Confidence interval for the two-parameter exponentiated Gumbel distribution based on record values

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1 Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp Witer 2014 Joural homepage: Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values M. Abdi * Received: 31 July 2013 ; Accepted: 4 December 2013 Abstract I this paper we study the estimatio problems for the two-parameter expoetiated Gumbel distributio based o lower record values. A exact cofidece iterval ad a exact joit cofidece regio for the parameters are costructed. A simulatio study is coducted to study the performace of the proposed cofidece iterval ad regio. Fially a umerical example with real data set is give to illustrate the proposed procedures. Keywords Cofidece Iterval Expoetiated Gumbel Distributio Exact Joit Cofidece Regio Record Values. 1 Itroductio The Gumbel distributio is perhaps the most widely applied statistical distributio for climate modelig. Some of its applicatio areas i climate modelig iclude: global warmig problems flood frequecy aalysis raifall modelig ad wid speed modelig. A recet book by Kotz ad Nadarajah [1] which describes this distributio lists over 50 applicatios ragig from accelerated life testig through to earthquakes floods horse racig raifall sea currets wid speeds ad track race records (to metio just a few). I literature expoetiated family of distributios defied i two ways. If G( x; ) is cumulative distributio fuctio (cdf) of a baselie distributio the by addig oe more parameter (say ) the cdf of expoetiated baselie distributio F( x; ) is give by a) F x; G x; 0 x R b) F x; 1 1 G x; 0 x R where is parameter space. Gupta et al. [2] itroduced the expoetiated expoetial (EE) distributio as a geeralizatio of the expoetial distributio. The two parameters EE distributio associated with defiitio (a) have bee studied i detail by Gupta ad Kudu [3] which is a sub-model of the expoetiated Weibull distributio itroduced by Mudholkar ad Shrivastava [4]. Nadarajah [5] itroduced expoetiated Gumbel distributio usig (b). Some of its applicatio areas i climate modelig iclude global warmig problem flood frequecy aalysis offshore modelig raifall modelig ad wid speed modelig. Persso ad Ryde [6] discussed * Correspodig Author. () me.abdi@bam.ac.ir. (M. Abdi) M. Abdi M.Sc Departmet of Mathematics ad Computig Higher Educatio Complex of Bam Bam Ira.

2 62 M. Abdi / IJAOR Vol. 4 No Witer 2014 (Serial #11) estimatio of T-year retur values for sigificat wave height i a case study ad compare poit estimates ad their ucertaities to the results give by alterative approaches usig Gumbel or Geeralized Extreme Value distributios. A radom variable X is said to have Gumbel distributio if its cdf is x ; e G x e x 0 0 By itroducig a shape parameter θ 0 ad usig defiitio (a) the cdf of the expoetiated Gumbel distributio is x e F x; G x; e x (1) th which is simply the θ power of cdf of the Gumbel distributio. The probability desity fuctio (pdf) correspodig to (1) is x x e f x; e x (2) We shall write X EG( ) to deote a absolutely cotiuous radom variable X havig the two-parameter expoetiated Gumbel distributio with shape ad scale parameters ad σ respectively whose pdf is give by (2). The purpose of this paper is to costruct the iterval estimatio for the two-parameter expoetiated Gumbel distributio based o lower record values. The rest of this paper is orgaized as follows. Sectio 2 provides some prelimiaries. I Sectio 3 we preset a exact cofidece iterval for scale parameter ad a exact joit cofidece regio for the parameters ( ) based o lower record values. I Sectio 4 a Mote Carlo simulatio is coducted to study the performace of the proposed cofidece iterval ad regio. Fially i sectio 5 a umerical example with real data set is preseted to illustrate the proposed methods. 2 Prelimiaries Let X1 X 2 be a sequece of idepedet ad idetically distributed (iid) cotiuous radom variables with cdf F( x) ad pdf f ( x ). A observatio X j is called a upper (lower) record value of this sequece its value exceeds (is lower tha) that of all previous observatios. Geerally let us defie T1 1 U1 X1 ad for 2 T mi j T : X X U X. -1 j T -1 T The the sequece { U }({ T }) is kow as upper record statistics (upper record times). Similarly the lower record times S ad the lower record values L are defied as follows:

3 Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values 63 S1 1 X1 ad for 2 S = -1 mi j S : X X L X. For more details o j S -1 S records ad its applicatios see Nevzorov[7] Ahsaullah[8] ad Arold et. al. [9]. The followig lemmas are useful i this paper. Lemma 2.1. Let L2 be the first m observed lower record values from a populatio with cdf F (.). Defie U l[ F( L )] i 1 2 m. i i The U1 U2 Um are the first m upper record values from a stadard expoetial distributio. Proof. From the joit pdf of L2 ad usig a simple Jacobia argumet we ca easily obtai the joit pdf of U1 U2 Um as f u u u e 0 u u... u -um U1 U2... Um 1 2 m 1 2 m which is the joit pdf of the first m upper record values from a stadard expoetial distributio (see Arold et al. [9]). The proof is thus obtaied. Lemma 2.2. If U1 U2 Um are the first m upper record values from a stadard expoetial distributio. The the spacigs U1 U2 U1 Um Um 1 are iid radom variables from a stadard expoetial distributio. Proof. The proof ca be foud i Arold et al. [9]. 3 Mai Result Let L2 be the first m observed lower record values from the expoetiated Gumbel distributio. I this sectio a 100(1 )% cofidece iterval for scale parameter ad a 100(1 )% joit cofidece regio for ( ) are costructed based o the observed lower records L2. Li Let us defie Yi exp i 1 2 m. The by Lemma 2.1 Y1 Y2 Ym are the first m upper record values from a stadard expoetial distributio. Moreover by Lemma 2.2 we ca observe that Z1 Y1 Z2 Y2 Y1 Z Y Y m m m 1 (3)

4 64 M. Abdi / IJAOR Vol. 4 No Witer 2014 (Serial #11) are iid radom variables from a stadard expoetial distributio. Hece V 2Z 2 Y (4) 1 1 has a chi-square distributio with 2j degrees of freedom ad m U 2 Z 2 Y Y i m 1 (5) i2 has a chi-square distributio with 2( m 1) degrees of freedom. We ca also fid that U ad V are idepedet radom variables. Let T U / 2( m 1) U 1 Y Y (6) m 1 V / 2 ( m 1) V m 1 Y1 Ad S U V 2 Y m (7) It is easy to show that T has a F distributio with 2( m 1) ad 2 degrees of freedom ad S has a chi-square distributio with 2m degrees of freedom. Furthermore T ad S are idepedet see Johso et al. ([10] P. 350). Let F ( 1 2 ) be the percetile of F distributio with right-tail probability ad 1 ad 2 degrees of freedom. Next theorem gives a exact cofidece iterval for the scale parameter base o lower record values. Theorem 3.1. Suppose that L2 be the first m observed lower record values from EG distributio i (1). The for ay 0 1 l[1 m 1 F (2( m 1) 2)] l[1 m 1 F (2( m 1) 2)] 1 is a 100(1 )% cofidece iterval for. Proof. From (6) we kow that the pivot T 1 e e 1 e 1 L 1 m 1 m 1 e has a F distributio with 2( m 1) ad 2 degrees of freedom. We ote that T ( ) is strictly

5 Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values 65 decreasig fuctio of σ. Hece for 0 1 we obtai F 2 m 1 2 T F 2 m is equivalet to the evet l[1 m 1 F (2( m 1) 2)] l[1 m 1 F (2( m 1) 2)] 1 this completes the proof. It should be metioed here that we ca also use T ( ) to test ull hypothesis H0 : 0. Let 2 ( ) deote the percetile of 2 distributio with right-tail probability ad degrees of freedom. Next theorem gives a exact joit cofidece regio for the parameters ad. Theorem 3.2. Suppose that L2 be the first m observed lower record values from EG distributio. The the followig iequalities determie 100(1 )% joit cofidece regio for ad : l[1 m 1 F (2( m 1) 2)] l[1 m 1 F (2( m 1) 2)] (2 m) (2 m) Proof. From (7) we kow that L m S 2 e 2 has a distributio with 2m degrees of freedom ad it is idepedet of T. Hece for 0 1 we have P [ F (2( m 1) 2) T F (2( m 1) 2)] ad P[ (2 m) S (2 m)] From these relatioships we coclude that

6 66 M. Abdi / IJAOR Vol. 4 No Witer 2014 (Serial #11) P F m T F m m S m or equivaletly l[1 m 1 F (2( m 1) 2)] l[1 m 1 F (2( m 1) 2)] 4 Simulatio study (2 m) (2 m) I this sectio a Mote Carlo simulatio is coducted to study the performace of the proposed cofidece iterval ad joit cofidece regio. I this simulatio we radomly geerated lower record sample L2... L m from the Gumbel distributio with the value of parameters ad the computed 95% cofidece itervals ad regios usig the results preseted i Sectio 3. We the replicated the process 5000 times. We preseted i Table 1 the simulated average cofidece legth for parameter cofidece area for the parameters ( ) ad the 95% coverage probabilities of the proposed cofidece itervals ad regios. From Table 1 we observe whe m icreases the average cofidece legth for ad the average cofidece area for ( ) are decreased. The simulatio results shows that the coverage probabilities of the exact cofidece itervals for parameter ad joit cofidece regios for parameters ( ) are close to the desired level of 0.95 for differet sample sizes. Hece our proposed methods for costructig exact cofidece itervals ad joit cofidece regios ca becused reliably.. Table 1 The simulated average cofidece legth (CL) cofidece area (CA) ad 95% coverage probabilities (CP) for the parameters. m CL ( ) CA( ) CP( ) CP( )

7 Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values 67 5 Numerical example I this sectio real example with climate record data are give to illustrate the proposed cofidece itervals ad joit cofidece regios. We preset a data aalysis ad illustrate applicatio of the results i Sectio 3 to the seasoal (July 1-Jue 30) raifall i iches recorded at Los Ageles Civic Ceter from 1962 to 2012 (see the website of Los Ageles Almaac: weather/we13.htm). The data are as follows: Here we checked the validity of the expoetiated Gumbel Model based o the parameters ˆ ˆ usig the Kolmogorov Smirov (K-S) test. It is observed that the K-S distace is K S with a correspodig P Value So the expoetiated Gumbel model provides a good fit to the above data. If oly the lower record values of the seasoal raifall have bee observed these are To fid a 95% cofidece iterval for ad a joit cofidece regio for ad we eed the followig percetiles: F F F (10 2) F (10 2) ad (12) By Theorem 3.1 the 95% CI for is ( ) with cofidece legth By Theorem 3.2 the 95% JCR for ad is determied by the followig iequalities: with area Figure 1 shows the above joit cofidece regio for the parameters.

8 68 M. Abdi / IJAOR Vol. 4 No Witer 2014 (Serial #11) Fig. 1 Joit cofidece regio for parameters ad. Ackowledgmets The authors would like to thak the Editor ad the referees for their useful commets which improved the paper. Refereces 1. Kotz S. Nadarajah S. (2000). Extreme Value Distributios: Theory ad Applicatios. Lodo: Imperial College Pres:. 2. Gupta R. C. Gupta P. L. Gupta R. D. (1998). Modelig failure time data by Lehma alteratives. Commuicatios i Statistics-Theory ad Methods 27: Mudholkar G. S. Srivastava D. (1993). The expoetiated Weibull family for aalyzig bathtub failure-rate data. IEEE Tras. Reliability. 42: Nadarajah S. (2005). The expoetiated Gumbel distributio with climate applicatio. Evirometrics. 17 (1) Persso K. Ryde J. (2010). Expoetiated Gumbel Distributio for Estimatio of Retur Levels of Sigificat Wave Height Joural of Evirometal Statistics. 1(3) Nevzorov V. B. (1988). Records. Theory of Probability ad its Applicatio Ahsaullah M. (1995). Record Statistics. New York : Nova Sciece Publishers Ic. Commack. 8. Arold B. C. Balakrisha N. Nagaraja H. N. (1998). Records. New York: Joh Wiley & Sos. 9. Gupta R.D. Kudu D. (2001). Expoetiated expoetial family: a alterative to gamma ad Weibull distributios. Biometrical Joural Johso N. L. Kotz S. Balakrisha N. (1994). Cotiuous Uivariate Distributios. New York: Joh Wiley & Sos.

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