How Has the Literature on Gini s Index Evolved in the Past 80 Years?


 Neal Charles
 2 years ago
 Views:
Transcription
1 How Has the Literature o Gii s Idex Evolved i the Past 80 Years? Kua Xu Departmet of Ecoomics Dalhousie Uiversity Halifax, Nova Scotia Caada B3H 3J5 Jauary 2004 The author started this survey paper whe visitig the Graduate School, the People s Bak of Chia ad the School of Fiace, Remi Uiversity i the summer of The author would like to thak the two istitutios for playig the hosts. The author also wishes to thak Professor Yag Yao at the Chia Ceter for Ecoomic Research of Pekig Uiversity for his ecouragemet ad suggestios, ad Zhiyog Huag, a studet at Na Kai Uiversity for correctig some errors i the earlier draft of this paper. I additio to what have bee writte here, the origial pla also icludes two sectios o the geeralizatio ad statistical ifereces of the Gii idex. This would make this paper too legthy. The author chooses to leave those topics i aother survey paper. A brief survey of this kid always rus the risk of ievitable omissios. The author wishes to thak all authors, cited ad ucited, for their cotributios to this large ad sigificat literature. 1
2 Abstract The Gii coefficiet or idex is perhaps oe of the most used idicators of social ad ecoomic coditios. From its first proposal i Eglish i 1921 to the preset, a large umber of papers o the Gii idex has bee writte ad published. Goig through these papers represets a demadig task. The aim of this survey paper is to help the reader to avigate through the major developmets of the literature ad to icorporate recet theoretical research results with a particular focus o differet formulatios ad iterpretatios of the Gii idex, its social welfare implicatio, ad source ad subgroup decompositio. 2
3 1 Itroductio Sice the Gii coefficiet or idex as a summary statistics bore Gii (1912, 1914, 1921) s ame as we ow kow it, the theoretical literature has evolved for more tha 80 years. Durig the past 80 years, the Gii idex gradually became oe of the pricipal iequality measures i the disciplie of ecoomics. This measure is uderstood by may ecoomists ad has bee applied i umerous empirical studies ad policy research. 1 As may are aware, research o iequality ad poverty measuremet cotiues to evolve. May ecoomists, experieced ad ewly mited, always wish to have the collectio of the theoretical results o the Gii idex hady ad accessible. Aad (1983) ad Chakravarty (1990) provided comprehesive surveys o the measures of iequality icludig Gii idex. But the literature is i a costat state of flux eve i the research area which is cosidered to be well established. Other authors such as Lambert (1989), Silber (1999), ad Atkiso ad Bourguigo (2000) also provided comprehesive refereces for icome iequality ad poverty with the Gii idex as oe of may iequality measures. This survey paper differs from those refereces i that it collects the theoretical results oly o the Gii idex, old ad more recet, i oe place. 2 1 May ecoomists i Chia such as Mr. Li Shi also worked o the Gii measures usig the Chiese data. Accordig to the ews coferece give by Premier Zhu Rogji of the People s Republic of Chia i March 15, 2001, the Gii idex calculated i 1999 i Chia reached 0.39 which was cosidered at a alarmig level by the iteratioal stadard. The Chiese Govermet vowed to solve the problem durig the developmet process. 2 Thak Buhog Zheg for poitig out a excellet techical survey paper o the Gii idex by Yizhaki (1998), which presets the results oly based o the cotiuous distributios. Hece the curret paper may still be justifiable give it focuses o both the history ad techical results ad cosiders both discrete ad cotiuous distributios. 3
4 This survey paper is a atural cotiuatio of Aad (1983) ad Chakravarty (1990) o this specific literature ad attempts to icorporate additioal research results o the Gii idex. It is hoped that this paper will ot oly provide readers a summary of mai theoretical literature but also cover some related issues such as differet formulatios ad iterpretatios of the Gii idex, its social welfare implicatio, ad source ad subgroup decompositio. The Gii idex ca be used to measure the dispersio of a distributio of icome, or cosumptio, or wealth, or a distributio of ay other kids. But the kid of distributios where the Gii idex is used most is the distributio of icome. For this reaso, ad for simplicity, this paper will focus o the Gii idex i the cotext of icome distributios although its applicatios should ot be limited to icome distributios. A icome distributio may be for differet icomes: household icomes or idividual icomes. The choice of the icome uit is ofte determied by the purpose of research. For simplicity, the discussio i this paper is based o icome distributios of the idividuals withi the populatio. Eve the cocept of icome distributio ca vary from the icomes that are pre tax ad other fiscal trasfers to those that are after tax ad other fiscal trasfers. As a covetio, this is ot the focus of the paper. Istead, the paper focuses o theoretical results ad iterpretatios. Similarly, the trasformatio from a family icome to idividual icomes via equivalet scale will ot be discussed here. This paper will show that the Gii idex has may differet formulatios ad iterestig iterpretatios. It ca be expressed as a ratio of two regios defied by a 45 degree lie ad a Lorez curve i a uit box, or a fuctio of Gii s mea differece, or a covariace betwee icomes ad their raks, 4
5 or a matrix form of a special kid. Each formulatio has its ow appeal i a specific cotext. The Gii idex was proposed as a summary statistics of dispersio of a distributio. It was viewed, for a quite log time, ot too differet from other dispersio measures such as variace ad stadard deviatio. But whe comig to a decisio as to which iequality measure should be adopted i a study, ecoomists foud that it was rather difficult to select oe statistics over others without ay justificatio i terms of social welfare implicatio. Thus, they started to search the lik betwee the existig iequality measures ad their uderlyig social welfare fuctios. It is ow kow that may wellkow iequality measures ideed have direct, although implicit, relatios with social welfare fuctios ad that the measured iequality ca be iterpreted as social welfare loss due to iequality. With this itellectual premise, the social welfare implicatio of a Gii idex value ca ow be iterpreted with a greater clarity. Ecoomists also examied how the Gii idex as a aggregate iequality measure could be decomposed accordig to either icome sources or subpopulatio groups. A great effort has bee made to specify the coditios uder which such decompositios are feasible. Eve whe decompositios are feasible, it is ot always clear what meaigful iterpretatio each ad every decomposed parts of the Gii idex has. More specifically, whe subgroup decompositio is made of the Gii idex, oe term called the crossover term appears difficult to iterpret. Over time, this view has chaged. Now may ecoomists foud that this term ca be viewed as a measure of icome stratificatio or the degree to which the icomes of differet social groups 5
6 cluster. The remaiig paper is orgaized as follows. Sectio 2 itroduces ecessary mathematical symbols ad basic defiitios. Sectio 3 reviews the evolutio of computatio methods of the Gii idex. Sectio 4 revisits the literature o PigouDalto s priciple of trasfers ad social welfare implicatio of the Gii idex. Icome source ad subgroup decompositio are discussed i Sectio 5. Fially, Sectio 6 cocludes. 2 Mathematical Symbols ad Defiitios There are geerally two differet approaches for aalyzig theoretical results of the Gii idex: oe is based o discrete distributios; the other o cotiuous distributios. The latter demads certai coditios o cotiuity while the former does ot require such coditios. The discrete icome distributio is easy to uderstad i some cases while the cotiuous icome distributio ca simplify some derivatios i some situatios. The two ca be uified as show i Dorfma (1979). Whe the distributio fuctio is discrete, y takes values that ca be deoted by a 1 colum vector y = [y 1, y 2,..., y ] such that the elemets i the vector are arraged i odecreasig order: y 1 y 2 y. The values of y are bouded below by a = 0 ad above by b < +, or y i [a, b] for i = 1, 2,...,. The otatio is used to sort y i the opposite order; i.e., the elemets of ỹ = [ỹ 1, ỹ 2,..., ỹ ] are arraged i oicreasig order: ỹ 1 ỹ 2 ỹ. The discrete cumulative distributio fuctio is F i = i, while the probability of y takig o the value y i is f i = 1 for i = 1, 2,...,. 6
7 F (y k ) = k i i=1 is the cumulative probability up to y k ad ca be iterpreted as the populatio share of those whose icomes are less tha or equal to y k. 3 The mea icome, µ y, is give by µ y = 1 i=1 y i. The cumulative icome shares of the populatio up to the idividual whose icome is raked ith from the lowest to the highest are give by L i = 1 µ y i y j, (1) for i = 1, 2,...,. L 0 is defied as zero while L = 1. Note that L i s are arraged i odecreasig order. Sometimes, ecoomists wish to use L i to idicate L i s that are arraged i oicreasig order [see equatio (29)]. Whe the icome distributio is cotiuous, y ca be viewed as a value of the cumulative distributio fuctio of icome F (y) or a distributio fuctio of icome f(y). j=1 I geeral, y is bouded below by a = 0 ad above by b < +. F (a) = 0 while F (b) = 1. F (y ) = y f(y)dy is the cumulative probability up to y. The mea icome, µ y, is give by µ y = b ydf (y) = b yf(y)dy. The cumulative icome share of the popula a a tio up to the idividual whose icome is y is give by L(p ) = L(F (y )) = 1 y yf(y)dy. (2) µ y The Lorez curve was hited by Sir Leo Chiozza Moey (1905) ad origially proposed by Mr. M. O. Lorez i It is deoted as L(p) = 3 Whe the complex samplig survey desigs are used to collect icome data, the researcher must deal with the issues of statistical iferece ad samplig weights. 4 See Publicatios of the America Statistical Associatio, Vol. ix, pp. 209ff. a a 7
8 L(F (y)), the proportio of the total icome of the ecoomy that is received by the lowest 100p of the populatio for all possible values of p. I other words, the graph of F ad L is the Lorez curve with 0 F 1 ad 0 L 1. For a discrete distributio, k = F (y k), F 1 ( k ) = y k, ad the Lorez curve is L k = L( k ) = L(F (y k)) for 0 k 1. For a cotiuous distributio, p = F (y), F 1 (p) = y, ad the Lorez curve is L(p) = L(F (y)) for 0 p 1. Figure 1 shows a Lorez curve is below the 45 degree lie. This reflects that the icome share grows at a much slower rate as the populatio share icreases ad that there exists a higher degree of icome cocetratio withi the populatio. 1 Figure 1. Lorez Curve % of the icome Lie of Perfect Equality Area A Area B Lorez Curve % of the populatio 3 Evolutio of Computatioal Methods The Gii idex as is called today was, accordig to Dalto (1920, p. 354), amed after the fact that a remarkable relatio has bee established be 8
9 twee this measure of iequality ad the relative mea differece, the former measure beig always equal to half the latter. This remarkable relatio was first give by Gii i Dalto (1920, p. 353) therefore called this mea differece as Professor Gii s mea differece. The computatioal methods for the Gii idex iclude the geometric approach, Gii s mea differece approach (or the relative mea differece approach), covariace approach, ad matrix form approach. Each approach has its ow appeal ad is desirable i some way but all ca be uified ad are cosistet with oe aother. These methods ad their techical justificatios are examied i the followig. 3.1 Geometric Approach The attractiveess of the Gii idex to may ecoomists is that it has a ituitive geometric iterpretatio. That is, the Gii idex ca be, as i Figure 1, defied geometrically as the ratio of two geometrical areas i the uit box: (a) the area betwee the lie of perfect equality (45 degree lie i the uit box) ad the Lorez curve, which is called Area A ad (b) the area uder the 45 degree lie, or Areas A + B. Because Areas A + B represets the half of the uit box, that is, A+B = 1, the Gii idex, G, ca be writte 2 as G = A A+B = 2A = 1 2B. If oe works with a discrete icome distributio, he or she ca compute (3) 9
10 F i s ad L i s ad the the area below the Lorez curve B = 1 1 (F i+1 F i ) (L i+1 + L i ). (4) 2 i=0 Substitutig equatio (4) ito equatio (3) yields the Gii idex G: 5 1 G = 1 (F i+1 F i ) (L i+1 + L i ). (5) i=0 To illustrate how to use the above defiitio, let the hypothetical icome distributio be y 1 = 0, y 2 = 1, y 3 = 2. For this distributio, the Lorez curve ca be described by the poits (L 1 = 0, F 1 = 1 3 ), (L 2 = 1 3, F 2 = 2 3 ), ad (L 3 = 1, F 3 = 1) i the uit box. As idicated i Figure 2, Area B is the sum of the area of a small triagle ( 1 ), the area of a square ( 1 ), ad the 18 9 area of a large triagle ( 1 9 ): B = = The Gii idex, as idicated i equatio (5), G = 1 [( ) 1 (0) + 3 ( 1 3 ) ( ) ( 1 3 ) ( )] 4 = There are various expressios of this defiitio. For example, Yao [1999, p. 1251, equatio (1)] adopted a spread sheet approach usig this method. Osberg ad Xu [2000, p. 59, equatio (14)] modified the defiitio to accommodate the complex samplig survey data. 10
11 Figure 2. Lorez Curve ad Gii Idex % of the icome Lie of Perfect Equality Area A Area B Lorez Curve % of the populatio Several alterative formulatios i fact follow the same traditio, Rao (1969) showed that the Gii idex ca be defied as 1 G = (F i L i+1 F i+1 L i ). (6) i=1 This formulatio ca be show to be cosistet to equatio (5): give F = L = 1 ad F 0 = L 0 = 0, the Gii idex defied i equatio (5) ca be rewritte as 1 1 G = 1 + (F i L i+1 F i+1 L i ) (F i+1 L i+1 F i L i ). i=0 i=0 Sice the last term o the righthad side is oe, we have equatio (6). 11
12 Se (1973) defied the Gii idex as G = µ y ( + 1 i) y i. (7) This formulatio illustrates that the icomerakbased weights are iversely associated with the sizes of icomes. That is, i the idex the richer s icomes get lower weights while the poor s icome get higher weights. Se s defiitio 6 i=1 ca be derived from equatio (6) by otig G = 1 i=1 (F il i+1 F i+1 L i ) = i=1 (F i 1L i F i L i 1 ) = i=1 (F i (L i L i 1 ) (F i F i 1 ) L i ) ( 1 = 2 µ y i=1 iy i ) i j=1 y j (8) give F i = i, L i = 1 µ y i j=1 y j, F i F i 1 = 1, ad L i L i 1 = y i µ y. The expressio for G ca be further maipulated as [ 1 G = 2 µ y i=1 iy i ] i i=1 j=1 y j = 1 2 µ y [ i=1 iy i i=1 ( + 1 i)y i] = 1 2 µ y [ i=1 ( + 1)y i 2 i=1 ( + 1 i)y i] = µ y i=1 ( + 1 i)y i. (9) The last equality is cosistet with equatio (7). Fei ad Rais (1974) ad Fei, Rais, Kuo (1978) defied the Gii idex 6 Se (1997) gives a slightly differet defiitio usig the icomes sorted i oicreasig order. 12
13 as a liear fuctio of a u y idex: where the u y idex is give by G = 2 u y + 1. (10) u y = i=i iy i i=1 y. i Substitutig u y = i=1 iy i i=1 y i ito equatio (10) yields 2 G = 2 µ y i=1 iy i +1 = +1 2(+1) µ y i=1 iy i = µ y i=1 ( + 1 i)y i, (11) which is cosistet with equatio (7). If oe deals with a cotiuous icome distributio, he or she ca express the area uder the Lorez curve as B = 1 0 L(p)dp. (12) Substitutig equatio (12) ito equatio (3) yields the Gii idex for the cotiuous icome distributio as G = L(p)dp. (13) Clearly, it is much simpler to uderstad the Gii idex geometrically. However, its computatio may be tedious. 13
14 3.2 Gii s Mea Differece Approach Differig from the geometric approach, Gii (1912) showed that the geometric approach is i fact related to the statistical approach via a cocept called the (absolute ad relative) mea differece. That is, the Gii idex as a ratio of two areas defied above is always equal to the half of the relative mea differece that will be explaied later. Accordig to David (1968), the relative mea differece discussed by ad amed after Gii (1912) was i fact discussed much earlier by F. R. Helmert ad other Germa writers i the 1870 s. I 1912, Gii s book was published i Italia ad hece was ot accessible to Eglishspeakig ecoomists at the time. I 1921 whe commetig o Dalto s (1920) work, Gii (1921) explaied his work (1912) ad related literature i Eglish i a short Ecoomic Joural article. Sice the, the Gii idex ad Gii s relative mea differece were made kow i the literature of icome iequality measuremet i the Eglishspeakig world. Followig Gii (1912), Kedall ad Stuart (1958) i their wellkow book Advaced Theory of Statistics stated the Gii idex as the half of Gii s relative mea differece because it was ideed a importat statistical result at that time. There is o doubt that may geeratios of statisticias leared this result through the classical work of Kedall ad Stuart. Gii s absolute mea differece for a discrete distributio is defied as = 1 2 y i y j. (14) i=1 j=1 where y i ad y j are the variates from the same distributio. The absolute 14
15 mea differece for a cotiuous distributio is defied similarly as the mea differece betwee ay two variates of the same distributios: = E y i y j (15) where E is the mathematical expectatio operator. The relative mea differece is defied as µ y = E y i y j µ y. (16) That is, the relative mea differece equals the absolute mea differece divided by the mea of the icome distributio. I additio to the above result, Shalit ad Yitzhaki (1984) also provided several alterative ways to express Gii relative mea differece. The Gii idex is the oehalf of Gii s relative mea differece The above expressio is also writte as G = G = 2µ y. (17) 1 2 i=1 i=1 max(0, y i y j ) (18) µ y because i=1 j=1 y i y j = 2 i=1 i=1 max(0, y i y j ) [see Pyatt (1976)]. Aad (1983) showed that equatio (17) is cosistet with the geometric defiitio give i equatio (5). For a discrete distributio, the absolute mea differece ca be rewritte as 2 2 i=1 j i (y i y i ) ad the Gii 15
16 idex ca be expressed as G = 1 2 µ y i=1 = 1 2 µ y i=1 j i (y i y i ) ( iy i i j=1 y j ). (19) This result is cosistet with equatio (8). I other words, Gii s mea differece approach is cosistet with the geometric approach. Followig the traditio of Kedall ad Stuart, Dorfma (1979) proposed a simple formula for the Gii idex for the cotiuous icome distributio, that is G = 2µ y = 1 1 µ y b a (1 F (y)) 2 dy (20) where Gii s absolute mea differece for a cotiuous icome distributio is give by equatio (15). They also oted that Gastwirth (1972) proposed a similar formula without a proof which was attributed to Kedall ad Stuart (1977) who also omitted the proof. This formula ca be derived as follows. Because y i y j = y i + y j 2 mi(y i, y j ), Gii s absolute mea differece is writte as = E y i y j = 2µ y 2E mi(y i, y j ). To lear more about the term E mi(y i, y j ), it is ecessary to kow the probability for mi(y i, y j ), that is Pr [mi(y i, y j ) y] = 1 Pr (y i > y) Pr (y j > y) = 1 (1 F (y)) 2. 16
17 Icorporatig this probability ito Gii s absolute mea differece yields = E y i y j = 2µ y 2 b Substitutig ito G yields yd ( 1 (1 F (y)) 2) = 2µ y +2 b a a yd (1 F (y)) 2. G = = 2µ b y + 2 yd (1 F (y))2 a. 2µ y 2µ y Sice a = 0 ad b is fiite, a(1 F (a)) 2 = b(1 F (b)) 2 = 0, G = 1+ 1 µ y (y(1 F (y)) 2 b a b a ) b (1 F (y)) 2 dy = 1 1µy (1 F (y)) 2 dy. a The defiitio of the Gii idex based o the relative mea differece has its root i the statistics. However, its computatio could be complex. It is the covariace approach, which will be discussed below, that ca facilitate the computatio of the Gii idex usig the commoly used covariace procedure i most statistical software packages. 3.3 Covariace Approach It was kow that the Gii s absolute mea differece ca be expressed as a fuctio of the covariace betwee variatevalues ad raks as oted i Stuart (1954, 1955): = 4 b a ( y F (y) 1 ) f(y)dy. (21) 2 17
18 But, fidig of the lik betwee this fact ad the computatio of the Gii idex occurred much later. I the cotext of discrete icome distributios, Aad (1983) showed the Gii idex ca be computed by 7 G = 2cov(y i,i) µ y ; (22) that is, the Gii idex ca be expressed as a fuctio of the covariace betwee icomes ad their raks. I the cotext of the discrete icome distributio, Aad demostrated how the defiitio of equatio (22) is justified. He oted that the mea of the raks is give by i = 1 i=1 i = ad the covariace is expressed as cov(y i, i) = 1 i=1 Thus, the Gii idex ca be writte as ( yi µ y ) ( i i ) = 1 i=1 iy i +1 2 µ y. G = 2cov(y i,i) µ y = 2 2 µ y which is cosistet with equatio (11). i=1 iy i + 1 Idepedetly, Lerma ad Yitzhaki (1984) also reported the same result 7 Based o the author s commuicatio with Sudhir Aad, the author leared that Sudhir Aad s thesis, which is the basis of Aad (1983), was completed i
19 for cotiuous icome distributios. Yitzhaki (1982) oted that accordig to Lomicki (1952) the Gii s absolute mea differece of the distributio F ca be writte as = b b a Lerma ad Yitzhaki (1984) 8 a x y f(x)f(y)dxdy = 2 b u = F (y) [1 F (y)] ad v = y, ca be writte as 9 = 4 b a a F (x) [1 F (x)] dx. showed that, by itegratio by parts, with ( y F (y) 1 ) f(y)dy. 2 By trasformatio of variables, write f(y)dy = df ad chage from the bouds [a, b] for y to the bouds [0, 1] for F, = ( y(f ) F 1 ) df. 2 Note that F is uiformly distributed betwee 0 ad 1 so that its mea is 1 2. Thus, Gii s absolute mea differece is give by = 4cov [y, F (y)] ad the Gii idex is defied as G = 2µ y G = 2cov [y, F (y)] µ y. (23) 8 This result i Charavarty (1990, p. 88) is called the Stuart(1954)LermaYitzhaki (1984) propositio. 9 This is cosistet to Stuart[1954, pp , equatios (13) (15)]. 19
20 I the cotext of cotiuous icome distributios, Lambert (1989) used a slightly differet approach to the same problem. He first oted that the Lorez curve L(p) has the followig property: L (p) = dl(p) dp = dl(p)/dy dp/dy = yf(y)/µ y f(y) = y µ y. (24) The usig itegratio by parts he rewrote the Gii idex from equatio (13) as G = L(p)dp = pl (p)dp 1 = 2 b a yf (y) µ y f(y)dy 1. (25) Sice cov[y, F (y)] = E[yF (y)] E(y)E[F (y)] ad E[F (y)] = 1, the Gii 2 idex ca be derived from equatio (25) as G = [ ] b 2 yf (y)f(y)dy µ y a 2 = µ y 2Cov[y, F (y)] µ y. (26) Although the derivatios of Aad (1983), Lerma ad Yizhaki (1984) ad Lambert (1989) differ, each result is a variat of the other. To compute the Gii idex usig Aad s approach, first obtai rak for each observatio y i, i ; the, compute the covariace betwee y i ad i, cov(y i, i). The resultig covariace must be divided by the umber of the observatios, cov(y i, i ) = 1 cov(y i, i), sice i/ terms are empirical cumulative distributio of F (y). Fially, the Gii idex is computed by G = 2 µ y cov(y i, i). This is cosistet with the result of Lerma ad Yitzhaki (1984) ad Lambert (1989). This approach is also exteded by Shalit (1985) so that a regressio ca be used 20
21 to compute the Gii idex. The advatage of the covariace approach is that the computatio of the Gii idex ca be facilitated by usig the covariace procedure i existig statistical software packages. 3.4 Matrix Form Approach I the literature, the matrix form approach was proposed by Pyatt (1976) ad Silber (1989) for decompositio purposes. 10 Pyatt (1976) focused o equatio (18) i which the Gii idex is iterpreted as a ratio of two terms: (a) the umerator 1 2 i=1 i=1 max(0, y i y j ) is the average expected gai to be expected by the populatio if each ad every idividual i the populatio is allowed to compare his or her icome y i with ay other chose idividual s icome y j ad to take y j whe y j is greater tha y i ad (b) the deomiator is the mea icome µ y. Cosider that the populatio ca be divided ito k subpopulatio groups with group i has p i proportio of the populatio. It is possible to express the average expected gai as k i=1 k E(gai i j) Pr(i j) (27) i=1 where Pr(i j) = p i p j for all i, j = 1, 2,..., k ad k i=1 p i = 1. Let E be a k k matrix with elemets beig E ij = E(gai i j). Stack p i s ito a k 1 colum vector p. Let the average icome for group i be m i. Stack m i s ito a k 1 colum vector m. Hece m p = k i=1 m ip i = µ y. Thus, the Gii 10 Also see Yao (1999) for a spread sheet applicatio. 21
22 idex, correspodig to equatio (18), ca be defied as G = (m p) 1 p Ep. (28) Silber (1989) proposed aother elegat approach for computig the Gii idex. The derivatio starts from the Gii idex s defiitio: G = j=1 L j [ ( j) ] (j 1), (29) where L i is the proportio of total icome eared by the idividual whose icome has the ith rak i the icome distributio with L 1 L 2 L j L. (This meas that the richest idividual is raked 1st while the poorest idividual is raked th.) First of all, examie how this defiitio is liked to the previous oes. Note that L j = y j j=1 y j implies that y j s are arraged i oicreasig order while y i s i our previous discussio are arraged i odecreasig order. Therefore, it is useful to fid out the system lik betwee two idex systems. It turs out that i = j + 1 ad that equatio (29) ca be rewritte as G = i=1 [ y i i 1 i=1 y i i ], 22
23 which i tur equals [ 1 G = i=1 y i i=1 y i i=1 = µ y i=1 (2 2i + 1) y i ( 2 2i+1 ) ] yi = µ y i=1 ( + 1 i) y i. The last equality i the above is cosistet with equatio (7). Secod, Equatio (29) ca be readily show to be G = [ L i j i i=1 1 j i ] 1. (30) Equatio (30) is equivalet to equatio (29) because the sum i the former is idetical to the sum i the latter as show below: j Lj [ ( j) ] (j 1) i Li [ j i 1 j i ] 1 j = 1 j = 2. j = L1 [ 1 L2 [ 2. L [ ] ( = 1 ) L1 ] = L2 ( 3 ) ] ( = 1 ) L [ i = 1 L1 ( 1 ] = 1 ) L1 [ i = 2 1 L2 ] ( 2 = 3 ) L2.. i = L [ 1 ] = L ( 1 ) Third, equatio (30) ca be readily writte compactly i matrix form as G = e G L (31) 23
24 or [ 1 G =, 1,..., 1 ] L 1 L 2. L. where e is a colum vector of elemets of 1/, L is a colum vector of elemets beig respectively equal to L 1, L 2,..., L, ad G is the Gmatrix whose elemets G ij are equal to 1 whe i < j, to 1 whe i > j, to 0 whe i = j. 4 Social Welfare Implicatio From the statistics poit of view, the Gii idex is a fuctio of Gii s mea differece ad hece it was iitially, ad still is, iterpreted a measure of dispersio. Pyatt (1976), however, wet a bit further ad gave the Gii idex a iterpretatio as the average gai to be expected, if each ad every idividual is allowed to compare his or her icome with the icome of aother idividual ad to keep the icome that is higher. But this iterpretatio is statistical i ature ad more coveiet for subpopulatio group decompositio rather tha for measurig social welfare (loss) due to iequality. I fact, Dalto i his 1920 paper, followig Pigou (1912, p. 24), had attempted to raise a miimum criterio for a iequality measure. It is ow called PigouDalto s priciple of trasfers. To establish this priciple, he said: 24
25 We have, first, what may be called the priciple of trasfers.... we may safely say that, if there are oly two icomereceivers, ad a trasfer of icome takes place from the richer to the poorer, iequality is dimiished. There is, ideed, a obvious limitig coditio. For the trasfer must ot be so large, as more tha to reverse the relative positios of the two icomereceivers, ad it will produce its maximum result, that is to say, create equality. Ad we may safely go further ad say that, however great the umber of icomereceivers ad whatever the amout of their icomes, ay trasfer betwee ay two of them, or, i geeral, ay series of such trasfers, subject to the above coditio, will dimiish iequality. (Dalto, 1920, p. 351) Dalto (1920) also oted that the Gii idex ca be viewed as half of the Gii s relative mea differece. Accordig to Dalto, as the relative mea differece satisfies the priciple of trasfers, the Gii idex must satisfies the same priciple ad be judged as a desirable iequality measure. Jekis (1991), amog others, used the total differetial approach to evaluate whether the Gii idex ideed satisfies the priciple of trasfers whe the trasfers are very small. To do so he assumed that the trasfer is meapreservig (i.e., µ y is fixed) ad that there is a trasfer from to the richer idividual i to the poorer idividual j but this trasfer will ot chage the fact the relative positios of the rich ad the poor i the icome distributio. Takig the total differetial of equatio (7) with respective to y i ad y j yields G = ( G/ y j )dy j ( G/ y i )dy i = 2(j i) 2 µ y dy < 0 (32) 25
26 give that dy i = dy j, j < i, ad dy i = dy j = dy. Thus, the Gii idex ideed satisfies the priciple of trasfer. That is, whe the trasfer occurs, the value of the Gii idex will decrease. Although the Gii idex ideed satisfies the priciple of trasfers, there was little discussio about the social welfare implicatio of iequality measures icludig the Gii idex after Dalto (1920). For example, Gii (1921) himself, i respose to Dalto s work (1920), suggested that the measure of iequality (such as the oe he proposed) was of icome ad wealth ot of ecoomic welfare. The ormative approach, which relates a iequality measure directly to a uderlyig social welfare fuctio, appeared much later. Kolm (1969) advocated the use of social welfare fuctio i measurig icome iequality. Atkiso (1970) oted that the social welfare implicatio was particularly importat whe oe came to select a summary statistics of icome iequality. He wrote: Firstly, the use of these measures ofte serves to obscure that fact that a complete rakig of distributios caot be reached without fully specifyig the form of the social welfare fuctio. Secodly, examiatio of the social welfare fuctios implicit i these measures shows that i a umber of cases they have properties which are ulikely to be acceptable, ad i geeral these are o grouds for believig that they would accord with social values. For these reasos, I hope that these covetioal measures will be rejected i favour of direct cosideratio of the properties that we should like the social welfare fuctio to display. 26
27 (Atkiso, 1970, p. 262) Se (1973) also discussed this approach as a geeralizatio of Atkiso s measure. It is Blackorby ad Doaldso (1978) who examied the issue further i a systematical fashio, established the geeral results ad applied them to iequality measures icludig the Gii idex. The way to lik the Gii idex to its uderlyig social welfare fuctio is to defie the Gii idex i terms of the equallydistributedequivaleticome (EDEI), or the represetative icome proposed by Atkiso (1970), Kolm (1969), ad Se (1973). Usig this approach, a iequality measure, I, ca be writte as a fuctio of the EDEI icome, ξ, ad the mea icome, µ y. I = 1 ξ. (33) µ y If I is defied based o the Gii social welfare fuctio, the I is deoted by I G or, simply, G. Give this setup, if ξ is idetical to µ y, the I is zero. That is, there is o iequality i the icome distributio from which ξ ad µ y are computed. If, o the other had, ξ is less tha µ y (say, the former is oly 70% of the latter), the I will be greater tha zero but bouded by 1 (I will take a value of 0.3). That is, there is some degree of iequality. Of course, it is crucial to kow how ξ is derived. Geerally speakig, for a particular social welfare fuctio or social evaluatio fuctio, a EDEI give to every idividual could be viewed as idetical i terms of social welfare to a actual icome distributio. To explai the idea further, let W (y) φ(w (y)) be a homothetic (ordial) social welfare fuctio of icome with φ beig a icreasig fuctio ad 27
NPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationDecomposition of Gini and the generalized entropy inequality measures. Abstract
Decompositio of Gii ad the geeralized etropy iequality measures Stéphae Mussard LAMETA Uiversity of Motpellier I Fraçoise Seyte LAMETA Uiversity of Motpellier I Michel Terraza LAMETA Uiversity of Motpellier
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationDescriptive statistics deals with the description or simple analysis of population or sample data.
Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 200617 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät LudwigMaximiliasUiversität Müche Olie at http://epub.ub.uimueche.de/940/
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationSubject CT5 Contingencies Core Technical Syllabus
Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationResearch Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More information, a Wishart distribution with n 1 degrees of freedom and scale matrix.
UMEÅ UNIVERSITET Matematiskstatistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 00409 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that
More informationDescribing Income Inequality
Describig Icome Iequality Module 051 Describig Icome Iequality Describig Icome Iequality by Lorezo Giovai Bellù, Agricultural Policy Support Service, Policy Assistace Divisio, FAO, Rome, Italy ad Paolo
More informationData Analysis and Statistical Behaviors of Stock Market Fluctuations
44 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Data Aalysis ad Statistical Behaviors of Stock Market Fluctuatios Ju Wag Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Email:
More informationBiology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships
Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationEstimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers
More informationBASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1.
BASIC STATISTICS 1.) Basic Cocepts: Statistics: is a sciece that aalyzes iformatio variables (for istace, populatio age, height of a basketball team, the temperatures of summer moths, etc.) ad attempts
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More information9.8: THE POWER OF A TEST
9.8: The Power of a Test CD91 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based
More information3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average
5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lieup for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationBENEFITCOST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEITCST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal  Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationResearch Method (I) Knowledge on Sampling (Simple Random Sampling)
Research Method (I) Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationEntropy of bicapacities
Etropy of bicapacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uivates.fr JeaLuc Marichal Applied Mathematics
More informationStandard Errors and Confidence Intervals
Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5yearold boys. If we assume
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationGibbs Distribution in Quantum Statistics
Gibbs Distributio i Quatum Statistics Quatum Mechaics is much more complicated tha the Classical oe. To fully characterize a state of oe particle i Classical Mechaics we just eed to specify its radius
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationDAME  Microsoft Excel addin for solving multicriteria decision problems with scenarios Radomir Perzina 1, Jaroslav Ramik 2
Itroductio DAME  Microsoft Excel addi for solvig multicriteria decisio problems with scearios Radomir Perzia, Jaroslav Ramik 2 Abstract. The mai goal of every ecoomic aget is to make a good decisio,
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea  add up all
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationZTEST / ZSTATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
ZTEST / ZSTATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large TTEST / TSTATISTIC: used to test hypotheses about
More informationCovariance and correlation
Covariace ad correlatio The mea ad sd help us summarize a buch of umbers which are measuremets of just oe thig. A fudametal ad totally differet questio is how oe thig relates to aother. Stat 0: Quatitative
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationUser manual and preprogrammed spreadsheets for performing revision analysis
User maual ad preprogrammed spreadsheets for performig revisio aalysis This documet describes how to perform revisio aalysis usig preprogrammed template spreadsheets based o data extracted from the OECD
More informationGrade 7. Strand: Number Specific Learning Outcomes It is expected that students will:
Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationHOSPITAL NURSE STAFFING SURVEY
2012 Ceter for Nursig Workforce St udies HOSPITAL NURSE STAFFING SURVEY Vacacy ad Turover Itroductio The Hospital Nurse Staffig Survey (HNSS) assesses the size ad effects of the ursig shortage i hospitals,
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationUnit 20 Hypotheses Testing
Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect
More informationProblem Set 1 Oligopoly, market shares and concentration indexes
Advaced Idustrial Ecoomics Sprig 2016 Joha Steek 29 April 2016 Problem Set 1 Oligopoly, market shares ad cocetratio idexes 1 1 Price Competitio... 3 1.1 Courot Oligopoly with Homogeous Goods ad Differet
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More informationCorrelation. example 2
Correlatio Iitially developed by Sir Fracis Galto (888) ad Karl Pearso (8) Sir Fracis Galto 8 correlatio is a much abused word/term correlatio is a term which implies that there is a associatio betwee
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationSwaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps
Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while
More informationArithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...
3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.
More informationBASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1)
BASIC STATISTICS. SAMPLES, RANDOM SAMPLING AND SAMPLE STATISTICS.. Radom Sample. The radom variables X,X 2,..., X are called a radom sample of size from the populatio f(x if X,X 2,..., X are mutually idepedet
More informationRepeated sampling in Successive Survey
Statistics istitutio Repeated samplig i Successive Survey (RSSS) Xiaolu Cao 15högskolepoägsuppsats iom Statistik III, HT011 Supervisor: Mikael Möller Abstract: I this thesis, the idea of samplig desig
More informationWe have seen that the physically observable properties of a quantum system are represented
Chapter 14 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More information