Estimation of Parameters in Weighted Generalized Beta Distributions of the Second Kind

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1 Journl of Sttisticl nd Econometric Methods, vol.1, no.1, 2012, 1-12 ISSN: (print), (online) Interntionl Scientific Press, 2012 Estimtion of Prmeters in Weighted Generlized Bet Distributions of the Second Kind Yun Ye 1, Broderick O. Oluyede 2 nd Mvis Prri 3 Abstrct This pper pplies the clss of weighted generlized bet distribution of the second kind (WGB2) s descriptive models for size distribution of income. The properties of WGB2 including men, vrince, coefficient of vrition(cv), coefficient of skewness(cs), coefficient of kurtosis(ck) re presented. Other properties including top-sensitive index, bottom-sensitive index, men logrithmic devition(mld) index nd Theil index obtined from generlized entropy(ge) re pplied in this pper. WGB2 proved to be in the generlized bet-f fmily of distributions, nd mximum likelihood estimtion(mle) is used to obtin the prmeter estimtes. WGB2 is fitted to U.S. fmily income ( ) dt with different vlues of the prmeters. The empiricl results show the length-bised distribution provides the best reltive fit. Mthemtics Subject Clssifiction : 62F10 Keywords: WGB2, Income distribution, Entropy, Bet-F fmily, MLE 1 Introduction Generlized bet distribution of the second kind (GB2) hs been widely used in income distribution. It provides good description of income distri- 1 Deprtment of Mthemticl Sciences, Georgi Southern University, Sttesboro, GA 30460, e-mil: yy00053@georgisouthern.edu 2 Deprtment of Mthemticl Sciences, Georgi Southern University, Sttesboro, GA 30460, e-mil: boluyede@georgisouthern.edu 3 Deprtment of Mthemtics, Indin University of Pennsylvni, Indin, PA 15705, e-mil: prrim@iup.edu Article Info: Received : November 29, Revised : December 26, 2011 Published online : Februry 28, 2012

2 14 Estimtion of Prmeters in WGB2 bution nd cptures the chrcteristics of size distribution of income such s: skewness, hs pek in low-middle rnge, nd long right hnd til [7]. Mc- Donld [7] dopted different distributions s models for size distribution of income, nd found tht GB2 provides the best reltive fit. Weighted distribution provides n pproch to deling with model specifiction nd dt interprettion problems. Fisher [3] nd Ro [12] introduced nd unified the concept of weighted distribution. Cox [1] nd Zelen [17] used it to present length bised smpling. The usefulness nd pplictions of weighted distribution to bised smples in vrious res including medicine, ecology, relibility, nd brnching processes cn lso be seen in Nnd nd Jin [9], Gupt nd Keting [2], Oluyede [10] nd in references therein. Suppose Y is non-negtive rndom vrible with its nturl pdf f(y; θ), θ is prmeter, then the pdf of the weighted rndom vrible Y w is given by: f w (y; θ, β) = w(y, β)f(y; θ), (1) ω where the weight function w(y, β) is non-negtive function, tht my depend on the prmeter β, nd 0 < ω = E(w(Y, β)) < is normlizing constnt. This pper pplies the clss of WGB2 s descriptive models for the size distribution of income. The properties of WGB2 such s moments nd generlized entropy (GE) re presented in Section 2. Section 3 pplies WGB2 in the size distribution of income. WGB2 is fitted to U.S. fmily income dt ( ) with different vlues of the prmeter k in Section 4. The prmeter estimtes of WGB2 re obtined nd empiricl results re presented. Section 5 contins n ppliction of the results to US fmily nominl income for the yers 2001 to The Distribution nd Specil Cses Weighted generlized bet distribution of the second kind (WGB2) with polynomil weight function w(y) = y k is very flexible five-prmeter distribution. The probbility density function (pdf) of WGB2 is given by: g W GB2 (y;, b, p, q, k) = y p+k 1 b p+k B(p + k, q k)[1 + ( y, (2) b ) ] p+q where y > 0,, b, p, q > 0 nd p < k < q. WGB2 includes GB2 s specil cse, it lso includes severl other weighted distributions s specil or limiting cses: WGG (weighted generlized gmm), WB2 (weighted bet of the second kind), WSM (weighted Singh Mddl),

3 Yun Ye, Broderick O. Oluyede nd Mvis Prri 15 Figure 1: Grph Tree WD (weighted Dgum), WG (weighted gmm), WW (weighted Weibull) nd WE (weighted exponentil). The distribution tree is given below: Jones nd Fddy [5] introduced the concept of generlized bet-f distribution: g F (y; α, β) = B(α, β) 1 f(y)[f (y)] α 1 [1 F (y)] β 1, where f(y) is the derivtive of F (y), B(α, β) = Γ(α)Γ(β) is the bet function. Γ(α+β) Sepnski nd Kong [14] pplied the generlized bet-f fmily in size distribution of income nd obtined the prmeter estimtes. They concluded tht log-f followed by GB2 provides the best reltive fit. 3 Relted Results From our previous work, we obtin the moments of WGB2 with weight function w(y) = y k, tht is E GW GB2 (Y j ) = bj B(p + k + j, q k j ) B(p + k, q k ). (3) The corresponding men nd vrince re given by µ GW GB2 = E GW GB2 (Y ) = bb(p + k + 1, q k 1 ) B(p + k, q k ), (4)

4 16 Estimtion of Prmeters in WGB2 nd nd where V r GW GB2 (Y ) = b 2 [ B(p + k+2, q k+2 ) B(p + k, q k ) ( p + k+1, q k+1 p + k, q k ) 2 ]. (5) respectively. The coefficient of vrition (CV) is given by B(p + k+2 CV W GB2 =, q k+2)b(p + k, q k) B 2 (p + k+1, q k+1) 1. (6) Similrly, the coefficient of skewness nd coefficient of kurtosis re µ = µ GW GB2, σ = CS = E[Y 3 ] 3µE[Y 2 ] + 2µ 3 σ 3, (7) CK = E[Y 3 ] 4µE[Y 3 ] + 6µ 2 E[Y 2 ] 3µ 4 σ 4, (8) V r GW GB2 (Y ), E[Y 2 ] = b2 B(p + k + 2, q k 2 ) B(p + k, q k ), E[Y 3 ] = b3 B(p + k + 3, q k 3 ) B(p + k, q k ), nd E[Y 4 ] = b4 B(p + k + 4, q k 4 ) B(p + k, q k ). The generlized entropy (GE) is widely used to mesure inequlity trends nd differences. It is primrily used in income distribution. Generlized entropy I(α) for WGB2 is given by: I(α) = B(p + k + α, q k α )B α (p + k + 1, q k 1) B1 α (p + k, q k) α(α 1)B 1 α (p + k, q k), (9) where α 0 nd α 1. The bottom-sensitive index is I( 1), nd the topsensitive index is I(2). Moreover, the men logrithmic devition (MLD) index nd Theil index re: nd k+1 B(p + I(0) = log, q k+1) B(p + k, q k) ψ(p + k) I(1) = respectively. k+1 ψ(p + ) ψ(q k), (10) k+1 ψ(q ) k+1 B(p + log, q k+1) B(p + k, q k). (11)

5 Yun Ye, Broderick O. Oluyede nd Mvis Prri 17 4 Estimtion of Prmeters For WGB2 with weight function w(y) = y k, the pdf cn be written s: g w (y;, b, p, q) = B 1 ( p + k, q k )( b )[( ) ] p+ k y b [ 1 + ( y b ) ] (p+q). If we set F (y) = 1 (1 + ( y b ) ) 1, then f(y) = ( y b b ) 1 [1 + ( y b ) ] 2 nd ( g w (y;, b, p, q) = B 1 p + k, q k ) f(y)[f (y)] p+ k 1 [1 F (y)] q k 1. (12) Clerly, this distribution belongs to the bet F clss of distributions with [ ( ) ] 1 y F (y) = = y b b + y. (13) Let θ = (, b, p, q) T be column vector of prmeters ssocited with the income distribution. The income distribution is given by: ( P i (θ) = B 1 p + k, q k ) F (yi ) t p+ k 1 (1 t) q k 1 dt, (14) F (y i 1 ) where P i (θ) denotes the estimted proportion of the popultion in the i th intervl of the r income groups defined by the intervl I i = [y i 1, y i ]. The multinomil likelihood function for the dt is given by: N! r [P i (θ)] n i, n i! i=1 where n i, i = 1, 2,..., r denotes the observed frequency in the i th group nd N = r i=1 n i. We mximize: L(θ) = r n i lnp i (θ), i=1 where P i (θ) = F (y i ) F (y i 1 ) h(t)dt, nd h(t) = B 1 (p + k, q k )tp+ k 1 (1 t) q k 1. Sepnski nd Kong [14] pointed out tht obtining P i by computing the cdf of bet rndom t F (y i 1 ) nd F (y i ) cn reduce the complexity of progrmming required to clculte the integrtions. The first derivtive with respect to θ = (, b, p, q) T re: dl(θ) dθ = r i=1 n i P i (θ) dp i(θ) dθ. (15)

6 18 Estimtion of Prmeters in WGB2 The prtil derivtive equtions of P i (θ) with respect to, b, p, q re given by: P i (θ) = h(f (y i )) b yi (lny i lnb) h(f (y (b + yi i 1 )) b yi 1(lny i 1 lnb) )2 (b + yi 1 )2 F (yi ) [ ( kh(t) + Ψ p + k ) ( Ψ q k ) + ln 1 t ] dt, (16) F (y i 1 ) 2 t P i (θ) b [ h(f = b 1 (yi ))yi (b + y h(f (y ] i 1))yi 1, (17) i )2 (b + yi 1 )2 P i (θ) p = F (yi ) F (y i 1 ) [ ( h(t) Ψ p + k ) ] + Ψ(p + q) + lnt dt, (18) nd P i (θ) q = F (yi ) F (y i 1 ) [ ( h(t) Ψ q k ) ] + Ψ(p + q) + ln(1 t) dt, (19) where Ψ(x) = d [Γ(x)]. Using the equtions (15)-(18) in eqution (14), we cn dx obtin the grdient functions of L(θ) with respect to prmeters, b, p, q. The prtil derivtive eqution (15) exists when k > 0. If k = 0, the prtil derivtive eqution of P i with respect to is given by: P i (θ) [ h(f = b (yi ))(y i ) (ln(y i ) ln(b)) h(f (y ] i 1))(y i 1 ) (ln(y i 1 ) ln(b)).(20) (b + yi )2 (b + yi 1 )2 5 Applictions In this section, we obtin prmeter estimtes bsed on our previous discussions nd results. WGB2 ws fitted to U.S. fmily nominl income for The groups consist of fmilies whose income re in the corresponding income intervl I i = [y i 1, y i ), the n i /N re the observed reltive frequencies (N = n i ). The common wy to obtin prmeter estimtes is to mximize the multinomil likelihood function. Since the likelihood function is nonliner nd complicted, we use MATLAB to serch for the mximum vlue of multinomil

7 Yun Ye, Broderick O. Oluyede nd Mvis Prri 19 likelihood function. 5 The results of this estimtion for 2001, 2005, nd 2009 re reported in Tbles 2-4. Bsed on the sum of squres error (sse) vlue we cn conclude tht: the length-bised WGB2 (k = 1) provides better fit thn GB2 (k = 0) nd other WGB2 (k = 2, 3, 4). If we plug the estimted prmeters in the prtil derivtive equtions in Section 4, we obtin the vlues of these prtil derivtive equtions in Tble 5-7. From the tbles, we find tht these vlues re close to zero or very smll, this mens tht the estimted prmeters tht we obtined re precise nd effective. Since we hve lredy obtined estimtes of prmeters for WGB2 in Section 4, nd found tht the length-bised WGB2 provides the best fit to income distribution, we cn pply these estimted prmeters from length-bised WGB2 model to obtin the estimtes of the men, vrince, coefficient of vrition, skewness nd kurtosis, bottom sensitive index, top-sensitive index, MLD index nd Theil index. The results re presented in Tble 8. 6 Concluding Remrks In this pper, the weighted generlized bet distribution of the second kind (WGB2) ws fitted to U.S. fmily income dt ( ). The mximum likelihood estimtion (MLE) is used for estimting the prmeters of the income distribution model. The results showed tht the length-bised WGB2 provides the best reltive fit to income dt. Bsed on previously obtined descriptive mesures for WGB2, we estimte the men, vrince, coefficient of vrition, coefficient of skewness, coefficient of kurtosis, bottom-sensitive index, top-sensitive index, MLD index nd Theil index of the income dt. ACKNOWLEDGEMENTS. The uthors wish to express their grtitude to the referees nd editor for their vluble comments. 4 The dt were tken from the Census Popultion Report 5 By setting different initil vlues nd using fminserchbnd to serch for the mximum log likelihood vlues

8 20 Estimtion of Prmeters in WGB2 References [1] D. R. Cox, Renewl Theory, Brnes & Noble, New York, [2] R.C. Gupt nd J.P. Keting, Reltions for Relibility Mesures Under Length Bised Smpling, Scndinvin Journl of Sttistics, 13(1), (1985), [3] R.A. Fisher, The Effects of Methods of Ascertinment upon the Estimtion of Frequencies, Annls of Humn Genetics, 6(1), (1934), [4] S.P. Jenkins, Inequlity nd GB2 Income Distribution, ECINQE, Society for study of Economic Inequlity, Working Pper Series, (2007). [5] M.C. Jones nd M.J. Fddy, A Skew Extension of the t distribution, with Applictions, Journl of the Royl Sttisticl Society. Series B, 65(1), (2003), [6] C. Kleiber nd S. Kotz, Sttisticl Size Distributions in Economics nd Acturil Sciences, Wiley, New York, [7] J.B. McDonld, Some Generlized Functions for the Size Distribution of Income, Econometric, 52(3), (1984), [8] J.B. McDonld nd Y.J. Xu, A Generliztion of the Bet Distribution with Appliction, Journl of Econometrics, 69(2), (1995), [9] K.A. Nnd nd K. Jin, Some Weighted Distribution Results on Univrite nd Bivrite Cses, Journl of Sttisticl Plnning nd Inference, 77(2), (1999), [10] B.O. Oluyede, On Inequlities nd Selection of Experiments for Length- Bised Distributions, Probbility in the Engineering nd Informtionl Sciences, 13(2), (1999), [11] G.P. Ptil nd R. Ro, Weighted Distributions nd Size-Bised Smpling with Applictions to Wildlife nd Humn Fmilies, Biometrics, 34(6), (1978), [12] C.R. Ro, On Discrete Distributions Arising out of Methods of Ascertinment, The Indin Journl of Sttistics, 27(2), (1965), [13] A. Renyi, On Mesures of Entropy nd Informtion, In Proceedings of the Fourth Berkeley Symposium on Mthemtics, Sttistics nd Probbility, 1(1960), University of Cliforni Press, Berkeley, (1961),

9 Yun Ye, Broderick O. Oluyede nd Mvis Prri 21 [14] J.H. Sepnski nd L. Kong, A Fmily of Generlized Bet Distribution For Income, Interntionl Business, 1(10), (2007), [15] C.E. Shnnon, A Mthemticl Theory of Communiction, The Bell System Technicl Journl, 27(7), (1948), [16] C.E. Shnnon, A Mthemticl Theory of Communiction, The Bell System Technicl Journl, 27(10), (1948), [17] M. Zelen, Problems in Cell Kinetics nd Erly Detection of Disese, Relibility nd Biometry, 56(3), (1974), pp

10 22 Estimtion of Prmeters in WGB2 Tble 1: U.S. fmily nominl income for [y i 1, y i ) observed reltive frequencies n i /N (thousnd) [0,15) [15,25) [25,35) [35,50) [50,75) [75,100) [100,150) [150,200) [200, )

11 Yun Ye, Broderick O. Oluyede nd Mvis Prri 23 Tble 2: Estimted prmeters of WGB2 for income distribution (2001) k = 0 k = 1 k = 2 k = 3 k = b p q sse* Tble 3: Estimted prmeters of WGB2 for income distribution (2005) k = 0 k = 1 k = 2 k = 3 k = b p q sse* Tble 4: Estimted prmeters of WGB2 for income distribution (2009) k = 0 k = 1 k = 2 k = 3 k = b p q sse* Tble 5: Vlues of prtil derivtive equtions of WGB2 (2009) k = 0 k = 1 k = 2 k = 3 k = b p q

12 24 Estimtion of Prmeters in WGB2 Tble 6: Vlues of prtil derivtive equtions of WGB2 for (2005) k = 0 k = 1 k = 2 k = 3 k = b p q Tble 7: Vlues of prtil derivtive equtions of WGB2 (2001) k = 0 k = 1 k = 2 k = 3 k = b p q Tble 8: Estimted sttistics for income distribution (k = 1 in WGB2) Yer Est.men Est.Vr Est.CV Est.CS Est.CK Yer Est.I(-1) Est.I(2) Est.MLE Est.Theil