Confidence Intervals for Double Exponential Distribution: A Simulation Approach

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1 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Physcal ad Mathematcal Sceces Vol:6, No:, 0 Cofdece Itervals for Double Expoetal Dstrbuto: A Smulato Approach M. Alrasheed * Iteratoal Scece Idex, Physcal ad Mathematcal Sceces Vol:6, No:, 0 waset.org/publcato/8809 Abstract The double expoetal model (DEM), or Laplace dstrbuto, s used varous dscples. However, there are ssues related to the costructo of cofdece tervals (CI), whe usg the dstrbuto.i ths paper, the propertes of DEM are cosdered wth teto of costructg CI based o smulated data. The aalyss of pvotal equatos for the models here comparsos wth pvotal equatos for ormal dstrbuto are performed, ad the results obtaed from smulato data are preseted. Keywords Cofdece tervals, double expoetal model, pvotal equatos, smulato T I. INTRODUCTION HERE are may dfferet probablty dstrbutos wth aalytcal probablty desty fuctos (PDF) that are used accordg to the research dscple. The most commoly appled PDF s ormal dstrbuto, whch descrbes a wde rage of dfferet processes. However, may processes ca be descrbed more precsely usg the double expoetal model (DEM) [-3]. Recetly, terest the Laplace dstrbuto has grow due to ts potetal trasformg applcato facal fuctos[3]. For stace, the dfferece of two depedet two parameter expoetal varables follows double expoetal dstrbuto, ad the logarthm of ratos of uform or Pareto varables follows the DEM, as well []. DEM has ot bee used extesvely as a model due, part, to the lack of avalable statstcal techques avalable for ths dstrbuto. From the expermeter s pot of vew, DEM s ofte ot used because of a sharp peak the ceter of the PDF; however,despte ths aomaly, t s ofte the preferred model wth expoetal tals [],[3]. I addto, t has bee suggested that to vestgate the propertes of real data,we eed to frst perform several goodess-of-ft tests. A example of oe such test s descrbed by [3]. Whle the evaluato of CI s mportat, t s trval cases of ormally dstrbuted values []. I such, we ca use auxlary dstrbutos, such as Studet, Laplace, ad. The goal of the preset study s to aalyze evaluato methods of CI for DEM. I addto, DEM data smulato, pvot costructo, ad aalyss of the results are also performed ths study. II. DOUBLE EXPONENTIAL DISTRIBUTION ANALYSIS The probablty desty fucto of DEM s defed as f x η ( x,, ) e () η * M. Alrasheed s wth the School of Busess, Kg Fasal Uversty, Hofuf, Alhasa 398 Saud Araba (phoe: ; fax: ; e-mal: malrasheedy@ kfu.edu.sa). DEM s specfc class of dstrbuto: x η + β f ( β ) exp () + ( + β ) Γ + ( + β ) Ths class of dstrbutos cludes ormal ( β 0 ) ad DEM ( β ), whch s symmetrcal scale ad locato parameters. Thus, we ca compare the PDFs of these two dstrbutos: Fg. Probablty desty fuctos of DEM ad Gauss dstrbutos Oe ca see from Fgure that the DEM has heaver tals tha the ormal dstrbuto. It s therefore evdet that creasg scale parameters leads to agradual decreasg of probablty desty fucto. III. SIMULATION OF DATA WITH DOUBLE EXPONENTIAL PROBABILITY DENSITY FUNCTION To geerate the umbers dstrbuted wth double expoetals, we requre a smple radom umber geerator Iteratoal Scholarly ad Scetfc Research & Iovato 6() 0 84 scholar.waset.org/ /8809

2 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Physcal ad Mathematcal Sceces Vol:6, No:, 0 ad the equato for the probablty desty fucto. The dstrbuto fucto for DEM s: x η e, x η F( x) (3) x η e, x > η Now, we ca fd x from the equato for F (x) : The correlato of the expermetal ad theoretcal results cofrms the accuracy of our approach. IV. ANALYSIS OF DEM DATA AND CONSTRUCTION OF CONFIDENCE INTERVALS A. Evaluato of Pvots After we have geerated the data wth DEM, cosder the tme seres obtaed from our data usg the followg equatos: l F + η, F < / x l( ( F) ) + η, F > / (4) W η, V X X X X X (5) Iteratoal Scece Idex, Physcal ad Mathematcal Sceces Vol:6, No:, 0 waset.org/publcato/8809 Fg. DEM geerato scheme For each radom umber, we smply calculate the x value usg the above equato. Ths scheme geerato s show Fgure. We ca use ths scheme cases whe the probablty desty fucto s defed aalytcally ad t s smple to derve the equato for varable x. Next, the radom umbers the terval ca be easly geerated. To check ths DEM geeratg algorthm, the theoretcal ad expermetal results for DEM wth η 5 ad 3, whch should geerate the theoretcal PDF as show (), are compared. For expermet, we ll use the method descrbed above. I ths expermet, X s the meda value. I the case of eve data, the array legth s defed as: X ( Array[ N / ] + Array[ N / + ] ) (6) The dstrbuto of W s show Fgure 4. A Fg. 3 Theoretcal ad expermetal probablty desty fuctos for η 5, 3 B Fg. 4 W-dstrbuto: A( η 5, 3 ), B( η 6, ) After comparg Fgure 4A ad Fgure 4B, we ca coclude that the W-dstrbuto s depedet of the DEM parameters Iteratoal Scholarly ad Scetfc Research & Iovato 6() 0 85 scholar.waset.org/ /8809

3 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Physcal ad Mathematcal Sceces Vol:6, No:, 0 Iteratoal Scece Idex, Physcal ad Mathematcal Sceces Vol:6, No:, 0 waset.org/publcato/8809 ( η, ). The percetles of W-dstrbuto are obtaed from the smulated data. The V-dstrbuto s show Fgure 5. B Fg. 5 V-dstrbuto: A( η 5, 3 ), B( η 6, ) A It s apparet from Fgure 5 that the V-dstrbuto doe sot η,, ether. deped o DEM parameters ( ) B. Cofdece Itervals for DEM Dstrbuto Cofdece tervalcostructo s a mportat part of the statstcal ferece thatrefers to obtag statemets such as P a X... X ) b( X... X ), where γ s typcally ( ) γ ( chose to be 0.9, 0.95, or I other words, we use the CI for the estmato of a ukow geeral parameter usg oly a sample of the data wth the cofdece probablty γ.the prcple of the CI s show Fgure 6. Oe of the most useful methods for costructg CI s the method of pvotal quattes. A pvot s a fucto of δ X, X.., whose dstrbuto does ot deped o the ( ), X parameterδ. I the case of DEM dstrbuto, the parameters are η ad. As log as the dstrbutos of W ad V do ot deped o correspodg parameters, W-dstrbuto ad V- dstrbuto should be used as pvots for estmato of the populato mea ad varace.ulke DEM dstrbuto, Normal dstrbuto (N) s well studed ad the CI algorthm costructo for the latter dstrbuto s smple. The pvots for N are the t-dstrbuto for the mea ad -dstrbuto for the varace, whch are a cosequece of the propertes of N [5]. The CI for the populato mea s defed as: s s P x tγ < η < x + tγ γ, (7) where x s sample mea, s s sample varace, ad t γ (,γ ) s the value from the t-dstrbuto table correspodg to the cofdece probablty ( γ ) ad umber of degrees of freedom ( ). The CI for the populato varace s defed as: where ( ) s ( ) s P < < γ, (8), are the values from the -dstrbuto table, whch correspod to the umber of degrees of freedom ( ) ad probabltes, +γ, γ [5]. To costruct the CI for DEM dstrbuto, the pvotal dstrbutos W ad V, whch are obtaed from the gve data sample usg equatos (5), should be used. The CI for the populato mea s defed as: Pa X η X X b γ I order to costruct CI wth the correspodg percetles of pvot dstrbuto W, the followg equato should be used: (9) Fg. 6 Cofdece terval ad true value of a ukow parameter Fg. 7 Percetles for the CI correspodg to γ probablty Iteratoal Scholarly ad Scetfc Research & Iovato 6() 0 86 scholar.waset.org/ /8809

4 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Physcal ad Mathematcal Sceces Vol:6, No:, 0 Iteratoal Scece Idex, Physcal ad Mathematcal Sceces Vol:6, No:, 0 waset.org/publcato/8809 Fally, we obta the equato for η estmato: X X + X X X X γ, X X γ (0) Smlar to the aforemetoed descrbed procedure, we ca costruct the CI for the populato varace usg the pvotal dstrbuto V: X X X + γ, X X X γ C. Asymptotc Cofdece Itervals for DEM Dstrbuto Based o -Dstrbuto () I ths secto, we wll costruct the asymptotc CI for the DEM dstrbuted data. Cosder the followg equatos: ad X η T X ± T + η The absolute value of Jacoba of trasformato s: d dt T ± + η () (3) (4) Usg the exstg dstrbuto of depedet value X wth PDF f (X ) ad the relatoshp wth depedet varable Y g(x ), we ca obta the equato for the PDF of depedet varable Y [5]: d fy ( y) f X ( g ( y)) g ( y) (5) dy For our case, g s defed (3). Thus, we obta: t f ( t ) exp + η η + t + η η t + exp exp( ) Ths equato s a desty fucto of (6) -dstrbuto wth degrees of freedom. We ca use the property of gamma dstrbuto [5] regardg the sum of depedet gammavarables. Cosderg that the -dstrbuto s a partal case of gamma dstrbuto, the same property s accurate for ths dstrbuto. X Therefore, η follows the -dstrbuto, ad ths dstrbuto ca be used as a pvot for the costructo of the asymptotcal CI. V. SIMULATION AND DISCUSSION A. Examg the pvotal quattes I order to prove the relablty of the descrbed techques for the costructo of CI, t s mportat to cosder a example wth a smulated data sample sze of 0x000. The obtaed expermetal dstrbuto from our data sample s show Fgure 3. The.5 ad 97.5 percetles of obtaed pvotal equatos (5) for smulated data wth η 5, 3 are: W.5, W % 97.5% The correspodg percetles of the t-dstrbuto wth 9 degrees of freedom are: t.5%.6, t97.5%.6 From these data, we ca see that the.5 percetle of W s greater tha the correspodg percetle of t-dstrbuto ad the 97.5 percetle of W s smaller tha the 97.5 percetle of t-dstrbuto. The cosdered percetles for V-dstrbuto are: V.04, V97.5%.5%.78 Now, we ca costruct the CI for the populato mea ad varace based o W, V, ad also o the t ad - dstrbutos. Cosder the followg small sample of smulato data: ,.705, 4.790,.899, 0.49, 5.304, 0.076, ,.909,.47 The meda, mea, ad stadard devato of the sample are.669,.439, ad 4.335, respectvely. The 95% CI for the populato mea ad varace based o both obtaed pvots Iteratoal Scholarly ad Scetfc Research & Iovato 6() 0 87 scholar.waset.org/ /8809

5 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Physcal ad Mathematcal Sceces Vol:6, No:, 0 Iteratoal Scece Idex, Physcal ad Mathematcal Sceces Vol:6, No:, 0 waset.org/publcato/8809 ad also o stadard t ad -squared, whch are used for ormal dstrbuto, ca be evaluated. For the smulated data, we obta the followg data usg equatos (7-): 0.6 η 7.794, based o W ad V. The CI, f we treat the data as ormallydstrbuted, are: based o t ad 9.39 η , squared dstrbutos. I last equato we σ have used the fact that after applyg (8). We ca see that, our case, the tervals for η are almost equal, but for we obta arrower tervals for W ad V dstrbutos. Ulke t ad -squared dstrbutos, W ad V dstrbutos are more precse for the DEM data because they are costructed drectly from the smulated sample. B. Costructo of Cofdece Itervals for the Populato Varace from the Smulated Data To demostratethat the use of pvotal equatos for ormal dstrbuto s correct for DEM cases, oe shouldcostruct 000 CI for each 0 values ofsmulated data. I ths study, we use the equatos (7) ad (8) to costruct the CI. After applyg these estmators, we obtaed 843 pots, whch captures the actualmea value ofη 5, thus, our case, the true coverage level s 84.3%. For the varace we obtaed 6 pots, whch captures the true value of varaceσ that correspods to 6.% of the true cofdece level. Now, let s exame the result obtaed 5.3. We ca calculate the CI for the populato varace based o - squared dstrbuto wth 0 degrees of freedom for 0 pots X from our smulato sample. Usg the pvot ηˆ we have: ˆ X η P,0.05 < <, Hece, the asymptotc cofdece terval for s: X ˆ η X, ,,0.975,0.05,0.5 ˆ η (.97,7.065) I ths example, we usethe percetles of the dstrbuto wth 0 degrees of freedom. It s therefore apparet that both asymptotc cofdece terval ad the oe based o bootstrappgprmarly agree ad capture the true value of the populato varace. VI. CONCLUSION I ths paper, we have vestgated the propertes of the DEM. The pvot equatos for ths dstrbuto were obtaed based o the smulated data. We have aalyzed the techques for the costructo of the CIfor both DEM ad ormal dstrbutos. Our study reveals that the trval usage of the pvots of ormal dstrbuto s correct f appled to DEM dstrbuted data. Also, we have demostrated that for populato varace estmato,the -dstrbuto wth 0 degrees of freedom, whch s a very mportat result for practcal applcatos, ca be used.these approachesmay be used as a theoretcal bass for the aalyss of expermetal data. REFERENCES [] S. Kotz, T. Kozubowsk, K. Podgorsk, The Laplace Dstrbuto ad Geeralzatos, Brkhauser Bosto, 00, Ch. -4. [] N. Johso, S. Kotz, N. Balakrsha, Cotuous Uvarate Dstrbutos, Vol., Wley-Iterscece, 995, Ch. 4. [3] Y. R. Gel, "Test of ft for a Laplace dstrbuto agast heaver taled alteratves", Computatoal Statstcs ad Data Aalyss, Vol. 54, o. 4, pp , 00. [4] N. Ekstrad, B. Smeets, "Weghtg of double expoetal dstrbuted data lossless mage compresso", Lud Uversty. Data compresso coferece, March 30-Aprl 0, 998. [5] J. Rce, Mathematcal statstcs ad data aalyss, 3 rd edto, Thomso Brooks/Cole, 007, p M. Alrasheed s PhD Operatos Research,Dept. of Quattatve Methods, School of Busess, Kg Fasal Uversty, Saud Araba. Research Iterests: Applcatos of Quattatve Methods Maagemet ad Face, Appled Operatos Research. Iteratoal Scholarly ad Scetfc Research & Iovato 6() 0 88 scholar.waset.org/ /8809