Some Statistical Inferences on the Records Weibull Distribution Using Shannon Entropy and Renyi Entropy

Size: px

Start display at page:

Download "Some Statistical Inferences on the Records Weibull Distribution Using Shannon Entropy and Renyi Entropy"

Patrick Russell
5 years ago
Views:

OPEN ACCESS Coferece Proceedgs Paper Etropy www.scforum.

1 OPEN ACCESS Coferece Proceedgs Paper Etropy Some Statstcal Ifereces o the Records Webull Dstrbuto Usg Shao Etropy ad Rey Etropy Gholamhosse Yar, Rezva Rezae * School of Mathematcs, Ira Uversty of Scece ad Techology, Tehra, IRAN; E-Mals: Yar@ust.ac.r School of Mathematcs, Ira Uversty of Scece ad Techology, Tehra, IRAN; E-Mals: Rezva_Rezae@mathdep.ust.ac.r * School of Mathematcs, Ira Uversty of Scece ad Techology, Tehra, IRAN; E-Mals: Rezva_Rezae@mathdep.ust.ac.r; Tel.: ; Fa: Receved: September 04 / Accepted: 30 September 04 / Publshed: November 04 Abstract: I ths paper, we dscuss dfferet estmators of the records Webull dstrbuto parameters ad also we apply the Kullback-Lebler dvergece of survval fucto method to estmate record Webull parameters. Fally, these methods have bee compared usg Mote Carlo smulato. Keywords: Webull dstrbuto; Upper record; Etropy; Parameter estmato; Smulato. PACS Codes: 6F, 6N0, 6F0, 6F5, 54C70, 37M05.. Itroducto I the commo practce, the commucato theory s ecessary order to quatfy ad fully study the formato mathematcally. Iformato theory s a brach of appled mathematcs, electrcal egeerg ad computer scece volvg the quatfcato of formato. lmted formatotheoretc deas had bee developed by Boltzma 896, Nyqust 94, Hartley 98, 930 ad etc. What s kow today as the formato theory has bee fouded by Shao 948 through troducg H as the etropy of radom varable. He for the frst tme troduced the qualtatve ad quattatve model of commucato as a statstcal process uderlyg formato theory. Later o, other measures were troduced such as Rey etropy, relatve etropy ad Mutual

2 formato. Relatve etropy also Kullback-Lebler dvergece was frst defed by Kullback ad Lebler 95 as the drected dvergece betwee two dstrbutos. Also, record values ad assocated statstcs have wdely bee used may real lfe applcatos, for eample sports, weather, busess, etc. Record values are very mportat case whe observatos are dffcult to obta or whe they are beg destroyed whe observatos are subected to a epermetal test. Observatos are obtaed by observg successve mamum mmum values. The term record value was frst troduced by Chadler 95. Some feretal statstcs based o record value have bee dscussed by Ahsaullah 988, 990, Balakrsha et al. 993, Balakrsha et al. 995, Arold et al. 998, Sulta et al. 999, Sulta et al. 000, Ahmad 000, Ahmad et al. 00a, 003, Raqab 00 ad Solma et al. 006, Ahsaullah 004 ad Ahsaullah et al. 006, Ahmad et al. 008, 009 ad etc. I statstcs, the Webull dstrbuto s oe of the most mportat cotuous probablty dstrbutos. It was, frst troduced by Webull 939 whe he was studyg the ssue of structural stregth. Teymor et al. 0 troduce a pot estmator for shape parameter of upper record Webull dstrbuto ad use mamum-lkelhood estmato MLE of scale parameter ther method. Also some feretal statstcs based o Webull dstrbuto have bee dscussed by Yar et al. 03. I ths paper we use Kullbak-Lebler method for estmato parameters of upper record Webull ad we dscuss o pot estmator method. The results show momet method estmato MME or Kullback- Lebler dvergece of survval fucto estmato DLS of are better tha mamum-lkelhood estmato MLE of, for estmato. Now we troduce some basc deftos that play a cetral role the preset paper, usg the otato of Arold et al. 998 ad Yar et al. 03. Defto. LetY ma{,,..., }, s a upper record of,,..., f. Y Y,.By ths defto s a upper record values. Sce we replace,...,, by,,..., or f P 0,, the,,..., upper record values of every ew sequece wll correspod to lower record values of the orgal sequece, so ths paper we shall study oly upper record values ad shall use the otato U for the th upper record statstc. The pdf of the U s obtaed by log[ F ] fu f,,,,...,, where s the gamma fucto. Defto. Suppose that be a radom varable wth pdf f ad wth support S, Shao etropy of s defed h f log f d. S Defto.3 Suppose that be a radom varable wth pdf f ad wth support S, Rey etropy of s defed

3 3 h, log, 0. f d Defto.4 Let,... be a sequece of postve, depedet ad detcally dstrbuted d, S radom varable wth a o-creasg survval fucto F, P wth the support S ad vector of parameters. Defe the emprcal survval fucto of a radom sample of sze by G I, 0 [, 3 4 where s the dcator fucto ad s the ordered sample.[] Defto.5 Let F, be the true survval fucto wth ukow parameters ad be the emprcal survval fucto of a radom sample of sze N from F,. Defe the Kullback-Lebler dvergece of Survval fucto DLS G G ad F by G F G l [ G F ] d. 5 0 F The rest of ths paper s orgazed as follows. I the et secto we state some propertes of record statstcs. Secto 3 preset the upper record value of Webull dstrbuto ad ts Etropy. Our approach s llustrated secto 4. Fally, we dscuss the estmato of parameters of the upper records Webull dstrbuto by Mote Carlo smulato secto 5.. Some propertes of record statstcs I curret secto we eame some propertes of upper record statstcs. See Arold et al. 99 ad Temour et al. 0. a Usg, the ch-square dstrbuto wth degree of freedom ad coverso of U log F, we have d F ep U b The 00 % cofdece terval for s U,, F ep, F ep c The p th qutle for 0<p< of s U F p F p F ep U See more detals Arold et al. 99 ad Temour et al. 0.I the remader of ths specfcato, we wll state a lemma ad two theorems whch has bee preseted the Baratpour et al Lemma. Ahmad, 000 For, we have G

4 4 e log d![ ], 6 where s the Euler s costat. Theorem. Let,... be a sequece of d radom varables from Cdf F wth pdf f ad etropy, H. We have H log,, U f z z z Where f e log f F e dz.! 0 7 Theorem.3 Let Y s a radom varable from cdf F wth pdf f. Uder the assumpto of Theorem, we have: log f y * H H B E[ Y ] log M, U U F y Y * H H log M, U U where M f m *, U ~,, B e., m s the mode of the dstrbuto ad Y=M 3. The upper record value of Webull dstrbuto ad ts Etropy Now we eame the Shao's Etropy o the upper record of the Webull dstrbuto. A cotuous radom varable s sad to be a two-parameter Webull dstrbuto wth shape parameter α ad scale parameter β deoted by ~ W,, f ts cdf s F ep[ ], 0. Now accordg to secto ad F y [ log F ], 0 we have: a d U, f U e 8 ad F U 0 e! 9

5 5 b The 00 % Cofdece terval for s U,,, c The p th qutle for 0<p< of U s q U. d H U l log, H U, [ log log log ] ad DLS G F l l d,! 0 Where 0., 0 4. Estmato of parameters of upper records Webull dstrbuto I ths paper, we use the otato th UW for the th upper record Webull dstrbuto ad curret secto we estmate shape parameter α ad scale parameter β by fve methods. Suppose that,,..., s a radom sample from 7 wth sample sze. a Momet method MME: Here we provde the MME method of the parameters of a th UW dstrbuto. To ths purpose, oe ca show that ts mea ad varace of the sample are respectvely : I, II s We cosder three cases: Whe α s kow, the from a we have a estmator for β, say ~ kow. ~ : kow Whe β s kow, ths case we eed to solve a wth respect to α, deoted by ~ kow: ~. kow **

6 6 Whe both α ad β are ukow, frst we obta the populato coeffcet of varato CV from a ad b var [ E The, equatg the sample CV wth the populato CV We observe that the populato CV s depedet of the β we have Where s s ad ] [ ]. We eed to solve to obta MME of α, deoted by ~.The substtutg α 0 we have a estmator for β, deoted by ~. 3 b Mamum-lkelhood method MLE: Here the mamum lkelhood estmators of s gve by th UW are cosdered. The log-lkelhood fucto l L, l [l l ]. 4 We cosder three dfferet cases: α s kow: dfferetate 3 wth respect to β the equatg to zero ad solvg wth respect to β. The the MLE of β, deoted by ˆ MLE, wll be ˆ MLE 5 β s kow: aga dfferetate 3 wth respect to α the equatg by equatg by zero solvg wth respect to α. The the MLE of α, deoted by ˆ MLE, wll be l L ˆ l l l 0 ˆ MLE 6 MLE Both α ad β are ukow: ths case, settg 4 3 ad the dfferetate o t wth respect to α the equatg to zero ad solvg wth respect to α. Here the MLE of α, deoted by ˆ,wll be

7 7 0. l l l l ] l [ l 7 c Bayesa method Here, we assume the prevous formato of α ad β are depedet of each other, so,. We represet ths approach for ormal, uform ad tragular pror dstrbuto by Mote Carlo smulato. d Pot estmator for the shape parameter PE Theorem 4. Temour et al. 0 Suppose that a sequece of th upper record from Webull famly are observed. A smple estmator of shape parameter s gve by, log log ˆ log ˆ U m PE 8 where ˆ U m s the sample meda of th upper record values.[5] e Kullback-Lebler dvergece of survval fucto method DLS: To estmate the parameters by ths method, we set8 4. The we have 0 0 l l! DLS G F d 9 where 0, 0. Now dfferetate!8 wth respect to α the equatg to zero ad solvg wth respect to α. The,. 0 But we do t have a good estmator for α by ths equato sce t does ot deped o the sample values. Now we suppose that α s kow. Now Equato 8 should be mmzed respect to β. For ths purpose, we have

8 8 k l l k0 k! 0. d 0 l l k k! We observe that a close form soluto of 9 for β s ot possble. k0 5. Smulato study Sce the MME, MLE, DLS ad Bayesa estmato of the parameters have ot closed form so checkg the performace of them, theoretcally s a dffcult task. Therefore, they must be solved umercally. We have doe ths work by Matlab software. Frst, we have geerated,,..., from a th 4 upper record Webull dstrbuto for more detals, see Teymor ad Gupta. I ths study we assumed, 3 ad both ad are ukow. Ths sample smulated was used to true true estmate the 4 th UW usg the MME, MLE, DLS, Bayesa ad pot methods. The above process was repeated 0,000 tmes. Cosequetly, we have a set of M= th UW parameter estmatos usg each method. The the mea values ad ad sample varaces S ad S were computed usg : Where k ad k are the estmated 4 th -UW shape ad scale parameters from th k sample. To llustrate the effect of sample sze, we carred out a smulato study for some levels of =5, 6,, 30. Frst, we study the behavor of β gve 0, 4 ad 9. Fgure shows the smulato results of. The followg results ca be cocluded from ths fgure: Accordg to fgure.a, the MME, DLS ad Bayesa wth pror ormal methods act better tha the rest terms of bas. MME ad Bayesa for are overestmated whereas DLS s uderestmated. Accordg to fgure.b, mea square error MSE of MME method s best for every. Now we study the behavor of gve, 6, 7 ad Bayesa method wth four pror dstrbuto. Fgure shows the smulato results for. The followg results ca be cocluded from ths fgure: Accordg to fgure.a, The MME ad pot estmator methods act better tha the rest ad are so smlar terms of bas. Also MLE values for are closed to true, the ths method ca be a good estmator for. Fgure.b shows MSE of MME, MLE ad pot estmator methods are best for every ad very closed to zero.

9 9 Fgure. a. b MSE as fuctos of sample sze. Fgure. a. b MSE as fuctos of sample sze. 6. Coclusos The curret paper cocered wth dfferet estmators of records Webull dstrbuto parameters ad dscussed how approprate ad approprate these estmators are. The smulato process, suggests

10 0 MME ad DLS methods for estmatg the parameter β. MME ad proposed pot estmators based o estmated β by MME dcated to be approprate estmators for α. Refereces ad Notes. Ahmad,J. Etropy propertes of Certa Record Statstcs ad Some Characterzato results, JIRSS Vol.7, 009, -, -3.. Ahmad,J.; Doostparast,M. Statstcal ferece based o K-records, Mashhad R.J.Math.Sc Vol., 008, Ahmad,J.; Balakrsha,N. Cofdece tervals for qutles terms of record rage, Statstcal probablty Letters, 004, Balakrsha,N.; Cha, P.s. Recorded ferece. Values from Raylegh ad Webull dstrbuto ad assocated ferece. Proceedgs of the cofdece o Etreme Value Theory ad applcatos, volume 3 Gathersbarg, Marylad, Baratpour,J.; Ahmad,J; Argham.R. Etropy propertes of record statstcs, Statstcal papers 48, 007, Cover,T.M.; Thomas, J.A. Elemets of formato theory, d ed.; Wley, New York, Lo, J. Iformato theoretc cotet ad probablty. Ph.D Thess, Uversty of Florda, USA, Perez-Cruz,F. Kullbak-Lbler dvergece estmato of cotues dstrbutos, IEEE Iteratoal symposum o Iformato Theory, Toroto, Caada, Temour, M; Gupta, A.K. O the Webull record statstcs ad assocated fereces, Statstca, ao LII, 0,.. 0. Wuu Wu,J.; Chao,T. Statstcal ferece about the shape parameter of Webull dstrbuto by record values, statstcal papers48, 006, Yar,G.H.; Mrhabb,A; Saghaf.A. Estmato of the Webull parameters by Kullback-Lebler dvergece of Survval fuctos, Appl.Math.If.sc.7, 03, by the authors; lcesee MDPI, Basel, Swtzerlad. Ths artcle s a ope access artcle dstrbuted uder the terms ad codtos of the Creatve Commos Attrbuto lcese

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information Malaysa Joural of Mathematcal Sceces (): 97- (9) Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Hadeel Salm Al-Kutub ad Noor Akma Ibrahm Isttute for Mathematcal Research, Uverst