ON STEREOGRAPHIC CIRCULAR WEIBULL DISTRIBUTION

Transcription

1 htt:// ISSN: 9- Reeived:..6 Published:.6.6 Year: 6 Number: Pages: -9 Origial Artile ** ON STEREOGRAPHIC CIRCULAR WEIBULL DISTRIBUTION Phai Yedlaalli * Sagi Vekata Sesha Girija Akkavajhula Vekata Dattatreya Rao 3 <haiyedlaalli3@gmail.om> <svs.girija@gmail.om> <avdrao@gmail.om> Deartmet of Basi Siee Shri Vishu Egieerig College for Wome Vishuur Bhimavaram- 53 Adhra Pradesh Idia. Deartmet of Mathematis Hidu College Gutur Adhra Pradesh Idia 3 Deartmet of Statistis Aharya Nagarjua Uiversity Gutur Adhra Pradesh Idia. Abstrat I this aer we itrodue Stereograhi- l -aial Weibull ad Stereograhi Cirular Weibull distributios for modelig l -aial ad irular data by etedig the Stereograhi Semiirular Weibull distributio [8]. Disussed the estimatio of arameters also ad derived the first two trigoometri momets for the roosed Stereograhi Cirular Weibull model. Keywords Semiirular models l -aial data Stereograhi rojetio Trigoometri momets. Itrodutio I some of the ases the diretioal / agular data does ot required full irular models for modelig this fat is oted i Guardiola [3] Joes [5] Byoug et al [] ad Yedlaalli et at [8]. For eamle whe sea turtles emerge from the oea i searh of a estig site o dry lad a radom variable havig values o a semiirle is well suffiiet for modelig suh data. Similarly whe a airraft is lost but its dearture ad its iitial headigs are kow a semiirular radom variable is suffiiet for suh agular data. Ad few more eamles of semiirular data is available i Ugai et al []. Toshihiro Abe et al [] ostruted some Uimodal ad symmetri distributios by alyig Iverse Stereograhi rojetio Yedlaalli et al [8] ostruted some semiirular distributios by alyig Iverse Stereograhi rojetio. I this aer we develoed the Stereograhi- l -aial Weibull distributio by etedig the Stereograhi Semiirular Weibull ad observed that Stereograhi Cirular Weibull distributio is a ** Edited by Oktay Muhtaroğlu (Area Editor) ad Naim Çağma (Editor-i-Chief). * Corresodig Author.

2 Joural of New Theory (6) -9 seial ase to roosed model ad also Stereograhi Cirular Eoetial [9] ad Stereograhi Cirular Rayleigh Distributios are seial ases to Stereograhi Cirular Weibull Distributio where the same is true i liear ase also. We lotted the grahs of the desity futio for various values of arameters. First two trigoometri momets for roosed model are derived we estimate arameters of the Stereograhi Cirular Weibull model by a maimum likelihood method. Stereograhi-l -aial Weibull Distributio Here we reall the defiitio of Stereograhi Semiirular Weibull Distributio [8]. Defiitio. (Stereograhi Semiirular Weibull Distributio) A radom variable o uit semiirle is said to have Stereograhi Semiirular Weibull distributio with shae arameter ad sale arameter deoted by SCW if the robability desity ad umulative distributio futios are give by. g. G se ta e ta for ad ta e We eted the above Stereograhi Semiirular Weibull model [8] to the l -aial distributio whih is aliable to ay ar of arbitrary legth say for l... so it l is desirable to eted the Stereograhi Semiirular Weibull distributio[8] to ostrut the Stereograhi- l -aial Weibull distributio we osider the desity futio of Stereograhi Semiirular Weibull distributio ad use the trasformatio l.... The robability desity futio of is give by l g l l l l se ta e ta ad l.. (..) l We all it as Stereograhi - l -aial Weibull distributio Case () Whe l i the robability desity futio (..) we get the desity futio g se ta e ta

3 Joural of New Theory (6) -9 3 ad (..) We all it as Stereograhi Cirular Weibull distributio. Case () Whe l the robability desity futio (..) is the same as that of Stereograhi Semiirular Weibull Distributio [8].. Stereograhi Cirular Weibull Distributio A radom variable X S o a uit irle is said to have Stereograhi Cirular Weibull Distributio with sale arameters shae arameter ad loatio arameter deoted by SCWD. If its robability desity ad umulative distributio futios are give by g se ta e ta ad (..) G e ta (..) Seial Cases Case: Whe i (..) we get the desity futio of Stereograhi-l - aial Eoetial Distributio[9]. i.e. l l l g se e ta ad l.. l (..3) Case: Whe & l i (..) we get the desity futio g se e ta (..) We all it as Stereograhi Cirular Eoetial Distributio [9]. Case3: Whe & l i (..) we get the desity futio of Stereograhi Semiirular Eoetial Distributio [8]. Case : Whe & l i (..) we get the desity futio

4 Joural of New Theory (6) -9 g se ta e ta (..5) We all it as Stereograhi Cirular Rayleigh Distributio The same is true i liear ase also..3 Grahs of Desity Futio of Stereograhi Cirular Weibull Distributio for Various Values of the Parameters 3 Trigoometri momets of Stereograhi Cirular Weibull Model Without loss of geerality here we assume that i (..).The trigoometri momets of the distributio are give by with E os ad E si i sie momets of the radom agle resetively. beig the : 3... where th order osie ad

5 Joural of New Theory (6) -9 5 Theorem 3. Uder the df of Stereograhi Cirular Weibull Model with the first four E os ad E si. are give as follows: C C C 33 Proof: i e g d g d i g d os si

6 Joural of New Theory (6) -9 6 E os i E si i for =. For the first osie ad sie momets use the trasformatio ta 8 os ad si []. Now 8 3 the results ad follows by the itegral formula ta Eos os se ta e d ad ta Esi si se ta e d ta os se ta e d 8 e d e d 8 + e d ta si se ta e d 8 3 e d

7 Joural of New Theory (6) e d e d 3 8 e d e d 8 e d e d 8 To obtai seod osie ad sie momets ad we use the trasformatio ta ad os si the results of ad follows by the same itegral formula of. ta os se ta e d 3 e d 3 3 e d e d 3 e d C 33 e d 3 C 33 ta si se ta e d

8 Joural of New Theory (6) e d e d e d e d e d e d e d C 33 e d C 33 e d C 33 C 33 I the similar roess we a obtai the higher order momets also. Estimatio of Parameter The mius log-likelihood for a radom samle of size 3... from the Cirular Weibull distributio is give by L i i i ; i se ta e ta i i g (.) i l ; log L ; log log log os i i log ta (.) i To fid maimum likelihood estimates we a use ay miimizatio subroutie for diret miimizatio of the mius log-likelihood itself. 5 Colusios I this aer we disussed irular distributio resultig from etedig Stereograhi Semiirular Weibull distributio o uit semiirle whih is obtaied by iduig Iverse Stereograhi Projetio o the real lie. The desity ad distributio futios of

9 Joural of New Theory (6) -9 9 Stereograhi Cirular Weibull distributio admit eliit forms as do trigoometri momets ad observed that i similar to liear ase Stereograhi Cirular Eoetial ad Stereograhi Cirular Rayleigh distributios are Seial ases to roosed Stereograhi Cirular Weibull Distributio. As this distributio is asymmetri romisig for modelig asymmetrial diretioal data. Referees [] Byoug J.A ad Hyoug M.K.(8). A New Family of Semiirular Models: The Semiirular Lalae Distributios Commuiatios of the Korea Statistial Soiety Vol [] Gradshtey ad Ryzhik (7). Table of Itegrals series ad roduts 7th editio Aademi Press. [3] Guardiola J.H. (). The Semiirular Normal Distributio Ph.D Dissertatio Baylor Uiversity Istitute of Statistis. [] Jammalamadaka S. Rao ad Se Guta A. (). Tois i Cirular Statistis World Sietifi Press Sigaore. [5] Joes T.A. (968). Statistial Aalysis of orietatio data Joural of Sedimetary Petrology [6] Kim H.M.(8). New family of the t distributios for modelig semiirular data Commuiatios of the Korea Statistial Soiety. Vol.5 N [7] Mardia K.V. ad Ju P.E. () Diretioal Statistis Joh Wiley Chihester. [8] Phai Yedlaalli Girija S.V.S. ad Dattatreya Rao A.V. (3). O Costrutio of Stereograhi Semiirular Models Joural of Alied Probability ad Statistis. Vol 8 No [9] Phai Yedlaalli R.V. Babu S.V.S.Girija ad A.V.Dattatreya Rao (5). Stereograhi -l-aial Eoetial ad Stereograhi Cirular Eoetial Distributios Iteratioal Joural of Sietifi ad Iovative Mathematial Researh (IJSIMR) Vol.3 Seial Issue [] Ugai. S.K. Nishijima M. ad Ka T. (977) Charateristis of raidro size ad raidro shae Oe symosium URSI Commissio F 5-3. [] Toshihiro Abe Kuio Shimizu ad Arthur Pewsey () Symmetri Uimodal Models for Diretioal Data Motivated by Iverse Stereograhi Projetio J. Jaa Statist. So. Vol. (No. ). 5-6.