A Note on a Characterization of J-Shaped Distribution. by Truncated Moment

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1 Applied Mahemaical Scieces Vol. 8 4 o HIKARI Ld hp://d.doi.or/.988/ams A Noe o a Characerizaio o J-Shaped Disribuio by Trucaed Mome Mohammad Ahsaullah Deparme o Maaeme Scieces Rider Uiversiy Lawreceville NJ 8648 USA Mohammad Shakil Deparme o Mahemaics Miami Dade Collee Hialeah Campus Hialeah Fl. 33 USA Copyrih 4 Mohammad Ahsaullah ad Mohammad Shakil. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese which permis uresriced use disribuio ad reproducio i ay medium provided he oriial work is properly cied. Absrac Characerizaio o a probabiliy disribuio plays a impora role i saisics ad mahemaical scieces. A probabiliy disribuio ca be characerized hrouh various mehods. The J-shaped disribuio is oe o he widely used disribuios i may ields o research where he shapes o probabiliy disribuios o o-ormal daa ehibi J-shaped disribuio. I his paper we ivesiae he characerizaio o J-shaped disribuio by rucaed mome. For he sake o compleeess some properies o J-shaped disribuio are provided. I is hoped ha he idis o his paper will be useul or researchers i diere ields o applied scieces. Mahemaics Subjec Classiicaio: Keywords: J-shaped disribuio characerizaio covariace momes order saisics

2 58 Mohammad Ahsaullah ad Mohammad Shakil. Iroducio I may ields o research such as such as bioloy ecoomics oresry eeics medicie psycholoy reliabiliy ec. he shapes o probabiliy disribuios o o-ormal daa ehibi J- shaped disribuios. May auhors have sudied he J-shaped or Topp Leoe disribuio ad is properies. A mahemaical ormulaio o he amily o J-shaped probabiliy disribuios was irs proposed by Topp ad Leoe [4]. They also derived is irs our momes ad showed is suiabiliy o model ailure daa. Furher developme coiued wih he coribuios o may auhors amo hem Nadarajah ad Koz [3] Ghiay e al. [9] Koz ad Nadarajah [] Zhou e al. [7] Zhoul [5 6] ad Geç [8] are oable. Characerizaio o a probabiliy disribuio plays a impora role i saisics ad mahemaical scieces. eore a paricular probabiliy disribuio model is applied o i he real world daa i is esseial o coirm wheher he ive probabiliy disribuio saisies he uderlyi requiremes by is characerizaio. I appears rom lieraure ha o aeio has bee paid o he characerizaios o he J-shaped disribuio. A probabiliy disribuio ca be characerized hrouh various mehods. I his paper we have esablished some characerizaios o he J- shaped disribuio by usi he rucaed mome where we have cosidered a produc o reverse hazard rae ad aoher ucio o he rucaed poi. The oraizaio o he paper is as ollows. For he sake o compleeess some properies o J-shaped disribuio are preseed i Secio. I Secio 3 we ivesiae he characerizaio o he J-shaped disribuio by usi he rucaed mome. Cocludi remarks are provided i Secio 4.. Disribuioal Properies o he J-Shaped Disribuio For he sake o compleeess some properies o J-shaped disribuio are preseed i his secio. For a absoluely coiuous radom variable he amily o cumulaive disribuio ucios cd s cosidered by Topp ad Leoe [4] ad ehibii J-shaped disribuios is deied as ollows: F where ad. The correspodi amily o probabiliy desiy ucios pd s obaied by diereiaio o q.. is ive by.

3 Characerizaio o J-shaped disribuio by rucaed mome For a absoluely coiuous radom variable whe we will say ha i has he eeral orm o J-shaped disribuio ha is i is disribuio ucio is ive by F.3 wih he probabiliy desiy ucio ive by..4 As poied ou by Topp ad Leoe [4] he disribuios as ive above are called J-shaped / // because ad or all where / ad // deoe he irs ad secod derivaives o he pd respecively. The shape o he pd as ive i q..4 is illusraed i he ollowi iure. or whe. I ca be easily see ha he probabiliy desiy ucio.4 o J-shaped disribuio saisies he ollowi eeralized Pearso sysem o diereial equaio a a a 3 b b b3 b where a a a b b b 3 b. 3

4 584 Mohammad Ahsaullah ad Mohammad Shakil Fiure.: Plos o he pd.4 or whe. s Momes: For ay real umber s o order s o he radom variable havi he eeral orm o J-shaped disribuio is derived as ollows. s s s usi he eeral orm.4 he mome d..5 Subsiui u i q..5 ad he usi he deiiio o he icomplee bea ucio a b which is deied as ollows

5 Characerizaio o J-shaped disribuio by rucaed mome 585 b a dz z z b a see or eample Abramowiz ad Seu [] ad Gradshey ad Ryzhik [] we obai aer simpliicaio he ollowi epressio or he mome: / / s s s s s..6 Ieer Order Momes: For his cosideri he q..5 ad aki s where is a ieer we obai aer simpliicaio he ollowi epressio or he h order ieer mome : j j j j j.7 rom which aki ad he irs ad secod momes are respecively ive by 3 3 ad Cosequely he variace is ive by 3 3 Var..9 I ca easily be see by direc diereiaio ha boh mome ad variace are icreasi ucios i or ied. Shao ropy: The Shao eropy o he o J-shaped disribuio whe i q..4 is ive by l l l l H d l l l l rom which o usi appropriae subsiuios applyi he quaios 4.53./P /P /P. 943 ad 8.37/P. 947 o Gradshey ad Ryzhik [] appropriaely ad simpliyi we obai he ollowi epressio or he Shao eropy:

6 586 Mohammad Ahsaullah ad Mohammad Shakil H l where m m m m m m l m m m.. deoes he psi ucio a b deoes he bea ucio ad he ucio z deied as ollows: z z z see or eample Gradshey ad Ryzhik [] amo ohers. I ca easily be see by direc diereiaio ha he Shao eropy o he J-shaped disribuio is a eaive icreasi cove ucio o ad as we have he eropy H. 3. A Characerizaio o he J-Shaped Disribuio is A probabiliy disribuio ca be characerized hrouh various mehods see or eample Ahsaullah e al. [] amo ohers. I rece years here has bee a rea ieres i he characerizaios o probabiliy disribuios by rucaed momes. For eample he developme o he eeral heory o he characerizaios o probabiliy disribuios by rucaed mome bea wih he work o Galambos ad Koz [4]. Furher developme o he characerizaios o probabiliy disribuios by rucaed momes coiued wih he coribuios o may auhors ad researchers amo hem Koz ad Shabha [] Gläzel [5 6] ad Gläzel e al. [7] are oable. However mos o hese characerizaios are based o a simple relaioship bewee wo diere momes rucaed rom he le a he same poi. As poied ou by Gläzel [5] hese characerizaios may also serve as a basis or parameer esimaio. I appears rom lieraure ha o aeio has bee paid o he characerizaios o he J-shaped disribuio by usi rucaed mome. I his secio we prese he characerizaio o he J-shaped disribuio by usi he rucaed mome. We irs prove a lemma Lemma 3. which will be useul i provi our mai characerizaio resuls. The mai

7 Characerizaio o J-shaped disribuio by rucaed mome 587 characerizaio resuls are proved i Theorems 3. ad 3.. Lemma 3.: Suppose ha is a absoluely coiuous wih respec o Lebesue measure radom variable wih cd F ad pd. We assume ha F F / eis or all ad. I where diereiable ucio o ad ' is a or all he we have F d ce where c is deermied by he codiio ha d. Proo o Lemma 3.: We have u u du. F F Thus u u du. Diereiai boh sides o he above equaio wih respec o we obai O simpliicaio we e ' '. ' '. Ierai he above equaio we obai ' d ce where c is deermied by he codiio ha d. This complees he proo o Lemma 3.. Theorem 3.: Suppose ha he radom variable has a absoluely coiuous wih respec o Lebesue measure cumulaive disribuio ucio cd F ad probabiliy desiy ucio pd. We assume ha F / F eiss or all ad

8 588 Mohammad Ahsaullah ad Mohammad Shakil. Le. F The has a J-shaped disribuio i ad oly i where 4 ad q p du u u q p called he icomplee bea ucio. Proo o Theorem 3.: Necessary Par: Suppose ha. The aer simpliicaio we easily have d 4 where q p du u u q p deoes he icomplee bea ucio. Suiciecy Par: We will prove ow he oly i codiio he Theorem 3.. Suppose ha. 4

9 Characerizaio o J-shaped disribuio by rucaed mome 589 The aer simple diereiaio ad simpliicaio i is easily see ha. Thus. Hece by Lemma 3. we have / rom which o ieraio we have d ce c where c is a cosa. Cosequely usi he boudary codiio d i he above equaio we obai. This complees he proo o Theorem 3.. Corollary 3.: Suppose ha he radom variable has a absoluely coiuous wih respec o Lebesue measure cumulaive disribuio ucio cd F ad probabiliy desiy ucio pd. We assume ha F F / eis or all ad. Le. F The has a J-shaped disribuio wih pd } { i ad oly i where. 4

10 58 Mohammad Ahsaullah ad Mohammad Shakil Proo: Taki i Theorem 3. he proo o Corollary 3. easily ollows. Corollary 3.: Suppose ha he radom variable has a absoluely coiuous wih respec o Lebesue measure cumulaive disribuio ucio cd F ad probabiliy desiy ucio pd. We assume ha F / F eis or all ad. Le. The has a J-shaped disribuio wih pd F i ad oly i where 4.. Proo: Taki ad i Theorem 3. he proo o Corollary 3. easily ollows. 4. Cocludi Remarks eore a paricular probabiliy disribuio model is applied o i he real world daa i is esseial o coirm wheher he ive probabiliy disribuio saisies he uderlyi requiremes by is characerizaio. Thus characerizaio o a probabiliy disribuio plays a impora role i saisics ad mahemaical scieces. A probabiliy disribuio ca be characerized hrouh various mehods. I his paper we ivesiae he characerizaio o J- shaped disribuio by rucaed mome where we have cosidered a produc o reverse hazard rae ad aoher ucio o he rucaed poi. For he sake o compleeess some properies o J-shaped disribuio are provided. We believe ha he idis o his paper would be useul or he praciioers i various ields o sudies ad urher ehaceme o research i disribuio heory ad is applicaios. Ackowledme. The auhors would like o hak he reerees ad he edior or helpul suesios which improved he qualiy ad preseaio o he paper. RFRNCS [] Abramowiz M. ad Seu I. A. 97. Hadbook o Mahemaical Fucios wih Formulas Graphs ad Mahemaical Tables. Dover New York.

11 Characerizaio o J-shaped disribuio by rucaed mome 58 [] Ahsaullah M. Kibria. M. G. ad Shakil M. 4. Normal ad Sude s Disribuios ad Their Applicaios. Alais Press Paris Frace. [3] Arold. C. alakrisha N. ad Naaraja H. N. 99. A irs course i order saisics. Wiley New York [4] Galambos J. ad Koz S Characerizaios o probabiliy disribuios. A uiied approach wih a emphasis o epoeial ad relaed models Lecure Noes i Mahemaics 675 Sprier erli Germay. [5] Gläzel W A characerizaio heorem based o rucaed momes ad is applicaio o some disribuio amilies Mahemaical Saisics ad Probabiliy Theory ad Tazmasdor 986 Vol Reidel Dordrech Germay. [6] Gläzel W. 99. Some cosequeces o a characerizaio heorem based o rucaed momes Saisics [7] Gläzel W. Telcs A. ad Schuber A Characerizaio by rucaed momes ad is applicaio o Pearso-ype disribuios Z. Wahrsch. Verw. Gebiee [8] Geç A. İ.. Momes o order saisics o Topp Leoe disribuio. Saisical Papers [9] Ghiay M.. Koz S. ad ie M. 5. O some reliabiliy measures ad heir sochasic orderis or he Topp Leoe disribuio. Joural o Applied Saisics [] Gradshey I. S. ad Ryzhik I. M. 98. Table o Ierals Series ad Producs 6h Prii. Academic Press Sa Dieo. [] Koz S. ad Nadarajah S. 6. J-shaped disribuio Topp ad Leoe s. cyclopedia o Saisical Scieces vol 6 d ed. Wiley New York p [] Koz S. ad Shabha D.N. 98. Some ew approaches o probabiliy disribuios. Advaces i Applied Probabiliy [3] Nadarajah S. ad Koz S. 3. Momes o some J-shaped disribuios Joural o Applied Saisics [4] Topp C.W. ad Leoe F.C A Family o J-shaped requecy ucios Joural o he America Saisical Associaio [5] Zhoul A. A.. Order saisics rom a amily o J-shaped disribuios. MTRON

12 58 Mohammad Ahsaullah ad Mohammad Shakil [6] Zhoul A. A.. Record values rom a amily o J-shaped disribuios. Saisica [7] Zhou M. Ya D. W. Wa Y. & Nadarajah S. 6. Some j-shaped disribuios: sums producs ad raios. I Reliabiliy ad Maiaiabiliy Symposium 6. RAMS'6. Aual pp I. Received: July 4