THEORETICAL PROPERTIES OF THE WEIGHTED FELLER-PARETO AND RELATED DISTRIBUTIONS

ASIAN JOURNAL OF MATHEMATICS AND APPLICATIONS Volume 204 Aricle ID ama073 2 pages ISSN 2307-7743 hp://scienceasia.asia THEORETICAL PROPERTIES OF THE WEIGHTED FELLER-PARETO AND RELATED DISTRIBUTIONS OLUSEYI ODUBOTE AND BRODERICK O. OLUYEDE Absrac. In his paper for he firs ime a new six-parameer class of disribuions called weighed Feller-Pareo WFP) and relaed family of disribuions is proposed. This new class of disribuions conains several oher Pareo-ype disribuions such as lengh-biased LB) Pareo weighed Pareo WP I II III and IV) and Pareo P I II III and IV) disribuions as special cases. The pdf cdf hazard and reverse hazard funcions monooniciy properies momens enropy measures including Renyi Shannon and s-enropies are derived.. Inroducion Pareo disribuions provide models for many applicaions in social naural and physical sciences and are relaed o many oher families of disribuion. A hierarchy of he Pareo disribuions has been esablished saring from he classical Pareo I) o Pareo IV) disribuions wih subsequen addiional parameers relaed o locaion shape and inequaliy. A general version of his family of disribuions is called he Pareo IV) disribuion. Pareo disribuion has applicaions in a wide variey of seings including clusers of Bose-Einsein condensae near absolue zero file size disribuion of inerne raffic ha uses he TCP proocol values of oil reserves in oil fields sandardized price reurns on individual socks o menion a few areas. Brazauskas 2002) deermined he exac form of he Fisher informaion marix for he Feller-Pareo disribuion. Rizzo 2009) developed a new approach o goodness of fi es for Pareo disribuions. Riabi e al. 200) obained enropy measures for he family of weighed Pareo-ype disribuions wih he he general weigh funcion wx; k i j) x k e x F i x) F j x) where F x) and F x) F x) are he cumulaive disribuion funcion cdf) and survival or reliabiliy funcion respecively... Some Basic Uiliy Noions. Suppose he disribuion of a coninuous random variable X has he parameer se { 2 n }. Le he probabiliy densiy funcion pdf) of X be given by fx; ). The cumulaive disribuion funcion cdf) of X is defined o be F x; ) x f; ) d. The hazard rae and reverse hazard rae funcions are given by hx; ) fx; ) and τx; F x; ) ) fx; ) respecively where F x; ) is he survival or reliabiliy funcion. The following useful funcions are applied in subsequen secions. The gam- F x; ) ma funcion is given by Γx) x e d. The firs and he second derivaive of he gamma funcion are given by: Γ x) 0 x log )e d and Γ x) x log ) 2 e d 0 0 200 Mahemaics Subjec Classificaion. 62E5. Key words and phrases. Weighed Feller-Pareo Disribuion Feller-Pareo disribuion Pareo Disribuion Momens Enropy. c 204 Science Asia

2 ODUBOTE AND OLUYEDE respecively. The digamma funcion is defined by Ψx) Γ x)/γx). The lower incomplee gamma funcion and upper incomplee gamma funcion are s x) x 0 s e d and Γs x) x s e d respecively. Le a b > 0 hen ) 0 x a x) b lnx) dx Γa)Γb) Ψa) Ψa + b)). Γa + b).2. Inroducion o Weighed Disribuions. Saisical applicaions of weighed disribuions especially o he analysis of daa relaing o human populaion and ecology can be found in Pail and Rao 978). To inroduce he concep of a weighed disribuion suppose X is a non-negaive random variable rv) wih is naural probabiliy densiy funcion pdf) fx; ) where he naural parameer is Ω Ω is he parameer space). Suppose a realizaion x of X under fx; ) eners he invesigaor s record wih probabiliy proporional o wx; ) so ha he recording weigh) funcion wx; ) is a non-negaive funcion wih he parameer represening he recording sighing) mechanism. Clearly he recorded x is no an observaion on X bu on he rv X w having a pdf wx )fx; ) 2) f w x; ) ω where ω is he normalizing facor obained o make he oal probabiliy equal o uniy by choosing 0 < ω EwX ) <. The random variable X w is called he weighed version of X and is disribuion is relaed o ha of X. The disribuion of X w is called he weighed disribuion wih weigh funcion w. Noe ha he weigh funcion wx ) need no lie beween zero and one and acually may exceed uniy. For example when wx; ) x in which case X X w is called he size-biased version of X. The disribuion of X is called he size-biased disribuion wih pdf f x; ) xfx;) where 0 < EX <. The pdf f is called he lengh-biased or size-biased version of f and he corresponding observaional mechanism is called lengh-biased or size-biased sampling. Weighed disribuions have seen much use as a ool in he selecion of appropriae models for observed daa drawn wihou a proper frame. In many siuaions he model given above is appropriae and he saisical problems ha arise are he deerminaion of a suiable weigh funcion wx; ) and drawing inferences on. Appropriae saisical modeling helps accomplish unbiased inference in spie of he biased daa and a imes even provides a more informaive and economic seup. See Rao 965) Pael and Rao 978) Oluyede 999) Nanda and Jain 999) Gupa and Keaing 985) and references herein for a comprehensive review and addiional deails on weighed disribuions. Moivaed by various applicaions of Pareo disribuion in several areas including reliabiliy exponenial iling weighing) in finance and acuarial sciences as well as in economics we consruc and presen some saisical properies of a new class of generalized Pareo-ype disribuion called he Weighed Feller-Pareo WFP) disribuion. The aim of his paper is o propose and sudy a generalizaion of he Pareo disribuion via he weighed Feller-Pareo disribuion and obain a larger class of flexible parameric models wih applicaions in reliabiliy acuarial science economics finance and elecommunicaions. This paper is organized as follows. Secion 2 conains some uiliy noions and basic resuls. The weighed Feller-Pareo disribuion is inroduced in secion 2 including he cumulaive disribuion funcion cdf) pdf hazard and reverse hazard funcions and monooniciy properies. In secion 3 momens of he WFP disribuion are presened. The

WEIGHTED FELLER-PARETO DISTRIBUTION 3 mean variance sandard deviaion coefficiens of variaion skewness and kurosis are readily obained from he momens. Secion 4 conains measures of uncerainy including Renyi Shannon and s enropies of he disribuion. Some concluding remarks are given in secion 5. 2. The Weighed Feller-Pareo Class of Disribuions In his secion he weighed Feller-Pareo class of disribuions is presened. Firs we discuss he Feller-Pareo disribuion is properies and some sub-models. Some sub-models of he FP disribuion are given in Table below. 2.. Feller-Pareo Disribuion. In his secion we ake a close look a a more general form of he Pareo disribuion called he Feller-Pareo disribuion which races i s roo back o Feller 97). Definiion 2.. The Feller-Pareo disribuion called FP disribuion for shor is defined as he disribuion of he random variable Y + X ) where X follows a bea disribuion wih parameers α and > 0 and > 0 ha is Y F P α ) if he pdf of Y is of he form 3) f F P y; α ) Bα ) y ) + y ) α+) for < < α > 0 > 0 > 0 > 0 and y > where Bα ) Γα)Γ) Γα+). The ransformed bea TB) disribuion is a special case of he FP disribuion. The pdf of he ransformed bea disribuion is given by 4) f T BD x; α ) x/) x 0. Bα )x + x/) α+ Therefore from 4) T B α ) F P 0 / α ). The family of Pareo disribuions Pareo I o Pareo IV) can be readily obained for specified values of he parameers α and. Table. Some Sub-Models of he FP Disribuion Family name Symbol Densiy funcion F P I) F P y; α ) ) + Bα) ) F P II) F P y; α ) + Bα) ) F P III) F P y; ) + ) F P IV ) F P y; α ) + Bα) ) α+) ) α+) ) 2 ) α+)

4 ODUBOTE AND OLUYEDE 2.2. Momens of Feller-Pareo Disribuion. The k h momen of he random variable under FP disribuion is given by: Y ) k Y E Bα ) y ) Le 0 < < hen dy ) k+ + ) d ) 2 and y ) α+) dy. ) k Y E 0 ) k+ ) α+) ) d Bα ) ) 2 k+ ) α k d Bα ) 0 Γk + )Γα k) Γα)Γ) for k 0... α k 0 2... and α k > 0. If 0 hen he k h momen reduces o EY k ) k Γk + )Γα k) Γα)Γ) for < k < α. Noe ha if 0 and hen we have EY k ) Γk + )Γα k) Γα)Γ) k 0 2... α k 0 2... The mean variance coefficien of variaion cv) coefficien of skewness cs) and coefficien of kurosis ck) of he Feller-Pareo disribuion when 0) are given by F P Γ + )Γα ) Γα)Γ) ) 2 σf 2 P 2 Γ2 + )Γα 2) Γ + )Γα ) Γα)Γ) Γα)Γ) ) 2 2 Γ2+)Γα 2) σ 2 Γ+)Γα ) Γα)Γ) Γα)Γ) CV F P F P F P CS F P EY 3 ) 3σ 2 3 σ 3 Γ+)Γα ) Γα)Γ) CK F P EY 4 ) 4EY ) + 6 2 EY 2 ) 4 3 EY ) + 4 σ 4 where F P σ σf 2 P EY 3 ) Γ3+)Γα 3) Γα)Γ) 983) Luceno 2006) and references herein. and EY 4 ) Γ4+)Γα 4). See Arnold Γα)Γ)

WEIGHTED FELLER-PARETO DISTRIBUTION 5 2.3. Weighed Feller-Pareo Disribuion. In his secion we presen he weighed Feller-Pareo WFP) disribuion. Some WFP sub-models are also presened in his secion. Table 2 conains he pdfs of he sub-models for he WFP disribuion. Firs consider he weigh funcion wy; k) y k. The WFP pdf f W F P y) when 0 and is given by for f W F P y) yk f F P y) EY k ) Γα + ) Γ + k)γα k) ) k+ y + < k < α. The lengh-biased k ) Feller-Pareo LBFP) pdf is ) Γα + ) y y f LBF P y) + Γ + )Γα ) ) α for < α and 0. In general wih he weigh funcion wy; k) he WFP pdf as: 5) f W F P y) Γα)Γ) Γk + )Γα k)bα ) y ) k+ + ) α y y ) k we obain ) α+) for k 0 2...; y > α k 0 2... and α k > 0. The cdf of WFP disribuion is given by: 6) F y; α k) y 0 Γα + ) Γ + k)γα k) Γα + )B + ) k+ x ) dx α+ x ) )/) ; k + α + k + )/) Γk + )Γα k) for α k 0 2... α+ k 0 2... where B; a b) 0 ya y) b dy is he incomplee bea funcion. The plos of he pdf and cdf of he WFP disribuion in Figures and 2 sugges ha he addiional parameers k conrols he shape and ail weigh of he disribuion. Figure. PDF of he Weighed Feller-Pareo wih k and differen values of k

6 ODUBOTE AND OLUYEDE Figure 2. CDF of he Weighed Feller-Pareo wih k and differen values of k Family name Table 2. Sub-Models of he WFP Disribuion Symbol W P I) F P α k) W P II) F P α k) W P III) F P k) W P IV ) F P α k) Densiy funcion Γα) Γk+)Γα k)bα) Γα) Γk+)Γα k)bα) ) Γk+)Γ k)b) ) Γα+) Γk+)Γα k) + + ) α+) y ) α+) + + ) 2 ) α+) 2.4. Hazard and Reverse Hazard Funcions. In his secion hazard and reverse hazard funcions of he WFP disribuion are presened. Graphs of he hazard funcion for seleced values of he model parameers are also given. The hazard and reverse hazard funcions are given by: h F y; k α k) fy; α k) F y; α k) Γα+) Γ+k)Γα k) Γα+) B Γk+)Γα k) ) k+ ) α + ) )/) ; k + α + k )/) +

and τ F y; α k) WEIGHTED FELLER-PARETO DISTRIBUTION 7 fy; α k) F y; α k) Γα+) Γ+k)Γα k) Γα+) B Γk+)Γα k) + ) k+ ) α + ) )/) ; k + α + k )/) for α k 0 2... α + k 0 2... respecively. Graphs of he WFP hazard rae funcion for seleced values of he model parameers are given in Figure 3. The graphs shows unimodal and upside down bahub shapes. Figure 3. Hazard funcion of he Weighed Feller-Pareo wih k0 and differen values of k 2.5. Monooniciy Properies. In his secion monooniciy properies of he WFP disribuion are presened. The log of he WFP pdf is given by: lnf W F P y)) n log Γα + ) n log Γk + ) n log Γα k) n log ) ) ) y y + + k log α + ) log +. Now differeniaing lnf W F P y)) wih respec o y we have ) lnf W F P y) k + α + ) y y + ) )

8 ODUBOTE AND OLUYEDE and solving for y we have Noe ha lnf W F P y; α k) y lnf W F P y; α k) y 0 y < 0 y > ) k +. k α ) k + k α and ) lnf W F P y; α k) k > 0 y < +. y k α k The mode of he WFP disribuion is given by y 0 k α) +. 3. Momens of WFP Disribuion Recall ha he k h momen of he random variable Y under he FP disribuion are given by ) k Y Γk + )Γα k) E Γα)Γ) for k 0... α k 0 2... α k > 0. Now we derive he r h momens of he random variable Y under WFP disribuion. This is given by ) r+k+ ) r Y Γα)Γ) E Γk + )Γα k)bα ) ) dy. α+) + ) Le 0 < < hen dy 7) ) r Y E 0 + ) d ) 2 and Γα)Γ) Γk + )Γα k)bα ) α+) ) d ) 2 ) r+k+ ) Γα)Γ) r+k+ ) α r k d Γk + )Γα k)bα ) 0 Γr + k + )Γα r k) Γk + )Γα k) for α k 0 2... α r 0 2... α r + k) 0 2... r 0 2... The mean variance coefficiens of variaion cv) skewness cs) and kurosis ck) can be readily obained from equaion 7).

WEIGHTED FELLER-PARETO DISTRIBUTION 9 4. Measures of Uncerainy for he Weighed Feller-Pareo Disribuion The concep of enropy was inroduced by Shannon 948) in he nineeenh cenury. During he las couple of decades a number of research papers have exended Shannon s original work. Among hem are Park 995) Renyi 96) who developed a one-parameer exension of Shannon enropy. Wong and Chen 990) provided some resuls on Shannon enropy for order saisics. The concep of enropy plays a vial role in informaion heory. The enropy of a random variable is defined in erms of is probabiliy disribuion and can be shown o be a good measure of randomness or uncerainy. In his secion we presen Renyi enropy Shannon enropy and s enropy for he WFP disribuion. 4.. Shannon Enropy. Shannon enropy 948) for a coninuous random variable Y wih WFP pdf f W F P y) is defined as 8) E logf W F P Y ))) logf W F P y)))f W F P y)dy. Noe ha ) Γα + ) logf W F P y)) log + logγα) + logγ) logγk + ) Γα)Γ) logγα k) + k + ) ) y log ) y α + )log +. Now Shannon enropy for he weighed Feller-Pareo disribuion is ) Γα + ) E logf W F P Y )) log + logγα) + logγ) logγk + ) Γα)Γ) logγα k) + k + ) ) Y E log ) Y 9) α + )E log +. Now wih he subsiuion E log is given by ) Y and E log + ) Y for 0 < < we can readily obain boh ) so ha Shannon enropy for he WFP disribuion 0) E logf W F P Y )) logbα ) + log) + Γα) + Γ) logγk + ) logγα k) + k + )ψ) ψα) α + )ψα) ψα + )) where ψ.) Γ.) Γ.) is he digamma funcion.

0 ODUBOTE AND OLUYEDE 4.2. s-enropy for Weighed Feller-Pareo Disribuion. The s-enropy of he WFP disribuion is defined as H s f W F P ) f s W s F P y)dy if s s > 0 E logf W F P Y )) if s. Noe ha fw s F P y)dy + ) and using he subsiuion ge ) f s W F P y)dy 0 Γα)Γ) Γk + )Γα k)bα ) y ) α+) y s dy for 0 < < so ha dy ) k+ s Bα ) k+ ) +α k ) d ) d ) 2 we ) s Γα + )s Γks + s s + )Γsα ks + s ) Γk + )Γα k) s Γs + sα) for s > 0 s. Consequenly s-enropy for he WFP disribuion is given by H s f W F P ) ) s Γα + )s Γks + s s + )Γsα ks + s ) s Γk + )Γα k) s Γs + sα) for s > 0 s α k 0 2... sα+s ks 0 2... ks+s s + > 0 and sα ks + s > 0. 4.3. Renyi Enropy. Renyi enropy 96) for he WFP disribuion is presened in his secion. Noe ha Renyi enropy is given by 2) H R f W F P ) ) s log f W F P x)) s dx s > 0 s. From equaion 0) we obain Renyi enropy as follows: H R f W F P ) s log Γα + ) s Γks + s s + )Γsα ks + s ) ) s Γk + )Γα k) s Γs + sα) for s > 0 s α k 0 2... sα+s ks 0 2... ks+s s + > 0 and sα ks + s > 0. 5. Concluding Remarks In his paper a new six-parameer class of disribuions called weighed Feller-Pareo WFP) disribuion is consruced and sudied. The pdf cdf hazard and reverse hazard funcions monooniciy properies are presened. Measures of uncerainy including Renyi Shannon and s-enropies are derived.

WEIGHTED FELLER-PARETO DISTRIBUTION Table 3. Shannon and s-enropies of he sub-models of he FP disribuion Family name P I) P II) P III) P IV ) Shannon enropy ) log Γα+) Γα) +α+)ψα) ψα + )) ) log Γα+) Γα) + α + ψα) + α + )ψα) ψα + )) s-enropy Γα+)s Γs+αs ) s Γα) s Γs+αs) Γα+)s Γs+αs ) s Γα) s Γs+αs) )ψα) ) ψα + )) Γ2) log 2 + 2ψ) ψ2)) s Γs s+)γs+s ) s Γ2s) ) Γα+) log Γα) )ψ) s α s Γs s+)γs+αs ) s Γs+αs) Table 4. Shannon and s-enropies of he WFP Disribuion Family name Shannon enropy s-enropy W P I) W P II) logbα )+log) Γα) +logγk+ )+logγα k) kψ) ψα)+α+ )ψα) ψα + )) logbα )+log) Γα) +logγk+ )+logγα k) kψ) ψα)+α+ )ψα) ψα + )) W P III) logb )+log) 2+logΓk +)+ logγ k) k+ )ψ) ψ)+ 2ψ) ψ2)) W P IV ) logbα ) + log) Γα) + logγk +)+logγα k) k + )ψ) ψα)+α+)ψα) ψα+)) s Γα+)s Γks+)Γsα ks+s ) Γk+) s Γα k) s Γs+sα) Γα+)s Γks+)Γsα ks+s ) s Γk+) s Γα k) s Γs+sα) s Γks+)Γs ks+s ) s Γk+) s Γ k) s Γ2s) s Γα+) s Γks+)Γsα ks+s ) s Γk+) s Γα k) s Γs+sα) References Arnold B. C. Pareo Disribuions Inernaional Cooperaive Publishing House Fairland Maryland 983). 2 Brazauskas V. Fisher Informaion Marix for he Feller-Pareo Disribuion Saisics and Probabiliy Leers 59 59-67 2002). 3 Block H.W. and Savis T.H. The Reverse Hazard Funcion Probabiliy in he Engineering and Informaional Sciences2 69-90 998). 4 Chandra N.K. and Roy D. Some Resuls on Reverse Hazard Rae Probabiliy in he Engineering and Informaion Sciences 5 95-02 200). 5 Feller W. An Inroducion o Probabiliy Theory and is Applicaions Vol. 2 2nd Ediion Wiley New York 97). 6 Gupa C. and Keaing P. Relaions for Reliabiliy Measures Under Lengh Biased Sampling Scandinavian Journal of Saisics 3) 49-56 985). 7 Luceno A. Fiing he Generalized Pareo Disribuion o Daa using Maximum Goodness-of Fi Esimaors Compuaional Saisics and Daa Analysis 5 2 904-97 2006). 8 Nanda K. and Jain K. Some Weighed Disribuion Resuls on Univariae and Bivariae Cases Journal of Saisical Planning and Inference 772) 69-80 999).

2 ODUBOTE AND OLUYEDE 9 Oluyede O. On Inequaliies and Selecion of Experimens for Lengh-Biased Disribuions Probabiliy in he Engineering and Informaional Sciences 32) 29-45 999). 0 Park S. The Enropy of Coninuous Probabiliy Disribuions IEEE Transacions of Informaion Theory 4 2003-2007 995). Pail P. Encounered Daa Saisical Ecology Environmenal Saisics and Weighed Disribuion Mehods Environmerics 24) 377-423 99). 2 Pail P. and Rao R. Weighed Disribuions and Size-Biased Sampling wih Applicaions o Wildlife and Human Families Biomerics 346) 79-89 978). 3 Riabi M.Y.A. Bordazaran G. R. M. and Yari G. H. - Enropy for Pareo-ype Disribuions and Relaed Weighed Disribuions Saisics and Probabiliy Leers 80 52-59 200). 4 Rao R. On Discree Disribuions Arising ou of Mehods of Ascerainmen The Indian Journal of Saisics 272) 320-332 965). 5 Renyi A. On Measures of Enropy and Informaion Berkeley Symposium on Mahemaical Saisics and Probabiliy ) 547-56 960). 6 Rizzo M. New Goodness-of-Fi Tess for Pareo Disribuions Asin Bullein 392) 69-75 2009). 7 Shannon E. A Mahemaical Theory of Communicaion The Bell Sysem Technical Journal 270) 379-423 948). 8 Wong K.M ChenS. The enropy of ordered sequences and order saisics IEEE Transacions of informaion Theory 36 276-284 990). Deparmen of Mahemaical Sciences Georgia Souhern Universiy Saesboro GA 30460 Unied Saes