Statistical Properties of the Mc-Dagum and Related Distributions

Transcription

1 Georga Southern Unversty Dgtal Southern Electronc Theses and Dssertatons Graduate Studes, Jack N. Avertt College of Sprng 2013 Statstcal Propertes of the Mc-Dagum and Related Dstrbutons Sasth Rajasoorya Georga Southern Unversty Follow ths and addtonal works at: Part of the Mathematcs Commons Recommended Ctaton Rajasoorya, Sasth, "Statstcal Propertes of the Mc-Dagum and Related Dstrbutons" (2013). Electronc Theses and Dssertatons Ths thess (open access) s brought to you for free and open access by the Graduate Studes, Jack N. Avertt College of at Dgtal Commons@Georga Southern. It has been accepted for ncluson n Electronc Theses and Dssertatons by an authorzed admnstrator of Dgtal Commons@Georga Southern. For more nformaton, please contact dgtalcommons@georgasouthern.edu.

2 STATISTICAL PROPERTIES OF THE MC-DAGUM AND RELATED DISTRIBUTIONS by SASITH RAJASOORIYA (Under the Drecton of Broderck O. Oluyede) ABSTRACT In ths thess, we present a new class of dstrbutons called Mc-Dagum dstrbuton. Ths class of dstrbutons contans several dstrbutons such as beta-dagum, beta-burr III, beta-fsk and Dagum dstrbutons as specal cases. The hazard functon, reverse hazard functon, moments and mean resdual lfe functon are obtaned. Inequalty measures, entropy and Fsher nformaton are presented. Maxmum lkelhood estmates of the model parameters are gven. KEY WORDS: Mc-Dagum Dstrbuton, Beta-Dagum Dstrbuton, Inequalty Measures, Informaton Matrx 2009 Mathematcs Subject Classfcaton: 62E15,60E05

3 STATISTICAL PROPERTIES OF THE MC-DAGUM AND RELATED DISTRIBUTIONS by SASITH RAJASOORIYA B.S n Busness Admnstraton A Thess Submtted to the Graduate Faculty of Georga Southern Unversty n Partal Fulfllment of the Requrement for the Degree MASTER OF SCIENCE STATESBORO, GEORGIA 2013

4 c 2013 Sasth Rajasoorya All Rghts Reserved

5 STATISTICAL PROPERTIES OF THE MC-DAGUM AND RELATED DISTRIBUTIONS by SASITH RAJASOORIYA Major Professor: Broderck O. Oluyede Commttee: Charles Champ Han Samaw Electronc Verson Approved: May 2013 v

6 DEDICATION Ths thess s dedcated to my beloved parents and specally my beloved wfe Pubudu for her exceptonal support and encouragement. v

7 ACKNOWLEDGMENTS My deepest grattude extends to Dr. Broderck Oluyede for hs gudance and encouragement gven to me throughout ths thess and also my entre studes at GSU. It s undoubtedly hs exceptonal support and drecton what made ths work a success. It was truly a prvlege to have hm as my advsor. My specal thank goes to my commttee members, Dr. Charles Champ and Dr. Han Samaw for ther valuable contrbuton towards the success of ths work. It s my pleasure to acknowledge all the members of the faculty and staff n the Department of Mathematcal Scences at Georga Southern Unversty, specally Dr. Sharon Taylor, Department Char, Martha Abell, Dean and former Department Char and Dr. Yan Wu, former Graduate Program Drector, for all ther utmost commtments towards the success of the students and the department. v

8 TABLE OF CONTENTS ACKNOWLEDGMENTS v CHAPTER 1 Introducton Revew for the Dagum and Mc-Donald Generalzed Dstrbutons Dagum Dstrbuton Mc-Donald Generalzed Dstrbuton Hazard and Reverse Hazard Functons Outlne of Results Introducng Mc-Dagum dstrbuton Mc-Dagum Dstrbuton Hazard and Reverse Hazard Functons Expanson of Dstrbuton Submodels v

9 2.3.2 Kum-Dagum Dstrbuton Concludng Remarks Moments and Inequalty Measures Moments Inequalty Measures Inequalty Measures for Some Sub models Concludng Remarks Entropy Reny and Shanon Entropy β-entropy Concludng Remarks Inference Maxmum Lkelhood Estmates (MLE) Fshers Informaton Matrx Concludng Remarks Future Works v

10 BIBLIOGRAPHY x

11 CHAPTER 1 INTRODUCTION 1.1 Revew for the Dagum and Mc-Donald Generalzed Dstrbutons Dagum Dstrbuton Dagum dstrbuton was proposed by Camlo Dagum n 1970 s (Dagum ). Hs proposals enable the development of statstcal dstrbutons used to ft emprcal ncome and wealth data, that could accommodate both heavy tals n emprcal ncome and wealth dstrbutons, and also permt nteror mode. Dagum dstrbuton has both Type-I and Type-II specfcaton, where Type-I s the three parameter specfcatons and Type-II deal wth four parameter specfcaton. Dagum n 1977 motvated hs model from emprcal observaton that the ncome elastcty η (F, x) of the cumulatve dstrbuton functon (cdf) F of ncome nto a decreasng and bounded functon of F. The cdf and pdf of Dagum (Type-I) dstrbuton are gven by G(x; λ, δ, β) = ( 1 + λx δ) β, (1.1) and g (x; λ, δ, β) = βλδx δ 1 ( 1 + λx δ) β 1, for λ, δ, β > 0, (1.2) respectvely, where λ s a scale parameter, and δ and β are shape parameters. Dagum (1980) refers to hs model as the generalzed logstc-burr dstrbuton. Actually when β = 1, Dagum dstrbuton was also referred to as the log-logstc dstrbuton. Also, generalzed (log-) logstc dstrbutons arse naturally n Burr s

12 2 (1942) system of dstrbutons. The most popular Burr dstrbutons are Burr-XLL dstrbuton, often called Burr dstrbuton wth cdf, F (x; δ, β) = 1 ( 1 + x δ) β, for x > 0, δ, β > 0, (1.3) and more mportantly the Burr-III dstrbuton wth cdf F (x; δ, β) = ( 1 + x δ) β, for x > 0 and δ, β > 0. (1.4) Thus, these dstrbutons are more popular n economcs, after the ntroducton of an addtonal parameter (λ as we can see above n the Dagum cdf and pdf). It s clear that the Dagum dstrbuton s a Burr III dstrbuton wth an addtonal scale parameter (λ). The k th raw or noncentral moments of Dagum dstrbuton are gven by E ( X k) = 0 ( = βλ k δ B x k βλδx δ 1 ( 1 + λx δ) β 1 dx β + k δ, 1 k δ ), (1.5) for δ > k, λ, δ, β > 0, where B(.,.) s the beta functon, (by settng t = (1 + λx δ ) 1 ). and The mean, mode and varance of the Dagum dstrbuton are gven by 1 λ µ X = Γ(β + ) ( ) 1 δ Γ 1 1 δ, (1.6) Γ (β) σ 2 X = Mode = 1 λ ( δβ 1 δ + 1 ) 1 δ, (1.7) 2 [ ( λ δ Γ (β) Γ β + 2 ) ( Γ 1 2 ) ( Γ 2 β + 1 ) ( Γ )], (1.8) Γ 2 (β) δ δ δ δ respectvely. The q th percentle of the Dagum dstrbuton s x(q) = λ 1 δ (q 1 β ) 1 δ 1. (1.9)

13 Mc-Donald Generalzed Dstrbuton Consder an arbtrary parent cdf G(x). The probablty densty functon (pdf) f(x) of the new class of dstrbutons called the Mc-Donald generalzed dstrbuton s gven by f(x; a, b, c) = cg(x) B(a, b) Gac 1 (x) (1 G c (x)) b 1, for a > 0, b > 0, and c > 0. (1.10) See Cordero et al.(2012) for addtonal detals. Note that g(x) s the pdf of parent dstrbuton, g(x) = dg(x)/dx, and a,b and c are addtonal shape parameters. Introducton of ths addtonal shape parameters s specally to ntroduce skewness. Also, ths allows us to vary tal weght. It s mportant to note that for c=1 we obtan a sub-model of ths generalzaton whch s a beta-generalzaton and for a=1, we have the Kumaraswamy (Kw),[Kumaraswamy (1980)] generalzed dstrbutons. For random varable X wth densty functon gven above n (1.10), we wrte X Mc-G(a,b,c). The cdf for ths generalzaton s gven by, F (x; a, b, c) = I G(x) c(a, b) = where I G c (x)(a, b) = B(a, b) 1 G(x) c 1 G(x) c ω a 1 (1 ω) b 1 dω, (1.11) B(a, b) 0 0 ω a 1 (1 ω) b 1 dω denotes ncomplete beta functon rato (Gradshteyn and Ryzhk, 2000). The same equaton can be expressed as follows: where F (x; a, b, c) = G (x)ac ab (a, b) [ 2F 1 (a, 1 b; a + 1; G(x) c )], (1.12) 1 2F 1 (a, b; c; x) = B (b, c b) 1 t b 1 (1 t) c b 1 0 (1 tz) a dt, (1.13)

14 4 s the well known hypergeometrc functon (Gradshteyn and Ryzhk, 2000), and B(a, b) = Γ(a)Γ(b) Γ(a + b). (1.14) One mportant beneft of ths class s ts ablty to ft skewed data that cannot properly be ftted by many other exstng dstrbutons. Mc-G famly of denstes allows for hgher levels of flexblty of ts tals and has a lot of applcatons n varous felds ncludng economcs, fnance, relablty and medcne Hazard and Reverse Hazard Functons In ths secton, some basc utlty notons are presented. Suppose the dstrbuton of a contnuous random varable X has the parameter set θ = {θ 1, θ 2,, θ n }. Let the probablty densty functon (pdf) of X be gven by f(x; θ ). The cumulatve dstrbuton functon of X, s defned to be F (x; θ ) = x f(t; θ ) dt. (1.15) The hazard functon of X can be nterpreted as the nstantaneous falure rate or the condtonal probablty densty of falure at tme x, gven that the unt has survved untl tme x. The hazard functon h(x; θ ) s defned to be h(x; θ P (x X x + x) ) = lm x 0 x[1 F (x; θ )] = F (x; θ ) F (x; θ ) where F (x; θ ) s the survval or relablty functon. = f(x; θ ) 1 F (x; θ ), (1.16) Reverse Hazard functon can be nterpreted as an approxmate probablty of a falure n [x, x + dx], gven that the falure had occurred n [0, x]. The reverse hazard functon τ(x; θ ) s defned to be τ(x; θ ) = f(x; θ ) F (x; θ ). (1.17)

15 Some useful functons that are employed n subsequent sectons are gven below. The gamma functon s gven by Γ(x) = The dgamma functon s defned by where Γ (x) = 0 t x 1 e t dt. (1.18) Ψ(x) = Γ (x) Γ(x), (1.19) 0 t x 1 (log t)e t dt s the frst dervatve of the gamma functon. The second dervatve of the gamma functon s Γ (x) = 0 t x 1 (log t) 2 e t dt. The lower ncomplete and upper ncomplete gamma functons are respectvely. γ(s, x) = x 0 t s 1 e t dt and Γ(s, x) = x 5 t s 1 e t dt (1.20) The hazard functon (hf) and reverse hazard functons (rhf) of the Mc-G dstrbuton are gven by and respectvely. h F (x) = cg (x) Gac 1 (x) {1 G c (x)} b 1 B (a, b) { 1 I G(x) c (a, b) }, (1.21) τ F (x) = cg (x) Gac 1 (x) {1 G c (x)} b 1, (1.22) B (a, b) I G c (x)(a, b)

16 Outlne of Results The outlne of ths thess as follows: In chapter 2, the Mc-Dagum dstrbuton and related famly of dstrbutons are ntroduced. The expanson for the densty, hazard and reverse hazard functons, and other propertes are presented. Chapter 3 presents the moments, and nequalty measures. Chapter 4 contans entropy measures of the Mc-Dagum dstrbuton. Chapter 5 contans nference for the model parameters as well applcatons of the results presented n earler chapters.

17 CHAPTER 2 INTRODUCING MC-DAGUM DISTRIBUTION In ths chapter, a new class of dstrbuton, called Mc-Dagum dstrbuton s ntroduced. Consderng the propertes and some useful features of both Dagum and Mc- Donald dstrbutons, a broad range of generalzaton s possble by combnng these dstrbutons. The new class of dstrbutons possess capabltes wdely applcable n several areas as we wll show n the next few chapters. 2.1 Mc-Dagum Dstrbuton In chapter 1, the cdf and pdf of Dagum dstrbuton were gven as G (x; λ, δ, β) = ( 1 + λx δ) β, (2.1) and g (x; λ, δ, β) = βλδx δ 1 ( 1 + λx δ) β 1, λ, δ, β > 0, (2.2) respectvely. The pdf for Mc-Donald dstrbuton s gven by f(x; a, b, c) = c B(a,b) g(x)gac 1 (x) (1 G c (x)) b 1, a > 0, b > 0, c > 0, (2.3) and the cdf s F (x) = I G(x) c(a, b) = 1 B(a,b) G(x) c 0 ω a 1 (1 ω) b 1 dω. (2.4) Now, combnng the denstes gven n equatons (2.2) and (2.3), we obtan the

18 8 pdf of the Mc-Dagum dstrbuton as follows: f(x; λ, δ, β, a, b, c) = c B(a, b) βλδx δ 1 ( 1 + λx δ) β 1 [ ( 1 + λx δ ) β ] ac 1 [1 ( 1 + λx δ) cβ ] b 1 (2.5) = cβλδx δ 1 B(a, b) ( 1 + λx δ ) βac 1 [ 1 ( 1 + λx δ) cβ ] b 1, for a, b, c, λ, β, δ > 0. The cdf of ths new dstrbuton s gven by F (x) = I G(x) c(a, b) = 1 B(a,b) = 1 B(a,b) G(x) c ω a 1 (1 ω) b 1 dω 0 (1+λx δ ) βc 0 ω a 1 (1 ω) b 1 dω = I βc (a, b), (1+λx δ ) (2.6) where I y (a, b) = 1 y B(a,b) 0 ωa 1 (1 ω) b 1 dω (2.7) s the ncomplete beta functon. The cdf can also be wrtten as follows: F (x) = ( 1 + λx δ ) βac ab(a, b) [ 2F 1 ( a, 1 b; a + 1; (1 + λx δ ) βc)], (2.8) where 1 1 y 2F 1 (a, b; c; x) = b 1 (1 y) c b 1 dy, (2.9) B(b,c b) 0 (1 y z ) a s the well-known hypergeometrc functon, (Gradshteyn and Ryzhk,(2000)).

19 9 2.2 Hazard and Reverse Hazard Functons The falure rate functon or hazard functon and reverse hazard functon are gven by h F (x; a, b, c, λ, β, δ) = cg(x)gac 1 (x)[1 G c (x)] b 1 B(a,b)[1 I G c (x) (a,b)] = cβλδx δ 1 (1+λx δ ) βac 1 [1 (1+λx δ ) cβ ] b 1 [1 I ], [(1+λx δ) βc ] (a,b) B(a,b) (2.10) and τ F (x; a, b, c, λ, β, δ) = cβλx δ 1 (1+λx δ ) βac 1 [1 (1+λx δ ) cβ] b 1 B(a,b)I (1+λx δ ) βc (a,b) (2.11) for a > 0, b > 0, c > 0, λ > 0, β > 0, δ > 0, respectvely. 2.3 Expanson of Dstrbuton In ths secton, we present a seres expanson of the Mc-Dagum cdf and pdf. Consder the Mc-Dagum cdf gven by F (x; λ, β, δ, a, b, c) = I G(x) c(a, b) = 1 B(a,b) = 1 B(a,b) G(x) c ω a 1 (1 ω) b 1 dω 0 (1+λx δ ) βc 0 ω a 1 (1 ω) b 1 dω. (2.12) Note that for ω < 1, (1 ω) b 1 = j=0 ( 1) j Γ (b) Γ (b j) j! ωj.

20 10 Therefore, the cdf can be expanded to obtan: F (x; λ, β, δ, a, b, c) = 1 (1+λx δ ) βc B(a,b) 0 ω a 1 ( 1) j Γ(b) j=0 = j=0 = j=0 = j=0 dω Γ(b j)j! ( 1) j Γ(b) G(x;λ,β,δ) c ω a+j 1 dω B(a,b) Γ(b j)j! 0 [ ] ( 1) j G(x;λ,β,δ) Γ(b) ω a+j 1+1 c B(a,b)Γ(b j)j! a+j 1+1 ( 1) j Γ(b) B(a,b)Γ(b j)j! 0 [G(x;λ,β,δ)] c(a+j) (a+j) = j=0 p jg (x; λ, βc(a + j), δ), (2.13) for b > 0, real non-nteger, where p j = Smlarly, the pdf s gven by ( 1)j Γ(a+b) j!γ(a)γ(b j)(a+j). f(x) = p j g(x; λ, βc(a + j), δ). (2.14) j=0 If b > 0 s an nteger, then b 1 F (x; λ, β, δ, a, b, c) = p j G (x; βc(a + j), λ, δ), (2.15) j=0 and b 1 f(x; λ, β, δ, a, b, c) = p j g(x; βc (a + j), λ, δ). (2.16) j=0 Ths s a fnte mxture of Dagum dstrbutons wth parameters λ, βc(a + j)andδ. The graphs below are the pdf of the Mc-Dagum dstrbuton for dfferent values of parameters λ, δ, β, a, b, and c.

21 11

22 12

23 The graphs below are the cdf and hazard functons of the Mc-Dagum dstrbuton for dfferent values of parameters a, b, c, λ, δ, β. 13

24 Submodels Wth ths generalzaton, we have several submodels that can be obtaned wth specfc values of the parameters λ, β, a, b and c. 1. When c = 1, the Mc-Dagum dstrbuton s the beta-dagum dstrbuton, wth the densty gven by: f(x; λ, β, δ, a, b) = βλδx δ 1 B(a, b) for x > 0, λ > 0, β > 0, δ > 0, a > 0, and b > 0. ( 1 + λx δ ) βa 1 [ 1 (1 + λx δ ) β] b 1, (2.17) 2. If a = b = c = 1, we have the Dagum dstrbuton wth the pdf, f D (x; λ, δ, β) = βλδx δ 1 ( 1 + λx δ) β 1, (2.18)

25 15 for λ, δ, β > If b = c = 1 and a > 0, then we have the Dagum dstrbuton wth parameters βa, λ and δ. The pdf s for λ, δ, β > 0. f (x; βa, λ, δ, ) = βaλδx δ 1 ( 1 + λx δ) βa 1, (2.19) 4. If a = c = 1 and b > 0, we have another Beta-Dagum dstrbuton wth parameters b, β, λ, δ and the pdf s gven by f BD (x; λ, δ, β, b) = bβλδx δ 1 ( 1 + λx δ) β 1 [ 1 ( 1 + λx δ) β ] b 1, (2.20) for λ, δ and β > If a = c = λ = 1, then we have the beta-burr III dstrbuton wth parameters b, β, δ and the pdf s gven by f BB (x; δ, βb, ) = bβδx δ 1 ( 1 + x δ) β 1 [ 1 ( 1 + x δ) β ] b 1, (2.21) for b, δ, β > If c = β = 1, then we have the beta-fsk dstrbuton wth parameters a, b, λ, δ and the pdf s gven by f BF (x; λ, δ, a, b) = λδx δ 1 B (a, b) for a, b, λ, δ > 0. ( 1 + λx δ ) a 1 [ 1 ( 1 + λx δ) 1 ] b 1, (2.22) Kum-Dagum Dstrbuton Kumaraswamy n hs paper (1980) proposed a two-parameter dstrbuton (Kumaraswamy dstrbuton) defned n (0, 1). Here we wll refer to t as Kum dstrbuton. Its cdf s

26 16 gven by: F (x; a; b) = 1 (1 x a ) b, x (0, 1), a > 0, b > 0. (2.23) The parameters a and b are the shape parameters. The Kum dstrbuton has the probablty densty functon (pdf) gven by: f(x; a, b) = abx a 1 (1 x a ) b 1, x (0, 1), a > 0, b > 0. (2.24) Note that the Kumaraswamy dstrbuton can be derved from the beta dstrbuton. The beta dstrbuton has the pdf: f(x; α, β) = Γ (α + β) Γ (α) Γ (β) xα 1 (1 x) β 1, where x (0 1), α > 0, β > 0. (2.25) Combnng cdf of Kum dstrbuton wth the Dagum dstrbuton dscussed n chapter 1, we obtan Kum-Dagum dstrbuton wth the cdf and pdf for ths dstrbuton gven by F Kum (x) = 1 [ 1 ( 1 + λx δ) βa ] b, (2.26) and f kum (x) = abβλδx δ 1 ( 1 + λx δ) β 1 [ 1 + λx δ ] β(a 1) [ 1 ( 1 + λx δ) βa ] b = abβλδx δ 1 ( 1 + λx δ) β βa+β 1 ( 1 [ 1 + λx δ] βa ) β 1 = abβλδx δ 1 ( 1 + λx δ) βa 1 ( 1 [ 1 + λx δ] βa ) β 1, (2.27) for a, b, β, λ, δ > 0, respectvely. We do not study the propertes of the Kum-Dagum dstrbuton n ths thess. 2.4 Concludng Remarks In ths chapter, we ntroduced a new class of dstrbutons called the Mc-Dagum dstrbuton. We obtaned the pdf, cdf, hazard functon, reverse hazard functon

27 17 for ths class of dstrbutons. We obtaned the seres expanson of the dstrbuton and presented plots of pdf, cdf and hazards functon for dfferent parameter values. Through these graphs we see that the dstrbuton possesses the ablty to ft for a large range of data sets. We noted that there are several submodels for selected values of the Mc-Dagum model parameters. Addtonally we ntroduced another new dstrbuton called Kum-Dagum dstrbuton but we do not dscuss ts propertes n ths thess.

28 CHAPTER 3 MOMENTS AND INEQUALITY MEASURES In ths chapter, we present moments and nequalty measures for the Mc-Dagum dstrbuton. Income dstrbuton and ts varaton s an mportant concern for economsts. We use the results presented n chapter 2 whch we obtaned by expandng the pdf. 3.1 Moments We can derve the k th moment of a Mc-Dagum dstrbuton usng propertes of the mxture dstrbuton. The k th raw or non-central moments are gven by, E(X k ) = ( x ) ( k cβλx δ 1 0 B(a,b) 1 + λx δ βac 1 1 ( 1 + λx δ) ) βc b 1 dx = cβλ B(a,b) 0 x k δ 1 ( 1 + λx δ) βac 1 ( 1 ( 1 + λx δ) βc ) b 1 dx. (3.1) Now let, y 1 = ( 1 + λx δ), then x = (1 y) 1 δ (λy) 1 δ, and we have E(X k ) = cβ δb(a,b) Usng the fact that (1 y βc ) b 1 = ( 1) j Γ(b) j=1 and y βc weobtan E(X k ) 1 k (1 y) δ (λy) k δ y βac 1 (1 y βc ) b 1 dy. (3.2) 0 = λ k δ cβ δb(a,b) 0 = λ k δ cβ δb(a,b) 0 j=0 p jβc(a+j)λ k δ Γ(b j)j! (yβc ) j, and for p j = ( 1) j Γ(b) Γ(b j)j! ( 1)j Γ(a+b) j!γ(a)γ(b j)(a+j), 1 0 y k δ +βac+βcj 1 (1 y) 1 k δ 1 dy ( 1) j Γ(b) Γ(b j)j! B(βc(a + j) + k δ, 1 k δ ) = B(βc(a + j) + k, 1 k ), δ > k. δ δ δ can obtan the k th ncomplete moment for a Mc-Dagum dstrbuton as follows: E [ X k X x ] = E X x [ X k ; λ, β, δ, a, b, c ] = x 0 uk f (u) du = x 0 uk = j=0 p j j=0 p jf(u; βc(a + j), λ, δ)du x 0 uk f(u; βc(a + j), λ, δ)du = 0 p j βc(a+j)λ k δ δ B((1 + λx δ ) 1 ; βc(a + j) + k δ, 1 k δ ), (3.3)We (3.4)

29 19 for δ > k, where B(t; c 1, c 2 ) = t 0 yc 1 1 (1 y) c 2 1 dy. gven by The mean resdual lfe (MRF) functon denoted by µ(x; λ, β, δ, a, b, c) = µ(x) s µ(x) = E[X x X x] = E(X) E(X X x) 1 F (x) x = k j=0 p j βc(a+j)λ δ B(βc(a+j)+ k δ δ,1 k δ ) 1 k 0 p j βc(a+j)λ δ δ B((1+λx δ ) 1 ;βc(a+j)+ k δ,1 k δ ) j=0 p jg(x;λ,βc(a+j),δ) x. (3.5) 3.2 Inequalty Measures Lorenz and Bonferron curves are the most wdely used nequalty measures n ncome and wealth dstrbuton (Kleber, 2004). Zenga curve was presented by Zenga n In ths secton, we wll derve Lorenz, Bonferron and Zenga curves for the Mc-Dagum dstrbuton. The Lorenz, Bonferron and Zenga curves are defned by L F (x) = x 0 tf(t)dt E(X) = E X x(x) E(X), (3.6) and B (F (x)) = x 0 tf(t)dt F (x)e(x) = E X x(x) F (x)e(x) = L F (x) F (x), (3.7) A(x) = 1 µ (x) µ + (x), (3.8)

30 respectvely, where µ (x) = x 0 tf(t)dt F (x) = E X(x) F (x) and µ + (x) = x tf(t)dt 1 F (x) 20 = E(X) E X>x(x) 1 F (x) are the lower and upper means. For Mc-Dagum dstrbuton, usng these results, we obtan the curves. Lorenz curve for Mc-Dagum dstrbuton s gven by L FG (x; λ, β, δ, a, b, c) = j=0 p jβc(a+j)λ 1 δ B((1+λx δ ) 1 ;βc(a+j)+ 1 δ,1 1 δ ) j=0 p jβc(a+j)λ 1 δ B(βc(a+j)+ 1 δ,1 1 δ ). (3.9) Bonferron curve for Mc-Dagum dstrbuton s gven by B(F G (x; λ, β, δ, a, b, c)) = j=0 p jβc(a+j)λ 1 δ B((1+λx δ ) 1 ;βc(a+j)+ 1 δ,1 1 δ ) j=0 p jg(x;λ,βc(a+j),δ) j=0 p jβc(a+j)λ 1 δ B(βc(a+j)+ 1 δ,1 1 δ ). (3.10) Zenga curve for the Mc-Dagum dstrbuton s gven by where E [X X x] = x E(X) = j=0 p jβc(a+j)λ 1 δ δ A(x; λ, β, δ, a, b, c) 0 p j βc(a+j)λ 1 δ F (x) = j=0 p jg (x; λ, βc(a + j), δ). = 1 [ E(X X x) F (x) E(X) E(X x) 1 F (x) ] (3.11) (1 F (x))e[x X x] = 1, F (x)[e(x) E(X X x)] δ B((1 + λx δ ) 1 ; βc(a + j) + 1 δ, 1 1 δ ), B(βc(a + j) + 1, 1 1 ), and δ δ Inequalty Measures for Some Sub models For varous submodels that we ntroduced n chapter 2, we can generate Lorenz, Bonferron and Zenga curves. Let ξ 1 =(λ, β, δ, a, b), ξ 2 =(λ, β, δ, b), ξ 3 =(λ, δ, a, b) and E= j=0 p jβ(a + j)λ 1 δ B(β(a + j) + 1 δ, 1 1 δ ) 1. If c = 1, we obtan the Lorenz and Bonferron curves for the beta-dagum dstrbuton: L FG (x; ξ 1 ) = j=0 p jβ(a + j)λ 1 δ B((1 + λx δ ) 1 ; β(a + j) + 1, 1 1) δ δ, E

31 21 and B(F G (x; ξ 1 )) = j=0 p jβ(a + j)λ 1 δ B((1 + λx δ ) 1 ; β(a + j) + 1, 1 1) δ δ j=0 p, jg (x; λ, β(a + j), δ) E respectvely. 2. If a = c = 1 and b > 0, then Lorenz and Bonferron curves for another Beta- Dagum dstrbuton wth parameters b, β, λ, δ gven by L FG (x; ξ 2 ) = j=0 p jβ(1 + j)λ 1 δ B((1 + λx δ ) 1 ; β(1 + j) + 1 δ, 1 1 δ ) j=0 p jβ(1 + j)λ 1 δ B(β(1 + j) + 1 δ, 1 1 δ ), and B(F G (x; ξ 2 )) = j=0 p jβ(a + j)λ 1 δ B((1 + λx δ ) 1 ; β(1 + j) + 1, 1 1) δ δ j=0 p, jg (x; λ, β(1 + j), δ) E respectvely. 3. If a = c = λ = 1, then we obtan the Lorenz and Bonferron curves for the beta-burr III dstrbuton wth parameters b, β, δ, that s L FG (x; β, δ, b) = j=0 p jβ(1 + j)b((1 + x δ ) 1 ; β(1 + j) + 1 δ, 1 1 δ ) j=0 p jβ(1 + j)b(β(1 + j) + 1 δ, 1 1 δ ), and B(F G (x; β, δ, b)) = j=0 p jβ(a + j)b((1 + x δ ) 1 ; β(1 + j) + 1, 1 1) δ δ j=0 p. jg (x; β(1 + j), δ) E 4. If c = β = 1, then we obtan the Lorenz and Bonferron curves for the beta-fsk dstrbuton wth parameters a, b, λ, δ. L FG (x; ξ 3 ) = j=0 p j(a + j)λ 1 δ B((1 + λx δ ) 1 ; (a + j) + 1, 1 1) δ δ, E and B(F G (x; ξ 3 )) = j=0 p j(a + j)b((1 + λx δ ) 1 ; (a + j) + 1 δ, 1 1 δ ) j=0 p jg (x; λ, (a + j), δ) E for a, b, λ, δ > 0.

32 Concludng Remarks In ths chapter, we presented the raw moments and the k th ncomplete moments for the Mc-Dagum dstrbuton. Inequalty measures for the dstrbuton are derved usng well known Lorenz and Bonferron curves. Addtonally, Zenga curve was also obtaned. Lorenz curve and Bonferron curves for some submodels of ths class of dstrbutons are also obtaned.

33 CHAPTER 4 ENTROPY In ths chapter, we dscuss the Reny entropy, Shannon entropy and β-entropy for the Mc-Dagum dstrbuton. The entropy of a random varable X s a measure of varaton of the uncertanty. 4.1 Reny and Shanon Entropy For a pdf f(x), Reny entropy (Reny, 1961) s gven by ( H R (f) = log f s (x)dx ), s > 0, s 1. (4.1) 1 s 0 As s 1, we obtan the Shanon entropy. Note that, f s (x) = (cβλδ)s x sδ s B s (a,b) and f s (x)dx = (cβλδ)s 0 B s (a,b) = (cβλδ)s B s (a,b) ( 1 + λx δ ) βacs s [ 1 ( 1 + λx δ) cβ ] bs s x ( sδ s 1 + λx δ) [ βacs s 1 ( 1 + λx δ) ] cβ bs s 0 dx ] [(1 y) 1 δ (λy) 1 sδ s y βacs+s (1 y βc ) bs s δ ] δ 1 dy 1 0 = (cβλδ)s 1 B s (a,b) 0 λ sδ s y sδ s δ [ y 2 λδ (1 y) 1 δ (λy) 1 δ +βacs+s 2+ 1 δ +1 (1 y βc ) sb s (1 y) Usng the fact that, (1 ω) b 1 = ( 1) j Γ(b) j=0 Γ(b j)j! ωj, and settng y = (1 + λx δ ) 1, so that x δ = y 1 1 = 1 y, and λ λy λδx δ 1 dx=y 2 dy, we obtan f s (x)dx = (cβλδ)s 1 0 B s (a,b) 0 λ sδ s y sδ s δ = (cβλδ)s λ sδ s λ 1+ 1 δ 1 B s (a,b) = (cβλδ)s λ 1+ 1 sδ s δ 1 B s (a,b) = (cβλδ)s λ 1+ 1 δ sδ s B s (a,b) = (cβλδ)s λ 1+ 1 δ sδ s B s (a,b) dy. (4.2) s 1 s 1+ δ +βacs+s 2+ 1 δ +1 (1 y βc ) sb s (1 y) s+ s δ 1 δ 1 dy 0 yβacs+s s s δ + 1 δ +1 2 (1 y) s+ s δ 1 δ 1 (1 y βc ) sb s dy 0 yβacs s δ + 1 δ 1 (1 y) s+ s δ 1 δ 1 j=0 j=0 j=0 ( 1) j Γ(sb s+1) Γ(sb s+1 j)j! ( 1) j Γ(sb s+1)y βcj Γ(sb s+1 j)j! 1 0 yβcj+βacs s δ +1δ 1 (1 y) s+ s δ 1 δ 1 dy ( 1) j Γ(sb s+1) Γ(sb s+1 j)j! B(βcj + βacs s δ + 1 δ, s + s δ 1 δ ). (4.3)

34 24 Therefore, Reny entropy for the Mc-Dagum dstrbuton s H R (f) = log [ (cβλδ) s λ 1+ 1 δ sδ s ( 1) j Γ(sb s + 1) 1 s B s (a, b) Γ(sb s + 1 j)j! j=0 ( B βcj + βacs s δ + 1 δ, s + s δ 1 ) ] δ (4.4) for s > 0 and s 1. If bs s s a postve nteger, then the sum n the Reny entropy stops at bs s. 4.2 β-entropy We also obtan β-entropy for the Mc-Dagum densty as follows. 1 H β(f) β 1 = [1 f β(x)dx], f β 0 1, β > 0, (4.5) E[ log(f(x))], f β=1. Therefore, f β 1, β > 0, H β(f) 1 = [1 (cβλδ) βλ 1+ 1 δ βδ β ( 1) j Γ( βb β + 1) β 1 B β(a, b) j=0 Γ( βb β + 1 j)j! ( B βcj + βac β β δ + 1 δ, β + β ) ] δ 1. δ 4.3 Concludng Remarks In chapter 4, measures of uncertanty, ncludng Reny, Shanon and β-entropy for the Mc-Dagum dstrbuton were presented.

35 CHAPTER 5 INFERENCE 5.1 Maxmum Lkelhood Estmates (MLE) Let Θ = (λ, β, δ, a, b, c) T. In order to estmate the parameters λ, β, δ, a, b and c of the Mc-Dagum dstrbuton, we use the method of maxmum lkelhood estmaton. Let X 1, X 2,..., X n be a random sample from f(x; λ, β, δ, a, b, c). The log-lkelhood functon L(λ, β, δ, a, b, c) s: ( ) ( n ) [ cβλδ n ] ( ) L(λ, β, δ, a, b, c) = nlog + log x δ 1 + log 1 + λx δ βac 1 B(a, b) =1 =1 n [ + log 1 ( ) 1 + λx δ cβ ] b 1 (5.1) =1 = nlog(c) + nlog(β) + nlog(λ) + nlog(δ) nlogb(a, b) (δ + 1) logx (βac + 1) log[1 + λx δ ] + (b 1) =1 =1 log[1 (1 + λx δ ) cβ ]. Dfferentatng L(λ, β, δ, a, b, c) wth respect to each parameter λ, β, δ, a, b and c and settng the result equals to zero, we obtan maxmum lkelhood estmates. The partal 1 (5.2) dervatves of L wth respect to each parameter or the score functon s gven by: ( L U n (Θ) = λ, L β, L δ, L a, L b, L ), (5.3) c where L λ = n ( x δ ) λ βac 1 + λx δ =1 =1 x δ + (b 1) (1 + λx δ ) =1 cβ(1 + λx δ ) cβ 1 x δ, [1 (1 + λx δ ) cβ ] (5.4)

36 L β = n β ac log(1 + λx δ ) + c(b 1) =1 L δ =1 = n δ logx + λ(βac + 1) =1 λcβ(b 1) =1 x δ (1 + λx δ ) cβ log(1 + λx δ [1 (1 + λx δ ) cβ ] =1 (1 + λx δ x δ log(x ) (1 + λx δ ) ) cβ 1 logx, [1 (1 + λx δ ) cβ ] =1 ) 26, (5.5) (5.6) L a = n(ψ(a) ψ(a + b)) βc log(1 + λx δ ), (5.7) L b = n[ψ(b) ψ(a + b)] + log[1 (1 + λx δ ) cβ ], (5.8) where ψ(.) s dgamma functon defned by ψ(x) = d dx logγ(x) = Γ (x) Γ(x), and L c = n c βa log(1 + λx δ ) + (b 1)β =1 =1 =1 (1 + λx δ ) cβ log(1 + λx δ [1 (1 + λx δ ) cβ ] ). (5.9) The MLE of the parameters λ,β,δ,a,b and c, say ˆλ, ˆβ,ˆδ,â,ˆb and ĉ are obtaned by solvng the followng equatons, L λ = L β = L δ = L a = L = L b c =0. There s no closed form soluton to these equatons, so numercal technque such as Newton-Rapson method must be appled. 5.2 Fshers Informaton Matrx To obtan the Fshers nformaton matrx (FIM), we derve the second partal dervatves and cross partal dervatves wth respect to each parameter λ,β,δ,a,b and c as follows: From equaton (5.4) we obtan λ = n 2 λ + (βac + 1) 2 =1 x 2δ A 2 + (b 1) =1 cβx 2δ A cβ 2 [A cβ cβ 1] [1 A cβ ] 2, (5.10)

37 where, A =(1 + λx δ ), λ β = ac =1 λ δ = ( βac 1) where, B = 1 λx δ cβa 1 x δ A + (b 1) =1 λx δ =1 x δ x δ log(x ) + (b 1) A 2 A 1 27 ca cβ 1 [cβlog(a ) 1 + A cβ ], (5.11) [1 A cβ ] 2 =1 +λx δ cβa cβ 1 (1+λx δ ) cβ, [1 A cβ ] 2 cβx δ A cβ 1 log(x )B, (5.12) λ a = βc x δ (1 + λx δ ), (5.13) =1 and λ b = =1 cβ(1 + λx δ ) cβ 1 x δ [1 (1 + λx δ ) cβ ], (5.14) λ c = βa =1 x δ A + (b 1) =1 βx δ A cβ 1 [cβloga 1 + A cβ ]. (5.15) [1 A cβ ] 2 From equaton (5.5), we obtan β = n 2 β + 2 c2 (b 1) β δ = ac C + (b 1) =1 where, C = λx δ log(x ) A, =1 =1 λca cβ 1 (1 + λx δ ) cβ [log(1 + λx δ )] 2, (5.16) [1 (1 + λx δ ) cβ ] 2 =1 x δ logx [1 cβloga A cβ ], (5.17) [1 A cβ ] 2 β a = c log(1 + λx δ ), (5.18) and β b = =1 c(1 + λx δ ) cβ log(1 + λx δ [1 (1 + λx δ ) cβ ] ), (5.19) β c = a loga + (b 1) =1 =1 A cβ loga [cβloga + A cβ 1] [1 A cβ ] 2. (5.20)

38 28 From equaton (5.6), we obtan where D = [1 λcβx δ δ 2 = n δ 2 (1+λx δ + λ(βac + 1) ) 1 λx δ (1+λx δ [1 (1+λx δ =1 x δ (logx ) 2 A cβ 1 D (5.21) ) 1 (1+λx δ ) cβ +λx δ (1+λx δ ) cβ 1 ]. ) cβ ] 2 Also, δ a = λβc x δ logx (1 + λx δ ), (5.22) =1 and δ b = λcβ δ c = λβc F λβ(b 1) =1 =1 =1 x δ (1 + λx δ [1 (1 + λx δ ) cβ ] A cβ 1 x δ ) cβ 1 logx, (5.23) logx [cβloga + A cβ 1] [1 A cβ ] 2, (5.24) where F = x δ logx. (1+λx δ ) From equaton (5.7), we obtan ] [(ψ(a a = n + b)) 2 Γ (a + b) 2 Γ(a + b) (ψ(a))2 + Γ (a), (5.25) Γ(a) and [(ψ(a a b = n + b)) 2 Γ =1 (a + b) Γ(a + b) ], (5.26) a c = β log(1 + λx δ ). (5.27) From equaton (5.8), we obtan ] [(ψ(a a = n + b)) 2 Γ (a + b) 2 Γ(a + b) (ψ(b))2 + Γ (b), (5.28) Γ(b)

39 29 and b c = =1 β(1 + λx δ ) cβ log(1 + λx δ [1 (1 + λx δ ) cβ ] ). (5.29) From equaton (5.9), we obtan c 2 = n c 2 + β(b 1) =1 β(1 + λx δ ) cβ [log(1 + λx δ )] 2. (5.30) [1 (1 + λx δ ) cβ ] 2 Fsher nformaton matrx for the Mc-Dagum dstrbuton s: I λλ I λβ I λδ I λa I λb I λc I βλ I ββ I βδ I βa I βb I βc I I(θ) = I(λ, β, δ, a, b, c) = δλ I δβ I δδ I δa I δb I δc I aλ I aβ I aδ I aa I ab I ac I bλ I bβ I bδ I ba I bb I bc I cλ I cβ I cδ I ca I cb I cc [ [ ],...,I λ 2 cc = E L ]. c 2 where, I λλ = E. (5.31) The elements of the 6 X 6 matrx I(λ, β, δ, a, b, c) can be approxmated by the elements of the nformaton matrx, where [ ] I j (θ) = E θ θ j 2 L θ θ j. (5.32) Applyng the usual large sample approxmaton, MLE of Θ, that s ˆΘ s approxmately N 6 (Θ, I 1 n (Θ)), where I n (Θ) s the 6X6 observed nformaton matrx. Under the regularty condtons and parameters n the nteror of the parameter space but not on the boundary, the asymptotc dstrbuton of n(( ˆΘ) Θ) s N 6 (Θ, I 1 (Θ)), where I(Θ) = lm n n 1 I n (Θ). Therefore, the approxmate 100(1-α)% two-sded confdence ntervals for λ, β, δ, a, b and c are gven by:

40 ˆλ ± Z α 2 â ± Z α 2 I 1 λλ (ˆθ), ˆβ ± Z α I 1 2 ββ (ˆθ), ˆδ ± Z α I 1 2 δδ (ˆθ), I 1 aa (ˆθ), ˆb ± Z α I 1 2 bb (ˆθ) and ĉ ± Z α I 1 2 cc (ˆθ), where, Z α 2 s the upper ( α 2 )th percentle of a standard normal dstrbuton Concludng Remarks In ths chapter, we presented log-lkelhood functon for the Mc-Dagum dstrbuton and obtaned partal dervatves wth respect to each parameter to estmate the model parameters. We notced that there are no closed form estmates of the parameters, so numercal methods must be appled. We also obtaned Fsher Informaton matrx; [ ] I j (θ) = E θ θ j 2 L θ θ j. (5.33) Fnally the approxmate confdence ntervals for each parameter was gven. 5.4 Future Works In the future, we wll nvestgate and obtan results on the Kumaraswamy-Dagum (Kum-Dagum) dstrbuton that was mentoned n the Chapter one. We wll also work on obtanng estmates of model parameters from the Bayesan vewpont for both Mc-Dagum and Kum-Dagum dstrbutons and conduct goodness-of-ft tests for these models.

41 31 BIBLIOGRAPHY [1] Cordero, G.M., and Castro,M., A New Famly of Dstrbutons, Journal of Statstcal Computaton and Smulaton, Vol.00, No.00, August (2009), [2] Cordero, G.M., Cnta,R.J., Rego,L.C., and Ortega,E.M.M., The McDonald Normal Dstrbuton, Statstcs n the Twenty-Frst Century: Specal Volume, PJ- SOR, Vol.8, No.3, pages , July (2012) [3] Dagum, C., A New Model of Personal Income Dstrbuton, Economc Applquee, 30, , (1977). [4] Dagum, C., The generaton and dstrbuton of ncome, the Lorenz curve and the Gn rato, Economc Applquee, XXXIII, pp , (1980). [5] Domma, F., and Condno, F., The Beta-Dagum Dstrbuton; defnton and Propertes,Communcaton n Statstcs-Theory and Methods,n press (2013). [6] Domma, F., and Gordano,S., The Fsher Informaton Matrx In Rght Censored Data From The Dagum Dstrbuton, Department of Economcs and Statstcs, Unversty of Calabra, workng paper n.04-(2011). [7] Gradshteyn,I.S., and Ryzhk,I.M., Table of Tntegral, Seres and Products, Seventh Edton,Elsever Inc. [8] Kleber, C., A Gude to the Dagum Dstrbutons, A publcaton of the Center of Busness and Economcs (WWZ), Unversty of Basel. Dec.04,(2004). [9] Kleber, C., Unversty of Dortmund, Germany, On the Lorenz Order wthn Parametrc Famles of Income Dstrbutons, Sankhya: The Indan Journal of Statstcs, Volume 61, Seres B, Pt.3, pp (1999). [10] Kumaraswamy, P., Generalzed probablty densty functon for double-bounded random process, Journal of Hydrology, 46,79-88, (1980) [11] MALA I.,(CZ), Dstrbuton Of Income Per Capta Of The Czech Households From 2005 To 2008, Journal of Appled Statstcs, volume IV (2011),number III.

42 [12] Marcano,F.W.P.,Nascmento,A.D.C., Santos-Neto, M. and Cordero,G.M., The Mc-Γ Dstrbuton and Its Statstcal Propertes:An Applcaton to relablty Data, Internatonal Journal of Statstcs and Probablty, Vol 1, No 1 (2012). 32 [13] Reny, A., On measures of Entropy and Informaton, Berkeley Symposum on Mathematcal Statstcs and Probablty :Unversty of Calforna Press.1(1), (1961).