if Di(x) Di(x) 2 0 for x =z, but not necessarily for all x <z. Hence, it could be that, for a given range of z, we find dominance of A by B according to A' and A3, but also find dominance of B by A according to A', a reversal which would not be possible with stochastic dominance relations. We could then define the thresholds z; and z: such that B dominates A according to AS only for z E [z;, z:]. More generally, we may only wish to check whether DJ(x) 2 Di(x) for x in our range of interest. For s = 1 or r = 2, this leads to the concept of restricted stochastic dominance defined in Atkinson (1987) for the headcount ratio and the mean poverty gap respectively (see his Conditions 1 and 2). It is clear that such restricted dominance conditions can be applied and generalized to any order s of the FGT index.
Other poverty indices can also be expressed in the additive form of (2), that is, as
for suitable choices of 6(y, z). This is the case for the Clark, Hamming, and Ulph (1981) second family of indices, for the Chakravarty (1983) index, for which 6(y,z)= 1 (y /z)' for 0 <e < 1, and for the Watts (1968) index, where 6(y, z) = log(z/yG). Bourguignon and Fields (1997) also propose an additive index that allows for discontinuities at the poverty line, with 6(y,z) = g(z, y)"ll + a?I(y IZ), where I(y sz) is an indicator function equal to 1 when J,5z, and 0 otherwise.
10 Wellknown arguments of the type found in Foster and Shorrocks (1988b) can be used to sho~v that this extends to nonadditive indices for s = 1, 2. In the teri~lil~ology of Jenkins and Lamhert (19971, dominance for s = 2 implies an ordering for all generalized (additive or nonadditive) poverty gap indices.
he original FGT indices are normalized by 2"'. We return to the interpretation of this normalization belo\<,.
STOCllASTIC DOMINANCE 1441
Stochastic dominance at first and second order can also be expressed in terms of quantiles. This is called the papproach to dominance. The indices in P' indicate at least as much poverty in A as in B if and only if, for all 0sp sI,
where Q,(p) and QB(p) are the pquantiles of the distributions A and B respectively. If z, =z,, condition (10) simplifies to checking if the quantiles of B's censored distribution are never smaller than those of A. As can be seen from Figure I, the condition (10) need only be checked for values of p less than the greatest value, p in the figure, for which z, Q,(p2) 2 0. It is also clear from the figure that, where the curves for A and B cross, at p =p,, the common value of Q,(p,) and QB(p,) is the z, defined in (6).
There also exists a papproach to secondorder dominance. To see this, define the cunaulntii'e porlerQ gap (CPG) curve (also called TIP curve by Jenkins and Lambert (1997), and poverty gap profile by Shorrocks (1998); see also Spencer and Fisher (1992)) by
It is clear that G(p; z)/p is the average poverty gap of the loop% poorest individuals. Typical CPG curves are shown in the upper panel of Figure 2. For values of p greater than F(z), the CPG curve saturates and becomes horizontal.
FIGLIRE and generalized Lorenz culves.
2.CPG
Since F(z) =D'(z), the abscissa at which the curve becomes horizontal is the headcount ratio. The ordinate for values of p such that F(z) sp I1 is readily seen to be Dyz), the average poverty gap.
To make the link with secondorder stochastic dominance, we quote a result of Jenkins and Lambert (1997) and Shorrocks (1998). They show that, for two distributions A and B and a common poverty line z, it is necessary and sufficient for the stochastic dominance of A by B at second order up to z that G,(p; z) 2 G,(p; z) for all p E[O, 11. The more general case with different poverty lines can be easily derived from Theorem 2 in Shorrocks (1983). Using Shorrocks' result, we find that poverty is greater in A than in B according to all indices in the class P' if and only if the CPG curve for A (using 2,) everywhere dominates the CPG curve for B (using z,).I2
CPG curves can be related to generalized Lorenz curves GL(p), defined by (see Shorrocks (1983)):
It is clear from this definition and (11) that G(p; z) =zp GL(p) for p ID1(z). Thus, as shown in Figure 2, for p 5D1(z),G(p; z) is the vertical distance between the straight line zp with slope z and GL(p). When G(p; z) saturates at 11=D1(z),its derivative with respect to p vanishes, and so we see that, at p =D1(z), GL1(p) =z. Thus, for p 2D1(z), G(p; z) is the vertical distance be tween the line zp and the tangent to GL(p) at p =D1(z). When we compare two distributions, A and B, this link between GL(p) and G(p; z) shows that the critical secondorder poverty line z, defined in (7) is given by the slope of the line that is sin~ultaneously tangent to both of the generalized Lorenz curves, at points n and b in Figure 2. This follows because, as can be seen in the figure, the vertical distances between the line z,p and the two generalized Lorenz curves at the points at which their slopes equal z2 are equal. Since these distances equal D2(z2), the result follows.
A popular class of poverty measures that are linear in incomes can be easily obtained from G(p; z). To see this, consider the class of indices O(z) that measure a weighted area beneath the CPG curve
for various choices of the fl~nctions r(.), q(.), and 8(.). O(z) is linear in incomes since G(p; z) is itself a linear (cumulative) function of incomes." Sen's (1976) index is given by setting O(p) = 2, ~(z)=D1(z), and q(z) =~'(z).8(p) = 2, r(z) =D2(z), and q(z) = 1 yield the Takayama (1979) index. 0(p) = 2, r(z) = 1, and q(z) = 1 give Thon's (1979)~ Shorrocks' (1995), and Chakravarty's (1997)
"see also Jenkins and Lamhert (1998) for an cxtension of this to generalized poverty gap indices. his is analogous to the definition of linear i~iequality indices in I\/Iehsan (1976).
STOCHASTIC DOMINAICE
poverty indices. Kakwani's (1980) index is obtained with
with k > 0, ~(z)= 1,and q(z) =D'(z). More generally, we can define any linear poverty index O(z) by defining O(p) as some particular nonnegative function of
p. As for the FGT indices, we might also wish to infer the restricted raliges [z, zf] over which the additive or linear indices A(z) and O(z) show more poverty in A than in B.
In the literature on the measurement of poverty, the poverty gap (4) is sometimes normalized by the poverty line.'" For this, absolute poverty gaps g(z, y) are replaced by relative poverty gapsi5 gl(z, y) =g(z, y)/z, in the defini tions of the poverty indices found in (5). We define classes P,' of relative poverty indices analogously to the classes P', with gl(z,y) in place of g(z,y). The stochastic dominance conditions are obviously unchanged if poverty lines are common. It can be seen that there will be more poverty in A than in B for all indices in P,' if and only if
for all x E [O,11. The theoretically equivalent papproach for class P,I is given by checking whether
For secondorder dominance, the papproach can be derived by redefining the CPG curve in terms of relative poverty gaps as follows:
and checking whether Gi(p) Gk(p) 2 0for all 0~p I 1.'"
Finally, for indices of relative inequality, observe that D'(x) can be used to check both equality and welfare dominance when means are the same. When A and B have different means, pA and p, say, we can study equality dominance by comparing the meannormalized distributions FA(xpA) and F,(X~~)." This implies checking whether
"~t is not clear that this is desirable \\,hen poverty lines tliffer across groups or societies; see Atkinson (1991, 13. 7 and footnote 3).
15 For a discussioll of absolute versus relative poverty gaps and indices, see Blackorby and Donaldson (1978, 1980).
16See also Jenkins and Larnbert (1998).
17This is also discussed in Foster and Shorrocks (1988c), Foster and Sen (1997). and Formby, Smith and Zheng (1998).
for all x 2 0. For .r = 2, this is equivalent to checking Lorenz dominance. Similarly to z,, we can define critical common proportions x, of the respective means up to which condition (16)is met at a given order s. When Lorenz curves cross, x, will give the slope of the line that is simultaneously tangent to both of the Lorenz curves.'"
3. ESTIMATION AND INFERENCE
Suppose that we have a random sample of N independent observations y,, 1 = 1,.. . ,N, from a population. Then it follows from (2)that a natural estimator of D'(x) (for a nonstochastic x) is
where $ denotes the empirical distribution function of the sample and I(.) is an indicator function equal to 1 when its argument is true and 0otherwise. For .r = 1, (17) simply estimates the population CDF by the empirical distribution function. For arbitrary s, it has the convenient property of being a sum of IID variables.
When comparing two distributions in terms of stochastic dominance, two kinds of situations typically arise. The first is when we consider two independent populations, with random samples from each. In that case,
The other typical case arises when we have N independent drawings of paired incomes, y," and y,", from the same population. For instance, y," could be beforetax income, and y," aftertax income for the same individual i, i = 1,.. .,N. The following theorem allows us to perform statistical inference in both of these cases.
THEOREM1: Let the jozrzt populatzon monzerlts of order 2s 2 ofJ>A nrzd y " be firute. Tlzerz ~'/~(fi;(x)D;(x)) u asymptot~cnlly normal with mean zero, for K =A, B, and wzth nsymptotzc cocarzarzce structure gcen by (K,L =A,B)
'"he zrrgument for this is analogous to that used above in discussing Figure 2.
STOCHASTIC DOMINANCE 1445
PROOF: For each distribution, the existence of the population moment of order s 1 lets us apply the law of large numbers to (17), thus showing that ES(x) is a consistent estimator of D'(x). Given also the existence of the population moment of order 2s 2, the cental limit tlleorem sllows that the estimator is rootN consistent and asymptotically normal with asymptotic covari ance matrix given by (19). This formula clearly applies not only for yA and yB separately, but also for the covariance of D; and 6;.
If A and l?are independent populations, the sample sizes NAand N, may be different. Then (19) applies to each with N replaced by the appropriate sample size. The covariance across the two populations is of course zero. Q.E.D.
REMARIG:This theorem was proved for the case of independent samples as early as 1989 in an unpublished thesis (Chow (1989)). The sampling distribution of the related estimator dY(x (see (8)) with a fixed x and independent samples is also found in Kakwani (1993), Bishop, Chow, and Zheng (1999, and Kongve (1997). For a different approach to inference on stochastic dominance, see Anderson (1996).
The asymptotic covariance (19) can readily be consistently estimated in a distributionfree manner by using sample equivalents. Thus D'(x) is estimated by 6'(s), and the expectation in (19) by
If B does dominate A weakly at order s up to some possibly infinite threshold z, then, for all x ~z, Dj(x) DA(x) 20. There are various hypotheses that could serve either as the null or the alternative in a testing procedure. The most restrictive of these, which we denote H,,: is that D,,;(x) Dh(x) = 0for all x Iz. Next comes HI, according to which D$x) D$(x) 2 0for x <z, and, finally H,, which imposes no restrictions at all on D;(x) Dl;(x). We observe that these hypotheses are nested: H, cH, cH,.
McFadden (1989) proposes a test based on sup,,,.(fi;(x) 6~(x>>
for the null of H,, against HI. For s = 1, this turns out to be a variant of the KolniogorovSniirnov test, with known properties, for the identity of two distri butions. Of higher values of s, McFadden considers only s = 2. Although it is easy to compute the statistic, its asymptotic properties under the null are not analytically tractable. However, a simulationbased method can provide critical values and P values.
In Kaur, Prakasa Rao, and Sing11 (1994) (henceforth KF'S), a test is proposed based on the minimum or infimum of the t statistic for the hypotl~esis that Dj(x) DA(x) = 0, computed for each value of x Iz. The minimum value is used as the test statistic for the null of ~zoizdominance,H,\H,, against the alternative of dominance, HI. Since the test can be interpreted as an intersec tionunion test, it is shown that the probability of rejection of the null when it is true is asymptotically bounded by the nominal level of a test based on the standard normal distribution.
Both the McFadden and the KPS statistics are calculated as the extreme value of the possibly very large set of values computed for x = yA and x = 7' for all i = 1,.. . ,N,, j = 1,.. . ,N,. Other procedures make use of a predetermined grid of a much smaller number of points, x, say, for j = 1,.. . ,m, at which d, Dj(xj) Di;(x,), or some quantity related to it, like the t statistic considered above, is evaluated. The arbitrariness of the choice of the number of points rn, and the precise values of the x,, is an undesirable aspect of all procedures of this sort. At the very least, it is necessary that the x, should constitute a grid covering the whole interval of interest.
Howes (1993) proposed an intersectionunion test for the null of nondomi nance, very much like the KPS test, except that the t statistics are calculated only for the predetermined grid of points. Its properties are similar to those of KPS.
The technique developed by Beach and Richmond (1985) allows us to test H, (dominance) against Hz (no restriction). The alternative is not the hypothesis that A dominates B. That hypothesis can of course play the role of H, and be tested similarly against Hz. This technique was originally designed by Richmond (1982) to provide simultaneous confidence intervals for a set of variables asymptotically distributed as multivariate normal with known or consistently estimated asymptotic covariance matrix. It was extended by Bishop, Formby, and Thistle (1992), who suggested a unionintersection test of the hypothesis that one set of Lorenz curve decile ordinates dominates another. For a test of stochastic dominance, one can use the t statistics for the hypotheses that the individual d,, j = 1,.. . ,in, are zero. The hypothesis HI,which implies that they are all nonnegative, is rejected against the unconstrained alternative, H,, if any of the t statistics is significant with the wrong sign (that is, in the direction of dominance of B by A), where significance is determined asymptotically by the critical values of the Studentized Modulus (SMM) distribution with nz and an infinite number of degrees of freedom.
None of the tests discussed so far makes use of the asymptotic covariance structure provided by Theorem 1. As a result, they can be expected to be conservative, that is, lacking in power, relative to tests that do exploit that structure. Such tests, to date at least, all rely on a predetermined grid, on which stochastic dominance implies the set of m inequalities d, 2 0. Methods, for testing hypotheses relating to such inequalities are developed in Robertson, Wright, and Dykstra (1988), in the context of orderrestricted inference. For our purposes, the relevant methods can be found in Kodde and Palm (1986) and Wolak (1989). Wolak provides a variety of asymptotically equivalent tests of H,, against H,, and of H, against Hz, and provides the joint distribution under H, of the statistics corresponding to the two tests. He also shows that this allows us to bound the size of the test asymptotically when H, is the null, because, for any nominal level, the rejection probability under HI is maximized when H, is
STOCFWSTIC DOMIKA~CF
true.'" The test of H,, against H, is of less interest, and in any case it is a perfectly standard test of a set of equality restrictions.
A feature of the order restrictcii approach is that, if m is large, the mixture of chisquared ciistributioils followed by the test statistics under H, can, as Wolak remarks, be difficult to compute. However, he also proposes a Monte Carlo approach that works independently of the magnitude of nz, and can be imple mented with sufficient accuracy easily enough on present day computers. The necessary ingredient for any of these procedures is the asymptotic covariance structure of the dj.
One should note that nonrejcction of the null of dominance by either the Wolak or BishopFormbyThistle approacl~ can occur along with nonrejection of the 11~111 of nondoniinance by the KPS or Howes approach. This occurs naturally if the fir(.)functions for the two populations are close enough over part of the relevant range. Such issues, and many others, are investigated in a valuable recent paper of Dardanoni and Forcina (1999), who also consider hypotheses according to which more than two distributions are ranked by a stochastic dominance criterion. They emphasize the intrinsically conservative nature of the KPS and Howes tests, and find that they are wholly lacking in power for comparisons with more than two distributions, although, as will be seen in our empirical illustration in Section 4, they remain useful with just two distributions when it is undesirable to infer donlinance unless there is very strong evidence for it.
Dardanoni and Forcina investigate, in a set of Monte Carlo experiments, the power gain achieved by the tests of Wolak and of Kodde and Palm relative to tests that do not take account of the covariance structure of the d,. They find that these are greatest when the dj are negatively correlated. Although this can occur naturally in comparisolls of more than two populations, the usual case with only two is that they are positively correlated. Even so, they find that methods like Wolak's are often worth the extra computational burden they imposethis conclusion is borne out by the results in our empirical illustration. They also advocate the use of tests that cornbine the information in a test of H, against HI with that in a test of H, against H2, bearing in mind that the statistics are not independent. Their very interesting analysis is however beyond the scope of this paper.
In Theorem 1, it was assumed that the argument x of the functions D'(x)was nonstochastic. In applications, one often wishes to deal with DS(zx), where z is the poverty line. In the next Theorem, we deal with the case in which z is estimated on the basis of sample information.
i'1 Note, however, that in the more general context of arbitrary nonlinear inccluality restrictioils on the parameters ol a nonlinear model, it is not ilecessarily true Illat the rejection proh;~bility is maxilnized at the point at which all the restrictions hold with equality: see Wolak (1991) for full discussion of this point.
THEOREM2: Let tlze joint population moments of older 2s 2 of y " and yB be finite. If s = 1, suppose in addition that F, and F, are diffelaiztiable and let DO(x)=F1(x). Assume first that N independent drawirzgs of pairs (yA, yR) hare been made ?om the joint distribution of A and B. Also, let the pocerty lines z, and z, be estimated by 2, aizd 2, ~sspecticely, where these estimates are expressible asymptotically as sunzs of iid rlariables drawn fiom the same sample, so that, for some fi~nction ["'A(.),
aizd sii?zilarly for B. Then N '/q(Lji(i,x) Di(z, x)),K =A,B, is asymptoti cally nornzal with mean zero, and with cocariaizce structure giceiz by (K, L =A,B)
If yA and yB are independently distributed, aizd if N, and N, iid dmwings are
respecticely made of these cariables, then, for K =L, N, replaces N in (22), while for K # L, the cor~ariance is zero.
PROOF: See Appendix.
REMARKS:The sampling distribution of the headcount when the poverty line is set to a proportion of a quantile is derived in Preston (1999, using results on the joint sampling distribution of quantiles. More generally, the sampling distribution of additive indices when the poverty line is expressed as a sum of iid variables is independently derived in Zheng (1997b) using the theory of U statistics.
Estimates of the poverty lines may be independent of the sample used to estimate the DYz x), as for example if they are estimated using different data. In that case the righthand side of (22) becomes
For indices based on relative poverty gaps, one needs the distribution of ds(&) for positive x; see (13) and (16). The result of Theorem 2 can be used by first eliminating the additive x in that result, and then replacing i by &.
STOCHASTIC DOMINANCE
The covariance (22)can, as usual, be consistently estimated in a distribution free manner, by the expression
The most popular choices of population dependent poverty lines are fractions of the population mean or median, or quantiles of the population distribution. Clearly any function of a sample moment can be expressed asymptotically as an average of iid variables, and the same is true of functions of quantiles, at least for distributions for which the density exists, according to the Bahadur (1966) representation of quantiles. For ease of reference, this result is cited as Lemma 2 in the Appendix. The result implies that &(p)is rootN consistent, and that it can be expressed asymptotically as an average of iid variables. When the poverty line is a proportion k of the median, for instance, we have that:
where Q(0.5) denotes the median. When z is k times average income, we have
This iid structure makes it easy to compute asymptotic covariance structures for sets of quantiles of jointly distributed variables.
For the purposes of testing for stochastic dominance, all the remarks follow ing Theorem 1 regarding possible procedures continue to apply here. Only the asymptotic covariance structure is different, on account of the estimated poverty lines.
We turn now to the estimation of the threshold z, definedAin(6).Assume that &(x) is greater than FB(x) for some bottom range of x.If F,(x)is smaller than FB(x) for large values of x, a natural estimator 2, for z, can be defined implicitly by
FAGl) =&(2J.
If FA(x) >FB(x) for all x sz, for some prespecified poverty line z, then we arbitrarily set 2,=z. If 2, is less than the poverty line z, we may define 2, by if this equation has a solution less than z, and by z otherwise. And so on for 2, for s >2: either we can solve the equation
or else we set 2, =z. Note that the second possibility is a mere mathematical convenience used so that 2, is always well definedwe may set z as large as we wish. The following theorem gives the asymptotic distribution of 2, under the assumption that z, <z exists in the population.
THEOREM3: Let the joint population nzoments of order 2s 2 of y A and y 'be finite. If's = 1, suppose ,fitrther fhat F, arzd FB are diferentiable, and let ~"(x)
=
F1(x). Suppose that there exists z, <z such that
and that Dj(x) >D$(x) ,for all x <z,. Assume that z, is a simple zero, so that the dericatir:e DjP'(z,) Di'(2,) is nonzero. Irz the case in which we coizsider N independent dmwings of pairs (r",y ") ,from one population in which y A and y " are joirztly distributed, ~'/'(f, z,) is asymptotically normally distributed with mean zero and asymptotic rlariance girlen by:
If yA and yB are independently distributed, and if N, aizd NB iid dmwiizgs are respecticely made of'tlilese ~~ariables, remaiizs constant as
where the ratio r N,/NB NA and N, teizd to infinity, then NAI2(i, z,) is asymptotically nonnal witlz mean zero and asymptotic oariaizce giceiz by
rzy +arz yB)il)
lim var( N;/Y~z)) =
,V,T ((sl)!D1(z)D~(~'(z)))~
'
PROOF:See Appendix.
REMARK:In this theorem, we assume that z, exists in the population, and is a simple zero of Dj(x) DA(x). Since the D;,,K =A, B, are consistent estimators of the D;<,this implies that, in large enough samples, 2, exists and is unique. In general, jn finite samples, it can happen that, although z, exists, the estimated curves Dj(x) and 6~(x)do not intersect. In such cases, our definition gives 2, =z, and no real harm is done. It may also happen that, even if no z, exists, the estimated curves cross. In that case, the regularity condition of the theorem is not satisfied, and nothing simple can be said of the spurious estimate f,,
STOCHASTIC DOIvIINANCE
1451
except of course that, in large enough samples, 2, =z with high probability. Clearly, no agmnptotic approach can handle these awkward cases, because asymptotically the true situation in the population is reflected in the sample. The situation is in fact analogous to what happens with parametric models for which the parameters may be identified asymptotically but not by a given finite data set, or vice versa.
The results of Theorems 1, 2, and 3 can naturally be extended to the additive poverty indices A(z) of (9) by using A(x) in place of D'(x),a(}), x) for ((s I)!)'(x y)\', and A1(x)for D" '(x).
In order to perform statistical inference for papproaches, we now consider the estimation of the ordinates of the cumulative poverty gap curve G(p; z) defined in (11).The natural estimator, for a possibly estimated poverty line 2, is
147
&p; I) =N 2 y,)I(y, 5 ~(p))
1=1
where &(p) is the empirical pquantile. The asymptotic distribution of this estimator is given in the following theorem.
THEOREM4: Let the joint population second moments of y" and yR be finite, and let FA and F, be differentiable. Let 2, and 2, be expressible asymptotically as sums of iid uariables, as in Theorem 2. If N independent dmwings of pairs (yz', y ") are made fionz the joint distribution of A and B, then N'/'(6K(p; 2) G,(p; z)), for K =A,B, is asymptotically ~zomzal with mean zero, and agmptotic co~>ariance structure given by
If y A and y 'are iizdepeizdently distributed, and if N, and N, iicl drawings are respectively made of these ~lariables, then, for K =L, N, replaces N in (25). For K f L, the cor'ariaizce is zero.
PROOF: See Appendix.
REMARKS:If iAand I, are independent of the drawings (yA, yB), then the righthand side of (25) can be modified as in (23).The result of Theorem 4 for the special case of a deterministic poverty line and for independent samples can also be found in Xu and Osberg (1998).
The arguments used in Theorems 14 can be used to obtain the asymptotic distribution of all those indices considered in the previous section not already covered by the earlier theorems. First, when z is deterministically set to a level exceeding the highest income in the sample, Theorem 4 yields the sampling distribution of the generalized Lorenz curves, and of the ordinaiy Lorenz curves when we also take into account the asymptotic distribution of the sample mean jl. Second, for the firstorder yapproach, based on quantiles (see (lo)), the asymptotic covariance structure is easy to derive because the quantiles can be expressed asymptotically as averages of iid variables, by Bahadur's Lemma, as can the estimated poverty lines, by (21). Third, for the indices based on relative poverty gaps, inference on the expressions in (13), (141, (151, and (16) can be performed by using the asymptotic joint distributions of objects like DY,~),iQ(~),i and ji. Fourth, the asymptotic distribution of estimates 6(i) of the general class of linear indices (12) can be readily obtained using the arguments of the proof of Theorem 4." Fifth, the asymptotic distribution of estimators of critical poverty lines z (for z =z,z+) for the linear indices O(z) can be obtained from Theorem 3 by replacing (s l)p\lar((zs y K);') by lim,, ,,~var(h,(z)) and Dip '(z,) by @;<(z), K =A, B. Sixth, the asymptotic distribution of estimators of critical relatille poverty lines x, in (16) can be derived from Theorem 3. Finally, the arguments found in the proof of Theorem 3 can also be used to provide the asymptotic distribution of the abscissae above which quantile, Lorenz, Generalized Lorenz, or CPG curves cross. For Lorenz curves, for instance, this would give the asymptotic distribution of the maximum proportion of the populatioil for which it can be said that the share of total income is greater in B than in A.
4.ILLUSTRATION
We illustrate our results using data drawn from the Luxembourg Income Study (LIS) data sets" of the USA, Canada, the Netherlands, and Norway, for the year 1991. The raw data were essentially treated in the same manner as in Gottschalk and Smeeding (1997). We take household income to be disposable income (i.e., posttaxandtransfer income) and we apply purchasing power parities drawn from the Penn World ~ables'~
to convert national currencies into 1991 US dollars. As in Gottschalk and Smeeding (1997), we divide household income by an adultequivalence scale defined as hoi,where h is household size, so as to allow comparisons of the welfare of individuals living in households of different sizes. Hence, all incomes are transferred into 1991 adultequivalent US$. All household observations are also weighted by the LIS sample weights "hweight" times the number of persons in the household. Sample sizes are
20 The statistical inference results frjr the special case of the Sen index with a deterministic poverty line can also he found in Bishop. Formhp, and Zheng (1997).
"See http://li,sy.ceps.lu for detailed infornlation on the structure of these data.
"see Summers and Heston (1991) for the methodology underlying the computation of these parities. and http://ww~v.nber.org/pwt56.htmlf or access to the 1991 figures.
STOCHASTIC DOMINAYCE 1453
16,052 for the US; 21,647 for Canada; 8,073 for Nonvay; and 4,378 for the Netherlands.
This illustration does not deal with important statistical issues. First, we assume that observations are drawn through simple random sampling. The LIS data, like most survey data, are actually drawn from a complex sampling structure with stratification, clustering, and nondeterministic inclusion rate^.'^ It would be possible, if messy, to adapt our methods to deal with complex sampling structures, provided of course that the design was known. Second, negative incomes are set to 0. This procedure, however, affects no more than 0.5% of the observations for all countries considered here. Finally, we ignore the measure ment errors due to contaminated data: see Cowell and VictoriaFeser (1996) for a discussion of how to minimize the consequences of these.
Table I shows the estimates ~'(x) and d2(x) for the selected countries and for poverty lines vaiying between US $2,000 and US $35,000 in adultequivalent units, along with their asymptotic standard errors. For the purpose of compar isons, since the samples for the different countries are independent, asymptotic variance estimates for the differences d;(x) d;(x) are obtained by adding the variance estimates for countries A and B. Comparing the US with the other countries, we find that firstorder dominance never holds everywhere in the samples.
It can be seen from the data for s = 1 that, with a conventional significance level of 556, Canada has a significantly lower headcount ratio for all x less than or equal to $25,000 (that is, a poverty line of $50,000 for a family of 4); in other words, Canada has less poverty than the US for all poverty lines equal to or below $25,000, and for all P' poverty indices. The American headcount is significantly lower than that of Nonvay only for those x no greater than $15,000. As for the Netherlands, its headcount is initially significantly greater than that of the US (for x equal to $2,000), it is lower than for the US for x between $4,000 and $8,000, and it is greater again subsequently. These results mean that, by use of Howes' intersectionunion procedure, the null of nondominance of the US by Canada can be rejected at the 5% level for all poverty lines up to $25,000. The corresponding hypotheses for Nonvay and the Netherlands cannot be rejected. The null of dominance of the US by Canada, on the other hand, cannot be rejected by the BishopFormbyThistle (BFT) unionintersection procedure until a poverty line of $35,000. By use of the Wolak procedure, a similar but more precise result is obtained. The Wald test statistic of Kodde and Palm was calculated for the set of incomes given in Table I up to $30,000 and up to $35,000. The weights for the mixture of chisquared distributions were obtained by running 10,000 simulations, and P values were calculated. The P value is
0.71 for the null of dominance of the US by Canada up to $30,000, but only 0.0001 up to $35,000.
"see Cowell (1989) and Howes and Lanjouw (1998) for the consideration of such issues in applied distributional analysis.
TABLE I HE~DCOLNTS POVERTY FOR V~RIOI'SPOVERTYLINES
AND AVERAGE G~PS
v
LISA Canadn Noiu,ay Netherlands
A'oic, Tlie fiift itern in each box is a'(.\):beneath is ilr asymptotic standard erlor Nest is ~'(1).nit11 it5 as)inptotlc 5tanddrd error undeincnth. All amounts ale in 1991 adultequ~valcrit USS. Data nrc for 1991. t~omthe LIS data base.
For s = 2, the major difference from the results for s = 1is that Canada now dominates the US for all values of x in the samples, and significantly so, so that Howes' procedure rejects the null of nondominance at order 2. Since there is dominance in the sample, both the BFT and the Wolak method fail to reject the null of dominance. As for Norway, the initial range of values of x for which the
STOCHASTIC DOMINANCE
US is dominated at second order 1s (as expected) larger for s = 2 than for s = 1. Compared to the US, the Netherlands have a significantly greater average poverty gap for x= $2,000, a statistically indistinguishable average poverty gap for x= $4,000, a lower one for x between $6,000 and $10,000, and a greater average poverty gap for x above $15,000. For both Norway and the Netherlands compared with the US, nondominance at second order is not rejected by the Howes procedure, and dominance is rejected by the BFT procedure. In this case, since the conclusions are clear, it is unnecessaiy to go to the trouble of using the Wolak procedure. The comparison of Canada and Norway, however, is less clear. By use of Wolak's procedure for x = $5,000 and x= $10,000, over which range Norway dominates Canada in the sample, the P value for the null that Canada dominates Norway is 0.078, so that the null cannot be rejected at the 5% level. If the income range is extended to $15,000, the P value grows to
0.14. If it is extended all the way to $35,000, the P value is 0.40. The closeness at the bottom of the range does of course rule out rejection of nondominance by Howes' procedure.
Table I1 shows estimates of the thresholds z, for dominance relations, and [z;, z,'] for restricted dominance relations, between the US and the other three countries for s = 1,2,3,4. Not surprisingly, we find that Canada stochastically dominates the US for s = 1 up to a censoring threshold of $27,840, with a standard error on that threshold of $1,575. For higher values of s, Canada dominates the US everywhere up to an arbitrarily large threshold. For Norway, dominance up to $35,000 is attained for s = 4, but with a threshold income barely significantly greater than $35,000. Regarding the comparison of the Netherlands and the US, we can conclude that there is firstorder dominance of the US for all poverty lines below $2,958 (with a standard error of $193), that there is restricted firstorder poverty dominance by the Netherlands over the US for poverty lines between $2,958 and $8,470, and restricted firstorder domi nance by the US over the Netherlands for poverty lines above $8,470 (with a standard error of $203). Similar results hold for higher values of s.
TABLE I1
ESTIMATES Z, FOR DOMINANCE OVERTHE US
OF THE THRESHOLDS BY THREECOUNTRIES
Netiiel.land\ s Canada Nor\v;~y LT I
[.Ls
,\'oics: 45)mptotii \t:indnid er~oliIn pnrentlir>es. All ,?mounts :ire ~n 19'21 adultequivalent US$. D,lta :ilc tor 1991. from the L1S data hie.
TABLE 111
s= 1 s=2
Mobt Rlediuni Ledst lvlost Medium Lca.1 Povci ty Po! ei ty Poverty Po!e~t> POFeit) I'o~ert!
USA CAN NL USA C.&V
0.012 0.0070 0 10.2 8.2

(0.001) (0.00oo (1.1) (0.8)
USA NL CAN USA CAN
0.020 0.012 0 0108 26.1 17.3
(0.001) (0.002) (0.0007) (2.1) (1.4)
USA NL C4N USA CAN
0.030 0.021 0.016 50.0 10.4
(0.002) (0.002? (0.001) (3.4) (2.1)
USA CAN NL USA CAN
0.050 0.027 0.024 88.9 51.2
(0.002) (0.001) (0.002) (5.3) (3.0)
USA CAN NL 1JSA CAN
0.077 0.041 0.028 152.1 84.2
(0.003) (0.002) (0.003) (7.7) (4.2)
USA CAN NL USA CAN
0.110 0.065 0.035 245 136.1
(0.004) (0.002) (0.003) (11) (5.8)
USA CAN NL USA CAN
0.144 0.088 0.045 172 211.6
(0.004) (0.002) (0.004) (14) (7.7)
11 SA CAN NL USA CAN
0.181 0.116 0.067 534 312.8
(0.005) (0.003) (0.00s) (18) (9.9)
A7orc\. Rankings based on D'lz r) for = I. 2 .i,)mprotic standard rrrois in parentheses. All amount5 arc in 1991
3
adultrquivnleiit USB Dnra dre for 1991, From the L.1S d;itn base.
Table 111 illustrates poverty rankings for the US, Canada, and the Nether lands when the poverty line is set to half median income in each countly. Even without use of the Wolak procedure, it can be seen. for s = 1, that the null hypotheses of do~nlnance of the US by Canada or by the Xetherlands cannot be rejected, and that by Howes' procedure the null of nondominance of the US by the other two countries can be rejected. Hence, there is more poverty in the US than in any of the other two countries for all P' indices, at poverty lines equal to half of med~an income and lor all other pairs of lower poverty lines such that the absolute difference between the two poverty lines remains constant. For s = 1, the rankings of Canada and the Netherlands switch twlce as x approaches
0. For s = 2, hobtever, poverty becomes everywhere s~gnificantly greater (except perhaps for x = $4.000 and r $3.000) in Canada than in the Ketherlands.
Dept. of Ecorzomics, Q~lee~zs Uniuersity, Kingston, Ontario K7L 3N6, Canada, and GREQAM, Centre de la Vie~lle Charit4 13002 Marseille, France;
STOCHASTIC DOMINANCE
[email protected]; http: // qed.econ.queerzsu.ca /pub /faculty / mini / dauidson. html and Dbpt. d'Economie, Pauillorz de Sue, Uniuersite' Lnual, SteFoy, Que'bec GIK 7P4 Canada; jduc@ecn. ulaual.ca; http: //www.ecn, uluval. ca /jyues
Mantrscript receiued March, 1998;jirzal revision receirled Nouenzbei; 1999.
APPENDIX
LEMMA1: If B dominates A for s = 1 up to some w > 0, with strict donzinurzce over at least part of that range, the17 for any finite threshold z, B donzinates A ut order. s up to z for s slrfjciently large.
PROOF:We have FA(x)F,(x) 2 0 for 0 sx < w, with strict inequality over some subintelval of [O, w].Thus
We wish to show that, for arbitrary finite z,we can find s sufficiently large that Dj(x) Di(x) > 0 for x sz, that is,
for x < z. For ease in the sequel, we have multiplied D'(x) by (s l)!/xF ', which does not affect the inequality we wish to demonstrate. Now the lefthand side of (26) can be integrated by parts to yield
We split this integral in two parts: the integral from 0 to w, and then from w to x. We may bound the absolute value of the second part: Since I FA(y)F,(y)l I1 for any y and 1y/x 2 0 for all y sx, we have
For the range from 0 up to w,we have, for s 2 2,
Putting (27) and (28) together, we find that, for x 2 w,
If we choose s to be greater than 1 + (z w)/u, then, for all w sssz,(n(s I) + W)/S 1 > 0. Thus for such s, the last expression in (29) is positive for all w <x Iz. For x < w, the dominance at first order up to w implies dominance at any order s > 1 up to w. The result is therefore proved.
LEMMA2: (Baharlur (1996)): Sc~plrose tlzat a polrulation is characterized by u twice cliflewntinble ilisrribcltion filrzctior~ F. Tlzer~,iJ tl7e pqunrltile is denoted bj' Q( ,I)), and tlze sarn,~rle 11quuntile,fr.orn n srinzple or N inileper7rlerzt rlrnwings y, Jron~ F by Q( ,I)), we 1zai.e
PROOFOF THEORE~I2: For diitributloni A and B. we have
(s 1)!1j3(ix) =/''(?,~j)''ri6(y) and
0
Thus
1; follows from (21)that iz = o(N'/'), and by standard properties of the empirical distribution, F F = O(AT'/?).Thus the first two terms and the fourth are of order N', and the others are of order AT' /'.
The third term can be expressed as
from which we see that it contributes asymptotically only if s = 1. In that case, the term is
since we made the definition D" = F'
STOCHASTIC DOMINANCE
The fifth term is obviously zero for s = 1. Fols > 1, it can be expressed ai
We see that expression (31)serves for the fifth term when s > 1 and for the third when s = 1. Finally, the sixth term is
and so it is the average of N iid variables of mean zero. Multiplying (30)by IV'/'. we see that
The result of the theorem follows from (32)by simple calculation.
PROOFOF THEOREM3: Consider the general problem in which, for some population, a value z is defined implicitly by 11(z)= 0. where the function 11 is defined in terms of the population distribution. For instance, if Q(p) is the pquantile of a distribution with cdf F, we have F(Q(p))=p, and we can set 11(z)= F(z)p.
For z,, the defining relationship, in terms of the population, A and B, is D,;(z,) = D;(z,), with ljj(.v) > D;(x) for all x < z,. Thus we set /7(x)= I);(x) Dh(s).According to (24), 2, is defined in terms of h(s)lji(r)lj;(.x). Under the assun~ption that z, exists in the population and is less than the poverty line z,2, is clearly a consistent estimator of z,, and, in particular, we need not consider the possibility that i,=z. since this will happen with vanishingly small probability as hl r.
The proof is similar for all values of s, and so we drop s from our notations. Since /I(z)= 0.we have by Taylor expansion that
(33) h(i)= /1'(2)(iz)
for some 2 such that 1 5 z 1 < 1 iz I. We will show later that
It was assumed that /zl(z)# 0, and, in fact, since /~(x)
> 0 for .x < z, and /l(z)= 0, it follows that ll'(z)< 0. Since itz as h1t r,we have that 22 as IVas well. Thus for large enough
^h
N, h'(2)+ 0. It follows from (33)and (34)that
Suppose first that the populations A and 4 are independent, and that we have IY., drawings from one and hZ, drawings from the other. For the purposes of the asymptotic analysis, we assume that the ratio i.= 1VA/lVB remains constant as N/, and h5 tend to infinity. We have that
because /I() = 0. It follows that
The expression (36) consists of two independent sums of iici valiables to which ue map apply the central limjt theorem since molilents of order 2s 2 are assumed to exist. It follows immediately that N,;/'II(Z) = O(1) in probability, and, from (351, that 2 = O(AT'/'). In addition, from (I ),
If s = 1, (37) remains correct becausc wc defined ~j'(z)=F;(z), the density associated wit11 the cdf F,$.TVe now scc from (35) arid (36) that
Next, suppose that we have A' paired observations j,"and j: from one aingle population. (36) continues to hold with 1% = N and r.== I. However. the two sulns of iid variables arc no longer independent in general, and so (38) must be replaced by
(39) lim var(h"l"2 z))
,\'+ ~L
It remains to prove (31). Note that, bccause /I(?) =11(z)= 0,
Consider the cxp~ession
for nonrandom 8. In the case of just one population and N paired drawings of y." and j",we can write
The expectation of this is hiz + 6)11(z), and so (11) is the avcrage of hounded iid variables with mean zero and finite variance of order 8'. Consequently, by the central limit tlieorcm, (41) times lv!/?has Incan zcro and variance of order 6'. Since iz = O(,\ri/') in probal)ility, it follows that
(40) times N'/' tends to zero in mean square, and hence in probability. An exactly silnilar argument applies when there are two populations.
STOCHASTIC DOMINANCE
PROOFOF THFORFM4: We have for both distributions A and A that
The second term on the righthand side of this is
and the first terrn is
This kind of integral can be expressed asymptotically as a sum of iid variables using a technique developed in Davidson and Duclos (1 997). The terrn becomes
which to leading ordel. is a deterministic term plus an average of iid random variables. We can combine the two terms in (42) using (21) to get
If z is known and not estimated, we call just set C(y,) = z, and the last term in the sum will vanish.
It is easy to check that, whether z < Q(p) or <> Q(p), the e5pectation of the leading term of the above expression is just G(p; 2). The fact that G,,(p; i.,)and G,,(p; i,])are sums of independently and identically distributed random variables with finite second moments leads to their asyn~ptotic normality by the central limit theorem. The covariance structure is obtained by simple calculation.
Q.E,D.
REFERENCES
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