Journal o Modrn Applid Statistical Mthods Volum 15 Issu 1 Articl 17 5-016 Charactrizations o Continuous Distributions by Truncatd Momnt M Ahsanullah Ridr Univrsity M Shakil Miami Dad Coll B M Golam Kibria lorida Intrnational Univrsity, kibria@iudu ollow this and additional works at: http://diitalcommonswayndu/jmasm Rcommndd Citation Ahsanullah, M; Shakil, M; and Kibria, B M Golam (016) "Charactrizations o Continuous Distributions by Truncatd Momnt," Journal o Modrn Applid Statistical Mthods: Vol 15 : Iss 1, Articl 17 DOI: 1037/jmasm/146076160 Availabl at: http://diitalcommonswayndu/jmasm/vol15/iss1/17 This Rular Articl is brouht to you or r and opn accss by th Opn Accss Journals at DiitalCommons@WaynStat It has bn accptd or inclusion in Journal o Modrn Applid Statistical Mthods by an authorizd ditor o DiitalCommons@WaynStat
Journal o Modrn Applid Statistical Mthods May 016, Vol 15, No 1, 316-331 Copyriht 016 JMASM, Inc ISSN 1538 947 Charactrizations o Continuous Distributions by Truncatd Momnt M Ahsanullah Ridr Univrsity Lawrncvill, NJ M Shakil Miami Dad Coll Hialah, L B M Golam Kibria lorida Intrnational Univrsity Miami, L A probability distribution can b charactrizd throuh various mthods In this papr, som nw charactrizations o continuous distribution by truncatd momnt hav bn stablishd W hav considrd standard normal distribution, Studnt s t, ponntiatd ponntial, powr unction, Parto, and Wibull distributions and charactrizd thm by truncatd momnt Kywords: Charactrization, ponntiatd ponntial distribution, powr unction distribution, standard normal distribution, Studnt s t distribution, Parto distribution, truncatd momnt Introduction Bor a particular probability distribution modl is applid to it ral world data, it is ssntial to conirm whthr th ivn probability distribution satisis th undrlyin rquirmnts o its charactrization Thus, charactrization o a probability distribution plays an important rol in statistics and mathmatical scincs A probability distribution can b charactrizd throuh various mthods, s, or ampl, Ahsanullah, Kibria, and Shakil (014), Huan and Su (01), Nair and Sudhsh (010), Nanda (010), Gupta and Ahsanullah (006), and Su and Huan (000), amon othrs In rcnt yars, thr has bn a rat intrst in th charactrizations o probability distributions by truncatd momnts or ampl, th dvlopmnt o th nral thory o th charactrizations o probability distributions by truncatd momnt ban with th work o Galambos and Kotz (1978) urthr dvlopmnt on th charactrizations o probability distributions Dr Ahsanullah is a Prossor o Statistics in th Dpartmnt o Inormation Systms and Supply Chain Manamnt Email him at: ahsan@ridrdu Dr Shakil is a Prossor o Mathmatics in th Dpartmnt o Arts and Scincs Email him at: mashakil@mdcdu Dr Kibria is a Prossor in th Dpartmnt o Mathmatics and Statistics Email him at: kibria@iudu 316
AHSANULLAH ET AL by truncatd momnts continud with th contributions o many authors and rsarchrs, amon thm Kotz and Shanbha (1980), Glänzl (1987, 1990), and Glänzl, Tlcs, and Schubrt (1984), ar notabl Howvr, most o ths charactrizations ar basd on a simpl proportionality btwn two dirnt momnts truncatd rom th lt at th sam point It appars rom litratur that not much attntion has bn paid on th charactrizations o continuous distributions by usin truncatd momnt As pointd out by Glänzl (1987) ths charactrizations may also srv as a basis or paramtr stimation In this papr, som nw charactrizations o continuous distributions by truncatd momnt hav bn stablishd Th oranization o this papr is as ollows: W will irst stat th assumptions and stablish a lmma which will b ndd or th charactrizations o continuous distributions by truncatd momnt Th ollowin sction contains our main rsults or th nw charactrizations o continuous distributions by truncatd momnt inally, concludin rmarks ar prsntd A Lmma This sction will stat th assumptions and stablish a lmma (Lmma 1) which will b usul in provin our main rsults or th charactrizations o continuous distributions by truncatd momnt Assumptions Lt X b a random variabl havin absolutly continuous (with rspct to Lbsu masur) cumulativ distribution unction (cd) () and th probability dnsity unction (pd) () W assum α = in{ () > 0} and β = sup{ () < 1} W din, and () is a dirntiabl unction with rspct to or all ral (α, β) Lmma 1 Suppos that X has an absolutly continuous (with rspct to Lbsu masur) cd (), with corrspondin pd (), and E(X X ) ists or all ral (α, β) Thn E(X X ) = ()η(), whr () is a dirntiabl unction and or all ral (α, β), i 317
CHARACTERIZATIONS O CONTINUOUS DISTRIBUTIONS c u ' u u du whr c is dtrmind such that d 1 Not: Sinc th cd () is absolutly continuous (with rspct to Lbsu masur), thn by Radon-Nikodym Thorm th pd () ists and hnc u u du ists Also not that, in Lmma 1 abov, th lt truncatd u conditional pctation o X considrs a product o rvrs hazard rat and anothr unction o th truncatd point Proo o Lmma 1 It is known that u u du Thus u u du Dirntiatin both sids o th quation producs th ollowin On simpliication, on ts Intratin th abov quation ivs c u u u du 318
AHSANULLAH ET AL whr c is dtrmind such that 1 d 1 This complts th proo o Lmma Charactrizations o som Continuous Distributions by Truncatd Momnts Standard Normal Distribution Th charactrization o standard normal distribution is providd in Thorm 1 blow Thorm 1 Suppos that an absolutly continuous (with rspct to Lbsu masur) random variabl X has cd () and pd () or - < < W assum that '(t) and E(X X t) ist or all t, - < t < Thn whr and () = -1, i and only i X X E, 1 1,, π which is th probability dnsity unction o th standard normal distribution Proo: Suppos 1 π 1 319
CHARACTERIZATIONS O CONTINUOUS DISTRIBUTIONS Thn it is asily sn that () = -1 Consquntly, th proo o th i part o th Thorm 1 ollows rom Lmma 1 W will now prov th only i condition o th Thorm 1 Suppos that () = -1 Thn it asily ollows that On intratin th abov quation, 1 c, whr c 1 This complts th proo o Thorm 1 π Studnt s t Distribution Th charactrization o th Studnt s t is providd in Thorm blow Thorm Suppos that an absolutly continuous (with rspct to Lbsu masur) random variabl X has th cd () and pd () or - < < W assum that '() and E(X X ) ist or all, - < < Thn X has th Studnt s t distribution i and only i whr and X X E, n 1, n1 n1 n 30
AHSANULLAH ET AL Proo: Suppos th random variabl X has th t distribution with n drs o rdom Th pd () o X is n 1 n1 1, n n n Thn it is asily sn that n 1 n1 u u 1 du n n n n 1 n 1 n1 n1 n 1 n n n Consquntly, th proo o i part o Thorm ollows rom Lmma 1 Now prov th only i condition o th Thorm Suppos that Thn, on asily has n 1 n1 n n 1 Thus, atr simpliication, on obtains th ollowin: 31
CHARACTERIZATIONS O CONTINUOUS DISTRIBUTIONS n 1 n 1 n1 n n 1 1 n 1 n n Thror, by Lmma 1, on has n1u n du n 1 u n1 1 ln 1 n n c c c1 n Now, usin th condition d 1, on obtains n 1 n1 1, n n n Not th condition n 1 is ndd or E(X X ) to ist This complts th proo o Thorm Eponntiatd Eponntial Distribution Th Charactrization o ponntiatd ponntial distribution is prsntd in th Thorm 3 blow Thorm 3 Suppos an absolutly continuous (with rspct to Lbsu masur) random variabl X has th cd () and pd () or 0 < < such that '() and E(X X ) ist or all, 0 < < Thn X has th ponntiatd ponntial distribution i and only i 1 1, 1, 0 3
AHSANULLAH ET AL X X E, whr and u 1 1 1 1 0 1 1 du Proo: Suppos Thn it is asily sn that 1 1, 1, 0 u 1 1 0 1 du 1 Consquntly, th proo o th i part o th Thorm 3 ollows rom Lmma 1 Now prov th only i condition o th Thorm 3 Suppos that u 1 1 0 1 du 1 Simpl dirntiation and simpliication ivs '() = + ()A(), whr 1 A, 1 1 33
CHARACTERIZATIONS O CONTINUOUS DISTRIBUTIONS Thus 1 A 1 On intratin th abov quation, i ollows that But A u du 0 0 Thus () = c -λ (1 -λ ) α 1, whr This complts th proo o Thorm 3 Powr unction Distribution A u du 0 c 1 u 1 u du 1 ln 1 1 1 1 1 d c 0 Th charactrization o th powr unction distribution is providd in Thorm 4 blow Thorm 4 Suppos an absolutly continuous (with rspct to Lbsu masur) random variabl X has th cd () and pd () or 0 < 1 Assum that '() and E(X X ) ist or all, 0 < < 1 Thn whr X X E, 34
AHSANULLAH ET AL and 1, i and only i 1, 1, 0 1, which is th pd o th powr unction distribution Proo: Suppos () = α α 1, α > 1, 0 < < 1 Thn it is asily sn that 1 I 1, thn 1, 1 1 Thus, by Lmma 1, 1 du u 1 c c, 1 35
CHARACTERIZATIONS O CONTINUOUS DISTRIBUTIONS whr c is a constant Usin th condition d 1, w obtain () = α α 1, α > 1, 0 < < 1 This complts th proo o Thorm 4 and on ts a charactrization o th uniorm distribution in [0, 1] Rmark 1 I α = 1, thn Parto Distribution Th charactrization o Parto distribution is providd in Thorm 5 blow Thorm 5 Suppos th random variabl X has an absolutly continuous (with rspct to Lbsu masur) cd () and pd () W assum that (1) = 0, () > 0 or all > 1, and E(X) ists Thn X has a Parto distribution i and only i whr and X X E, 1, 1, 1 1 Proo: Suppos th random variabl X has th Parto distribution Th pd () o X is ivn by Thn it is asily sn that, 1, 1 1 36
AHSANULLAH ET AL u u 1 du 1 1 1 1 Consquntly, th proo o th i part o Thorm 5 ollows rom Lmma 1 Now prov th only i condition o th Thorm 5 Suppos that 1, 1, 1 1 Thn it is asy to show that 1 1 and 1 1 Consquntly, Thror, by Lmma 1, on obtains 1 1 du c u 1 1 c Now, usin th condition d 1,, 1, 1 1 37
CHARACTERIZATIONS O CONTINUOUS DISTRIBUTIONS This complts th proo o Thorm 5 Wibull Distribution Th charactrization o Wibull distribution is providd in Thorm 6 blow Thorm 6 Suppos an absolutly continuous (with rspct to Lbsu masur) random variabl X has th cd () and pd () or 0 < <, and that '() and E(X X ) ist or all, 0 < < Thn whr and with n 1 whr 0 X X E 1 h, 1 h,, u, n u du, i and only i 1, 0, 0 which is th pd o th Wibull distribution Proo: Not that 38
AHSANULLAH ET AL u u u du du 0 0 1 1 1 1, Suppos that 1 1, Thn 1 1 1 1 1, 1 1 1 1 1, Also, 1 1 1 1 1, 1 1 Thus 1 1 1 On intratin with rspct to rom 0 to, on obtains, whr c is constant On usin th boundary conditions (0) = 0 and ( ) = 1, w hav c = λ and This complts th proo o Thorm 6 39
CHARACTERIZATIONS O CONTINUOUS DISTRIBUTIONS Conclusions Som continuous probability distributions, namly, standard normal, Studnt s t, ponntiatd ponntial, powr unctions, Parto, and Wibull distributions, ar considrd Thir corrspondin charactrizations ar providd by truncatd momnts, which may b usul in applid and physical scincs Acknowldmnts Th authors would lik to thank th rrs and th ditor or hlpul sustions which improvd th quality and prsntation o th papr Author B M Golam Kibria ddicats this papr to th prson h rspcts most Hazrat Gausul Azam Shah Sui Syd Mahtab Uddin Ahmd (R) or his lov, action, and showin him th ood path in li Rrncs Ahsanullah, M, Kibria, B M G, & Shakil, M (014) Normal and Studnt s t Distributions and Thir Applications, Atlantis Prss, ranc Galambos, J & Kotz, S (1978) Charactrizations o probability distributions A uniid approach with an mphasis on ponntial and rlatd modls, Lctur Nots in Mathmatics, 675 Brlin, Grmany: Sprinr doi: 101007/Bb0069530 Glänzl, W (1987) A charactrization thorm basd on truncatd momnts and its application to som distribution amilis In P Baur, Koncny, & W Wrtz (Eds), Mathmatical Statistics and Probability Thory (Vol B) (75-84) Dordrcht, Nthrlands: Ridl doi: 101007/978-94-009-3965- 3_8 Glänzl, W (1990) Som consquncs o a charactrization thorm basd on truncatd momnts, Statistics, 1(4), 613-618 doi: 101080/03318890088073 Glänzl, W, Tlcs, A, & Schubrt, A (1984) Charactrization by truncatd momnts and its application to Parson-typ distributions, Zitschrit ür Wahrschinlichkitsthori und Vrwandt Gbit, 66(), 173-183 doi: 101007/B0053157 330
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