ON SIZE BIASED KUMARASWAMY DISTRIBUTION

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1 ON SIZE BIASED KUMARASWAMY DISTRIBUTION rxiv: v [stt.me] 9 Sep 06 Dremlee Shrm, * nd Tpn Kumr Chkrbrty Deprtment of Sttistics, North-Estern Hill University, Shillong , Meghly, Indi Contct: *dremleeshrm@yhoo.in, tpnkumrchkrbrty@gmil.com September 30, 06 Abstrct In this pper, we introduce nd study the size-bised form of Kumrswmy distribution. The Kumrswmy distribution which hs drwn considerble ttention in hydrology nd relted res ws proposed by Kumrswmy [7]. The new distribution is derived under sizebised probbility of smpling tking the weights s the vrite vlues. Vrious distributionl nd chrcterizing properties of the model re studied. The methods of mximum likelihood nd mtching quntiles estimtion re employed to estimte the prmeters of the proposed model. Finlly, we pply the proposed model to simulted nd rel dt sets. Key Words: Kumrswmy distribution; size-bised distribution; quntile function; regulrized bet function. AMS 00 subject clssifictions: 60E05, 6F0 Introduction The concept of weighted distribution ws first introduced by Fisher [4] to model scertinment bis, nd ws lter formlized in unifying theory by Ro [3]. Let X be rndom vrible of interest such tht X fx; θ), where θ is vector of prmeters. Under equl probbility smpling, the estimtion of the prmeter θ cn be mde with n bundnce of methods. However, under size-bised schemes, the probbility of smpling n individul is proportionl to X r provided tht E θ X r ) < for ll θ. In situtions like this, the weighted

2 probbility density function is defined s f r x, θ) = xr fx, θ) µ r ) where µ r = x r fx; θ)dx in plce of fx; θ) cn be used. The weighted distributions hve vrieties of uses in vrious fields. A number of ppers hve ppered implicitly using the concepts of weighted nd size-bised smpling distributions. Ptil nd Ro [] hve briefly surveyed the pplictions of weighted nd size-bised distributions. Size-bised distributions rise nturlly in rnge of smpling nd modeling problems in forestry [6]. They lso occur in pplictions spnning domins including environmentl sciences, econometrics, humn demogrphy nd biomedicl sciences [, 6]. To hve n ide of their pplictions, one cn refer to, [, 3, 8, 9, 4, 7, 8, 9]. When the probbility of observing positive-vlued rndom vrible is proportionl to the vlue of the vrible the resultnt is size-bised distribution. Size-bised distributions of order is specil cse of the weighted distribution defined in ) with weight s x. In this pper, the term size-bised distribution will be used to indicte the size-bised distribution of order. Thus tking r =, in ) we obtin the size bised distribution which is given by the p.d.f. gx, θ) = xfx, θ) µ ) The Size Bised Kumrswmy Distribution The Kumrswmy distribution [7] is similr to the Bet distribution, but much simpler to use especilly in simultion studies due to the simple closed form of both its probbility density function nd cumultive distribution function. This distribution is minly used for vribles tht re lower nd upper bounded. The probbility density function pdf) of the Kumrswmy distribution Kum) is given by gx;, b) = bx x ) b, for o < x < = 0, otherwise 3)

3 where, > 0 nd b > 0 re the two shpe prmeters. The r th order rw moment of the Kum is given by µ r = bb + r, b) where B + r, b) is bet function defined by the integrl Bα, β) = 0 x α x) β dx Thus, the expecttion of the Kum is given by µ = bb +, b) = µ sy) 4) Thus using the reltion ) nd 4), the pdf of the SBKD is obtined s: fx;, b) = x x ) b B +, b) ; 0 < x < 5) Ducey nd Gove [3] hve obtined the weighted distribution of the Generlized Bet I GBI), the Generlized Bet II GBII) nd the Generlized Gmm GG) distributions nd hve shown tht the GBI, the GBII, the GG distributions re form invrint under size bised scheme. The Kumrswmy distribution is distribution in the GBIα, β, p, q) fmily of distributions [0]. So tht the SBKD is lso specil cse of the GBI distribution for α = > 0, β =, p = +, q = b.. Specil Cses. Tking = in 5) we get, fx; b) = B, b) x x) b ; 0 < x < Thus the SBKD reduces to Bet-I distribution with prmeters nd b.. Tking b = in 5) we get fx; ) = + )x ; 0 < x < 3. Tking = nd b = in 5), the SBKD reduces to specil cse of the Tringulr distribution fx) = x; 0 < x < 3

4 . Shpe of the distribution The SBKD is Bet, b) distribution for =. Hence, for ny b nd fixed =, the distributionl shpe of SBKD will be like tht of Bet, b) distribution. therefore for =, the following shpes will be obtined.. We know, Bet I distribution is lwys symmetric if both the prmeters re equl. Hence for =, the SBKD is symmetric if b =.. We know the Bet, ) distribution is the Right-Tringulr distribution with right ngle t the right end, t x = nd is stright line with slope +. Hence the SBKD, ) is lso Right Tringulr distribution. 3. For Bet, b < ), the Bet distribution is negtively skewed J-shped curve. Hence the SBKD, b < ) is J-shped negtively skewed curve. 4. The Bet, b) is unimodl nd positively skewed for b > nd negtively skewed for < b < nd hence the SBKD, b) is lso positively skewed for b > nd negtively skewed for < b <. Figure gives plot of the possible shpes of the distribution for =. Figure : Density plot of SBKD, b) for vrious vlues of b 4

5 Some of the possible density plots of the SBKD for < nd > is given respectively in Figure nd 3. Figure : Density plot of SBKD for < The possible shpes of the SBKD for < nd > is discussed below: For <, b <, the SBKD hs J-shped negtively skewed density. For <, b =, the SBKD hs n incresing density. For <, b >, the SBKD hs either unimodl positively skewed density or reverse J-shped positively skewed decresing density. For >, b <, the SBKD hs J-shped negtively skewed density. For >, b =, the SBKD hs negtively skewed incresing density. For >, b > the SBKD hs unimodl skewed density. 5

6 Figure 3: Density plot of SBKD for > 3 Properties of the Size-Bised Kumrswmy Distribution 3. Cumultive distribution function of SBKD Theorem. Let X SBKD, b), then its cumultive distribution function c.d.f.)is given by 6) F x) = I x + ), b 6) where, +, b ) = B x ; +, b) I x B +, b) is the regulrized incomplete bet function nd is defined s the rtio of n incomplete bet function, Bz; α, β) = z 0 xα x) β dx nd the complete bet function, Bα, β). Proof. As X SBKD, b) so its p.d.f. is given by 5). Let F x) be the c.d.f. of SBKD then by definition, F x) = x 0 fy)dy; 0 < x < 6

7 Thus substituting fy) from 5) we hve x F x) = B +, b) 0 = I x + ), b y y ) b dy 3. Quntile function of SBKD Theorem. Let X SBKD, b), then its quntile function is given by 7) [ Qp) = I p + )], b 7) where, I p α, β) is the inverse regulrized bet function defined s I p α, β) = w such tht I w α, β) = p Proof. Let F x) = p be c.d.f. function, Qp) is defined s then the corresponding quntile Qp) = F p) 8) Therefore by using the reltion 6) nd 8) the quntile function of the SBKD is [ Qp) = I p + )], b Corollry.. The medin of the SBKD is Q0.5) = [ I )], b 3.. Rndom number genertion Using the quntile function of the SBKD s defined in 7), rndom smple of size n cn be simulted. Let U be uniform U0, )) r.v. nd let Qp), 0 p be the quntile function of SBKD, then by uniform trnsformtion rule, [5] the vrible X, where x = Qu), hs distribution with quntile function Qp). Thus, by using the uniform trnsformtion rule, rndom smple of size n cn be esily simulted from the SBKD by generting rndom smple of the sme size from U0, ) distribution. 7

8 3.3 Moment generting function of SBKD Theorem 3. Let X SBKD, b) then the moment generting function, M X t) of X is given by M X t) = i=0 t i B + i+, b) i! B + 9), b) Proof. By definition, the moment generting function m.g.f. of r.v. X is given by M X t) = Ee tx ) = Thus, for SBKD, the m.g.f. is e tx fx)dx M X t) = B +, b) e tx x x ) b dx 0 = B +, b) + tx + t x + t3 x tn x n 0! 3! n! ) + i+, b = i=0 t i B i! B +, b) ) +... x x ) b dx Corollry 3.. The cumulnt generting function, K X t) of the SBKD is given by t i B + i+ K X t) = ln, b) i! B +, b) i=0 Corollry 3.. The r th order rw moment of SBKD is µ r = B + r+, b) B + 0), b) Corollry 3.3. The men i.e. the st order rw moment of SBKD is µ = B +, b) B + ), b) 3.4 Moments of SBKD Theorem 4. Let X SBKD, b), then the r th order rw moment µ r nd centrl moment µ r re defined respectively by ) nd 3) µ r = B + r+, b) B + ), b) 8

9 [ B µ r = B +, b) + r + + ) k r C k µ k B ), b rµb + r ), b + + r k +, b rr ) µ B + r ) ) r µ r B + )], b where, µ is the men of the SBKD nd is given by ). ), b... 3) Proof. The proof for ) follows directly from the Corollry 3.. Now, the r th order centrl moment is defined s µ r = E[X EX)) r ] = x EX)) r fx)dx 4) The EX) or the first order rw moment of the SBKD is given by ). Let this be denoted by µ. Thus using the reltion 5), ) nd 4), the r th order centrl moment of the SBKD is obtined s µ r = B +, b) x µ) r x x ) b dx = B +, b) 0 [ = B +, b) B + r + ), b rµb + r ), b rr ) + µ B + r + ) k r C k µ k B + r k + ), b ) r µ r B + )], b [ x r r C x r µ ) k r C k x r k µ k ) r µ r] x x ) b dx ), b... Corollry 4.. The first four centrl moments re µ = 0 µ = B + 3, b) [ B + B +, b), b) ] B +, b) µ 3 = B + 4, b) B +, b) 3 B +, b) B + 3, b) [ B + B +, b) +, b) ] 3 B +, b) µ 4 = B + 5, b) B +, b) 4 B +, b) B + 4, b) B +, b) + 6 B +, b) B + 3, b) [ B + B +, b) 3, b) ] 4 3 B +, b) 3.5 Skewness nd kurtosis of SBKD The skewness of the SBKD is given by Men Medin S k = 3 µ = 3 B +, b) [ I 0.5 +, b)] B +, b) [ B + 3, b) B +, b) B +, b) ] 9

10 The kurtosis of the SBKD is given by β = µ 4 µ = B + 5, b) 4µB + 4, b) + 6µ B + 3, b) 3µ 4 B +, b) B+ 3,b) B+,b) µ B + 3, b) + µ 4 B +, b) where, µ = B +, b) B +, b) 3.6 Hrmonic men of SBKD Theorem 5. Let X SBKD, b), then the hrmonic men of X is given by H.M. = b B + ), b Proof. The hrmonic men of r.v X is given s H.M. = Thus for SBKD, the H.M. is H.M. = B +, b) 0 = b B +, b) or H.M. = b B + ), b x fx)dx x x x ) b dx 3.7 The survivl nd hzrd function The survivl function of SBKD is given by St) = F t) = I t + ), b 5) The hzrd function of the SBKD is given by ht) = ft) St) = t t ) b B +, b) B t ; +, b) 0

11 4 Prmeter estimtion of SBKD 4. Method of mximum likelihood estimtion The method of mximum likelihood estimtion MLE) selects the set of vlues of the model prmeters tht mximizes the likelihood function. By definition of the method of mximum likelihood estimtion, it is required to first specify the joint density function for ll observtions. For rndom smple of size n from SBKD, the likelihood function is given by L = n x i x i ) b i= or equivlently, ) n x i x i ) b lnl) = ln B +, b) i= = n ln) + i B +, b) lnx i ) + b ) i ln x i ) n ln B + ), b 6) To obtin the MLE of the SBKD, 6) is differentited w.r.t. nd b nd then equted to 0. Hence the likelihood equtions re n â + lnx i ) â b ) i i [ ln x iâ) n ψb) ψ i x â i x â + ṋ i + )] â + b = 0 [ ψ + ) ψ + )] â â + b = 0 7) where, ψ.) is the digmm function given by the logrithmic derivtive of the gmm function. The set of equtions 7) cn be solved by using numericl methods. 4. Method of quntile mtching estimtion The method of mtching quntiles, n itertive procedure bsed on the ordinry lest squres estimtion OLS) computes mtching quntile estimtion MQE). The method of mtching quntiles is bsed on mtching theoreticl quntiles of the prmetric distribution ginst the empiricl quntiles for specified probbilities, [5]. The bsic ide is to mtch the distribution of totl counterprt portfolio by tht of selected portfolio. We choose the representtive portfolio to minimize the men squred difference between the quntiles of the two

12 distributions cross ll levels. This leds to the mtching quntiles estimtion MQE). If Qp is the p th smple quntile, then the equlity of theoreticl nd empiricl quntiles is expressed by Qp k ; θ) = Q pk for k =,,..., d with d, the number of prmeters to be estimted. The MQE is vilble in the r pckge, fitdistrplus []. A numericl optimiztion is crried out to minimize the sum of squred differences between observed nd theoreticl quntiles. Thus, using the R-pckge, fitdistrplus the MQE of the SBKD cn be obtined. 5 Appliction to dt 5. Simultion study It hs been discussed under Subsection 3. tht rndom smple of size n cn be generted from SBKD using its quntile function. In this section some rndom smples with known prmeters hve been generted nd the smples hve been fitted to SBKD, Kumrswmy distribution nd Bet distribution respectively, by using the method of mximum likelihood estimtion. The R pckge fitdistrplus hs been used to obtin the MLE for the 3 distributions. The result obtined is summrized in Tble.

13 SBKD, b) Distribution Estimte Estimte Log-likelihood AIC BIC SBKD = b = Kum Bet SBKD =.3, b = 0.75 Kum Bet SBKD = 3, b = Kum Bet SBKD = 0.65, b =.6 Kum Bet Tble : MLE of the simulted dtsets for SBKD, Kumrswmy nd Bet distributions Tble clerly shows tht, in cse of simulted dt from SBKD, the estimtes re more closer to the ctul vlues. The SBKD lso gives mrginlly better fit thn the Kum nd the Bet distribution in terms of the log-likelihood function. This is quite obvious s becuse the smple hs been drwn from the SBKD. Figure 4 gives plot the stndrd error of the estimtes, â nd b of the simulted smples for incresing smple size. Figure 4: Stndrd error of the estimtes of nd b for n incresing smple size 3

14 Figure 4 indictes tht for ll the simulted smples the stndrd error of the estimtes decreses with incresing smple size. Hence the method of estimtions s discussed in Section 4) cn be prcticlly used to fit some rel life dt. 5. Fitting to rel life dt In this section tensile strength dt hs been fitted to the size bised Kumrswmy distribution by the method of MLE nd MQE. The dt is vilble in the R pckge gmlss.dt nd it contins the mesurements of tensile strength of 30 polyester fibres. R pckge fitdistrplus hs been used to obtin both the MLE nd MQE. The bove dt fitted to the SBKD by the method of MLE nd MQE is shown respectively in Figure 5 nd 6. Figure 5: Tensile strength dt fitted to SBKD by the method of MLE 4

15 Figure 6: Tensile strength dt fitted to SBKD by the method of MQE The tensile dt hs lso been fitted to the Kumrswmy distribution nd the bet distribution by the corresponding methods nd the respective log-likelihood functions, the Akike Informtion Criteri AIC) nd the Byesin Informtion Criteri BIC) hve been obtined. The results obtined for the three distributions, viz., SBKD, Kum nd Bet hve been summrized in Tble. Method used Distribution Estimte Estimte Log-likelihood AIC BIC SBKD MLE Kum Bet SBKD MQE Kum Bet Tble : Summry of the fitted dtsets for SBKD, Kumrswmy nd Bet distributions Tble clerly shows tht in terms of the log-likelihood, the SBKD gives mrginlly better fit to the tensile strength dt s compred to the Kum nd Bet. 6 Conclusions We hve proposed size-bised version of Kumrswmy distribution which cn be employed in modeling dt from hydrology, forestry nd vrious other relted fields. Specil cses of the SBKD hve been 5

16 discussed. The structurl properties including cumultive distribution function, the Quntile function, moments, nd shpe of the model for vrying vlues of the prmeters hve been discussed nd derived. Two methods for estimtion of the prmeters of the model viz, MLE nd MQE ws studied. Using simulted dt we hve shown tht the methods cn provide resonbly good estimtes of the prmeters; it ws shown tht the stndrd devitions of the estimtes decrese with increse in the smple size. The model hs been pplied to rel dtset which is indictive of potentilly better cndidte thn either bet or Kumrswmy distribution in terms of greter likelihood. Acknowledgments: The first uthor cknowledges the Deprtment of Science nd Technology DST), Government of Indi for her finncil support through DST-INSPIRE fellowship with wrd no. IF References [] Delignette-Muller, M.L.; Dutng, C. fitdistrplus: An R pckge for fitting distributions. Journl of Sttisticl Softwre 05, 644), -34. [] Dennis, B.; Ptil, G. The gmm distribution nd weighted multimodl gmm distributions s models of popultion btidnce. Mthemticl Biosciences 984, 68), 87-. [3] Ducey, M.J.; Gove, H.G. Size-bised distributions in the generlized bet distribution fmily, with pplictions to forestry. Forestry 05, 88), [4] Fisher, R.A. The effects of methods of scertinment upon the estimtion of frequencies. Annls of Eugenics 934, 6), 3-5. [5] Gilchrist, W.G. Sttisticl modelling with quntile functions; Chpmn nd Hll: New York, 000; [6] Gove, J.H. Estimtion nd pplictions of size-bised distributions in forestry. In Modeling Forest Systems; Amro, A., Reed, D., Sores, P., Eds.; CABI Publishing: 003; pp. 0-. [7] Kumrswmy, P. A Generlized probbility density function for double-bounded rndom processes. Journl of Hydrology 980, 46), [8] Lppi, J.; Biley, R.L. Estimtion of dimeter increment function or other tree reltions using ngle-count smples. Forest Science 987, 333), [9] Mgnussen, S.; Eggermont, P.; Lricci, V.N. Recovering tree heights from irborne lser scnner dt. Forest Science 999, 453),

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