Interval Estimation of a P(X 1 < X 2 ) Model for Variables having General Inverse Exponential Form Distributions with Unknown Parameters

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1 Amerca Joural of Theoretcal ad Appled Statstcs 08; 7(4): do: 0.648/j.ajtas ISSN: (Prt); ISSN: (Ole) Iterval Estmato of a P(X < X ) Model for arables havg Geeral Iverse Expoetal Form Dstrbutos wth Ukow Parameters N. A. Mokhls, E. J. Ibrahm, D. M. Ghareb Departmet of Mathematcs, A Shams Uversty, Caro, Egypt Emal address: To cte ths artcle: N. A. Mokhls, E. J. Ibrahm, D. M. Ghareb. Iterval Estmato of a P(X < X ) Model for arables havg Geeral Iverse Expoetal Form Dstrbutos wth Ukow Parameters. Amerca Joural of Theoretcal ad Appled Statstcs. ol. 7, No. 4, 08, pp do: 0.648/j.ajtas eceved: February 8, 08; Accepted: March 6, 08; Publshed: May 3, 08 Abstract: I ths artcle the terval estmato of a P(X < X ) model s dscussed whe X ad X are o-egatve depedet radom varables, havg geeral verse expoetal form dstrbutos wth dfferet ukow parameters. Dfferet terval estmators are derved, by applyg dfferet approaches. A smulato study s performed to compare the estmators obtaed. The comparso s carred out o bass of average legth, average coverage, ad tal errors. The results are llustrated, usg verse Webull dstrbuto as a example of the geeral verse expoetal form dstrbuto. Keywords: Geeral Iverse Expoetal Form Dstrbuto, Coverage Probablty, Bootstrap Cofdece Iterval, Geeralzed Pvotal Quatty. Itroducto There has bee cotuous terest the problem of estmatg the stress stregth relablty, PX X, where X ad X are depedet radom varables. The parameter s referred to as the relablty parameter. Ths problem arses the classcal stress stregth relablty, where the radom stregth X of a compoet exceeds the radom stress X to whch the compoet s subjected. If X X, the the compoet fals. Kotz et al.[] have surveyed most all the theoretcal ad the practcal results o the theory ad applcatos of the stress-stregth relablty problem up to year 003. However, after year 003 several authors have cosdered the stress-stregth relablty problem by dfferet approaches for example, Al-Mutar et al. [], Amr et al. [3], ezae et al. [4], amog others. ecetly Mokhls et al. [5] have obtaed the pot ad the terval estmato of PX X by dfferet methods, uder the assumpto that X ad X are o-egatve depedet ad cotuous radom varables, havg the geeral verse expoetal form wth the cumulatve dstrbuto fuctos (CDFs) ad probablty desty fuctos (pdfs) gve respectvely by ; ;, ; ; ; ;,. where c s a commo kow parameter, η ζ ;,, are ukow parameters,ζ s the parametrc space,gx; c s a cotuous, mootoe decreasg, dfferetable fucto, such that, gx; c as x 0 ad gx; c 0 as x. They proved that the relablty fucto,, s gve by. / 0., f ad oly f, X ad X have CDFs as (). Also, Mokhls et al. [6] obtaed terval estmators of, whe c s ukow, ad η b, c s a fucto of the ukow parameters b ad c;,. The relablty,, takes the form.., / /, 0.., The preset paper, presets estmato of, whe X ad X follow the geeral verse expoetal form dstrbutos wth CDFs ad pdfs gve respectvely by () ()

2 33 N. A. Mokhls et al.: Iterval Estmato of a P(X < X ) Model for arables havg Geeral Iverse Expoetal Form Dstrbutos wth Ukow Parameters F 4 x;η,c = exp η b,c gx;c,ad f 4 x; η,c = η b,c g < x;c exp η b,c gx;c ; =,. (3) whereη = η b, c s a dfferetable fucto two ukow parameters b ad c, where c, b B, ad ad B are the parametrc spaces of c ad b, respectvely. The fucto gx; c s a cotuous, mootoe decreasg, dfferetable fucto, such that, gx; c as x 0 ad gx; c 0asx, g < x; c s the frst dervatve of gx; c w.r.t x. Usg (3), the relablty s gve =? η b, c A The above tegral ca be evaluated umercally. If c = c = c, the the relablty ca be expressed as () as Mokhls et al. [6]. Dfferet terval estmators are costructed by applyg dfferet approaches. () A approxmate cofdece terval for s costructed; usg the maxmum lkelhood estmator (ME) of. () A geeralzed cofdece terval s obtaed, usg the geeralzed varable (G) approach. () Two bootstrap cofdece tervals (percetle ad t) are also preseted. (v) Two Bayesa credble tervals of are obtaed, usg Markov cha Mote Carlo (MCMC) method, wth dfferet prors. The dfferet terval estmators obtaed are llustrated usg verse Webull dstrbutos as examples of the uderlyg dstrbutos. A comparso s performed, by meas of smulato amog the estmators obtaed o the bass of average legth, average coverage, ad tal errors. Ths paper s orgazed as follows: the approxmate cofdece terval for s obtaed, Secto. I Secto g z; c expd η b, c gz; c η b, c gz; c E dz (4) 3, usg geeralzed varable approach, the geeralzed cofdece terval of s derved. The bootstrap tervals are obtaed, usg percetle ad t-bootstrap methods, Secto 4. I Secto 5, usg MCMC method, two Bayesa credble tervals of are preseted by applyg two dfferet sets of prors. I Secto 6, takg the verse Webull dstrbuto as a example of the uderlyg dstrbutos, the results obtaed are llustrated ad a umercal comparso of the terval estmators s performed.. Approxmate Cofdece Iterval (ACI) of et X = FX, X,, X H; =,, be two depedet radom samples from populatos wth geeral verse expoetal form dstrbutos gve by (3). The lkelhood fucto s ( η η,, ) = η ( b,c ) + ( ) η ( b,c ) j j = = = = j= j (5) x,x, c c exp l l g (x ;c ) g(x ;c ), where x j s the jth observato the sample X ; j =,,. For smplcty wrte η = η b, c ; =,, the log-lkelhood fucto s ( ) ( ) η η = η + j η j = = j= = j= l = l x,x,, c, c l l g (x ;c ) g(x ;c ). (6) bj To derve the MEs,ηI, ci, ad bj of η,c, ad b ; =,, respectvely, solve the followg system of equatos for ηi, ck, ad l η = = 0,ad b η g(x j;c ) j= ˆ b b = b ˆ,c = c ˆ, η =ηˆ b = b,c = c ˆ, η =ηˆ (7) l η η = + ( j ) g(x ;c ) η j j c j j j ˆ η c = b c = c = c ˆ = b,c = c ˆ, η =ηˆ b = b,c = c ˆ, η =ηˆ l g (x ;c ) g(x ;c ) = 0. (8) η As kow 0, from (7) the ME, ηi of η ; =,, s b expressed as η (9) ˆ = ; =,. g(x ˆ j;c ) j= The ME, ci of c ; =,, ca be obtaed by substtutg (9) (8), ad the solvg (8) umercally. Fally, the ME, bj of b ; =,, s obtaed by usg the relato ηi = η bj, ci. Cosequetly, the ME, of ca be obtaed by replacg the parameters (4) wth ther MEs. Ths =? ηi A g < z; ci exp ηi gz; ci ηi gz; ci dz. (0) Clearly, bj does't eed to be. The ME, s asymptotcally ormal wth mea ad varace σ N = S P T S, where T s the verse of the Fsher

3 Amerca Joural of Theoretcal ad Appled Statstcs 08; 7(4): formato matrx, T, of θ = θ, θ,θ S, θ T = c, c, b, b, S P s the traspose of matrx S, where S = θ, ad l T E =, wth θ θ j l η η = + ( j ) j η j c j j j η c = c c = c = c l g(x ;c ) g(x ;c ) g(x ;c ), l l η η η g(x ;c ), η = = j c j b b c b = c b c l η = ;,, = b η b l l l l = = = = 0;, j =,, j. b b c c b c c b j j j j Clearly, the explct expresso of U N depeds o η, g < Fx ; c H, ad gfx ;c H ; j =,,, =,. The α 00% ACI for s F ±z [ σk N H, where σk N = S P T S, I, K s the estmator of U N. 3. Geeralzed Cofdece Iterval (GCI) of et X = FX, X,, X H; =,, be two depedet radom samples from populatos wth dstrbutos gve (3), havg ukow parameters c ad b ; =,. It s kow that, a GPQ of ay parameter s a fucto of observed statstcs ad radom varables whose dstrbuto s free of ukow parameters. For costructg the GCI, applyg a useful feature of the G approach, whch states that f G ad G are GPQs of b ad c, the η G, G s a GPQ of η ; =,. Ths feature eables us obtag a GPQ for gve as G N = G /, G., G /, G. by replacg the parameters (4) wth ther GPQs, the usg G N costructg cofdece terval for. The α 00% GCI for s obtaed asfg N[, G N[ H, where G N[ ad G N[ are the α th ad α th quatles of. 4. Bootstrap Cofdece Iterval (boot) of There are several ways to costruct bootstrap cofdece tervals. Clearly, the percetle ad t-bootstrap cofdece tervals are commoly used for the relablty, (see, Efro [7]). Algorthm s appled for costructg the bootstrap cofdece tervals. Algorthm.. From the orgal data _`, compute the MEs, ci ad ηi of c ad η ; =,, respectvely, ad the ME, of.. esample two depedet radom samples _`; =,, from x ; =,, respectvely; compute the MEs, ci, ci, ηi, ηi, ad of c, c,η, η, ad, respectvely. 3. epeat step, N tmes to derveb ; j =,, Nd, ad order g ; j =,, N, lke that. 4. Costruct the percetle ad t-bootstrap cofdece tervals of, as follows: a. Percetle bootstrap (P-boot) The α 00% P-boot for s gve by F [, [ th ad α α H, where [ ad [ th quatles of, respectvely. are the b. T-bootstrap (T-boot) The α 00% T-boot for s expressed as F t [ S, t [ S H, where S s the sample stadard devato of b ; j =,, Nd ad t [ s the α th quatle of j N k N ;j =,, Nm. l 5. Bayesa Credble Iterval (BCI) of a. Gamma prors (G-BCI) et X ; =, be two depedet radom samples from geeral verse expoetal form dstrbutos (3) wth ukow parameters b ad c. Cosder, η = η b, c, as a sgle parameter, assume that the pror dstrbutos of η ; =, are depedet wth pdfs β π ; =,, η, α, β > 0. α α βη ( η ) = η e Γα () Moreover assume that, c ; =, have depedet gamma prors wth pdfs λ π = ; =,, c, µ, λ > 0. () Γµ µ µ λc (c ) c e The jot posteror dstrbuto of η ad c ; =, s as follows

4 35 N. A. Mokhls et al.: Iterval Estmato of a P(X < X ) Model for arables havg Geeral Iverse Expoetal Form Dstrbutos wth Ukow Parameters (,,c,c x,x ) π η η = ( x,x η, η,c,c ) ( η η ) x,x,,c,c π ( η ) π ( η ) π (c ) π (c ) π ( η ) π ( η ) π (c ) π (c )dη dη dc dc (3) It s observed that (3) caot be obtaed a closed form. The BCI ca be computed by a combato of Metropols- Hastgs ad Gbbs samplg. Moreover the margal posteror dstrbuto of η s ( +α ) β + g(x j;c ) j= ( +α ) π ( η c,x,x ) = η exp g(x j;c ), ( η β + Γ + α ) j= (4) whch s Gamma + α, pβ + gfx ;c Hst; =,. The margal posteror dstrbuto of c s π ( ) = ( ) µ λ + ( ) ( + α ) β + c x,x A exp lc c l g (x ;c ) l g(x ;c ), j j j= j= A = exp µ l c cλ + l g (x j;c ) + α l β + g(x j;c ) dc ; =,. 0 Notce that the margal posteror dstrbuto of c ; = η g pz;c sˆ dz.,, s ot kow ad so the Metropols-Hastgs wth Gbbs samplg algorthm ca be used to solve t as follows (see, Asgharzadeh et al. [8]). Algorthm. where ( ) ( ) ( ) ( ) j= j= A. Choose a startg values c ; =,.. For j= to N tmes. 3. Geerate η g px ; c s from Gamma + α, pβ + st; =,, respectvely. 4. Geerate c from (5) usg the Metropols-Hastgs algorthm wth the ormal proposal dstrbuto ρ Nc, ; =,. a. Geerate ξ from the proposal dstrbuto ρ ; =,. b. Defe Q = m y, z k~/ F{ /,. H} p s ; =,. z p k~/ /,. s} { c. Geerate u from Uform (0, ). Take The jot posteror dstrbuto of η ad c ; =, ξ ; u Q, caot be obtaed a closed form. The margal posteror c = y. c ; otherwse of η s obtaed as (4), whle the margal posteror 5. Calculate the dstrbuto of c s gve by, usg =? η 0 g < pz;c s exp η g pz; c s π ( c x,x ) = B exp l ( g (x j;c ) ) ( ) l + α β + g(x j;c ), (6) j= j= The margal posteror dstrbuto of c ; =,, s ot a kow form. To obta the α 00%M-BCI for of mxed prors, use Algorthm wth a dfferece step 4 by where ( ) ( ) B = exp l g (x ;c ) + α l β + g(x ;c ) dc ; =,. j j 0 j= j= geeratg c from (6) stead of (5). The α 00%M-BCI for sf Š[, Š[ H, where Š[ ad Š[ are the α th ad α th quatles of, respectvely. (5) 6. Ed j loop. 7. Order ; j =,, N, ascedg ordered to obta g. 8. Costruct the α 00%G-BCI for of gamma prors as F [, [ H, where [ ad [ are the α th ad α th quatles of, respectvely. b. Mxed prors (M-BCI) Assume that X, =, are two depedet radom samples from populatos wth (3) havg ukow parameters b ad c ; =,, ad also assume that, η ; =, has depedet gamma pror as (), ad c has uform mproper pror dstrbuto wth pdf π (c ) = ; =,,c > Numercal Illustrato Ths secto, presets a umercal llustrato of the results obtaed. The cofdece tervals; ACI, GCI, P-boot, T- boot, G-BCI, ad M-BCI for wth some geeral verse expoetal form dstrbutos are compared. 000 samples of sample szes (, ) = (0, 0) ad (30, 30) from the uderlyg dstrbutos of X ad X, wth ukow

5 Amerca Joural of Theoretcal ad Appled Statstcs 08; 7(4): parameters are geerated. Takg α = 0.05, average legth, average coverage probablty, left ad rght tal errors of the α 00% cofdece tervals are calculated. The parameter values that produce the values of to be approxmately 0.6, 0.7, 0.8, 0.9, 0.95, 0.97, ad 0.99 are selected. The verse Webull dstrbuto s chose as a example of the geeral verse expoetal form. The verse Webull dstrbuto s flexble ad cludes a varety of dstrbutos. For the verse Webull dstrbutos, the CDFs wth η b, c = are, gx; c =, ad g< x; c F 4 x; b, c = Œ/ ; =,, = exp p s ˆ, (7) ad the relablty s obtaed by substtutg these values (4) c =? c b z c c exp b A z c c b z c dz. (8) As otced from (9), ηi = I = hece bj = kž/ ~ I k ~ I kž/ k ; =,, ad / I. Usg the Newto aphso teratve method ci s obtaed from (8) after applyg the sutable substtutos. () The α 00% ACI of ca be obtaed as F ±z [ σk N H, where s obtaed from (0), ad σk N = S P T S, I, K s obtaed from the followg equatos. ( ( j )) l = c c j l b x, c c b = xj l l = = + c ( j ) + c c + c j j l b x, c b b c b b = xj b = xj c c c (c + ) l = c c j ; = +, b b b = xj () As metoed Secto 3, takg G = p I sci A = I I, ad G = p s / bj A = t / bj A ; =,, ad ci A ad bj A are the observed values of ci ad bj. The dstrbutos of ci = p I s ad bj = p s ; =,, do ot deped o ay ukow parameters, ad so they are pvotal quattes (see, Thoma et al. [9]).cI ad bj are the MEs of c ad b, respectvely, based o two depedet radom samples from stadard verse expoetal dstrbutos (see, Krshamoorthy et al. [0]). The cosequetly the GPQs of η are G =. The GCI s obtaed by computg the t G N = G /, G., G /, G.. The ext Algorthm 3 s used to estmate the GCI of (see, Krshamoorthy ad []). Algorthm 3.. et _`; =,, be two depedet radom samples from (7), respectvely. Compute the ci A ad bj A of c ad b ; =,.. Geerate two depedet radom samples _` ; =,, from stadard verse expoetal dstrbutos. Compute the ci ad bj of c ad b. 3. Compute the G, G, G, ad heceg N ; =,. 4. epeat the steps -3, N tmes to obta a set of values of G N, say G Nk ; j =,, N. Order G Nk ; j =,, N, ascedg to obta G Nk g G Nk. 5. Costruct the α 00% GCI of as FG N[, G N[ H. () The α 00% P-boot ad T-boot of are obtaed, usg Algorthm.. Table. Average legth of the cofdece tervals of for verse Webull dstrbutos, = ;œ =,. c =, c = 5, b =., b = 0.9 c = 3, c = 4, b =., b = 0.9 c =, c = 3, b =, b = c =, c = 3, b =.8, b = 0.9 c =, c = 3, b = 4, b = 0.9 c =, c = 3, b = 6, b = 0.9 c =, c = 3, b = 9, b = 0.9 Average legth =0 =30 P T Gamma Mxed P T Gamma Mxed

6 37 N. A. Mokhls et al.: Iterval Estmato of a P(X < X ) Model for arables havg Geeral Iverse Expoetal Form Dstrbutos wth Ukow Parameters Table. Average coverage probablty of the cofdece tervals of for verse Webull dstrbutos, = ; œ =,. c =, c = 5, b =., b = 0.9 c = 3, c = 4, b =., b = 0.9 c =, c = 3, b =, b = c =, c = 3, b =.8, b = 0.9 c =, c = 3, b = 4, b = 0.9 c =, c = 3, b = 6, b = 0.9 c =, c = 3, b = 9, b = 0.9 Average coverage probablty =0 =30 P T Gamma Mxed P T Gamma Mxed Table 3. eft tal error of the cofdece tervals of for verse Webull dstrbutos, = ; œ =,. c =, c = 5, b =., b = 0.9 c = 3, c = 4, b =., b = 0.9 c =, c = 3, b =, b = c =, c = 3, b =.8, b = 0.9 c =, c = 3, b = 4, b = 0.9 c =, c = 3, b = 6, b = 0.9 c =, c = 3, b = 9, b = 0.9 eft tal error =0 =30 P T Gamma Mxed P T Gamma Mxed Table 4. ght tal error of the cofdece tervals of for verse Webull dstrbutos, = ; œ =,. c =, c = 5, b =., b = 0.9 c = 3, c = 4, b =., b = 0.9 c =, c = 3, b =, b = c =, c = 3, b =.8, b = 0.9 c =, c = 3, b = 4, b = 0.9 c =, c = 3, b = 6, b = 0.9 c =, c = 3, b = 9, b = 0.9 ght tal error =0 =30 P T Gamma Mxed P T Gamma Mxed

7 Amerca Joural of Theoretcal ad Appled Statstcs 08; 7(4): (v) Fally, the α 00% G-BCI ad α 00%M-BCI of, are obtaed usg Algorthm. The hyper-parameters of prors for G-BCI ad M-BCI are chose o bass of the same mea but dfferet varaces as the follow G-BCI: let α, β =, 4, α, β =,, μ, λ = p S, 3s, ad μ, λ = p, s. M-BCI: let α, β =, 4, ad α, β =,. The comparsos o the bass of average legth, average coverage, ad left ad rght tal errors are troduced Tables (-4), respectvely. From Tables -4, the followg results are observed:. As expected, the average legth of all cofdece tervals decreases whe ad crease.. Whe = , the average legth of boot ad the left tal error of T-boot are smallest, ad the left tal error of all cofdece tervals decrease whe ad crease. 3. The average coverage of GCI s approxmately aroud the α 00%. 4. The average coverage of ACI, boot, ad G-BCI decrease whe creases. But the average coverage of ACI ad boot crease whe creases. 5. For all the cofdece tervals, the rght tal error s decreasg whe s creasg. Moreover the rght tal error of all cofdece tervals except BCI are decreasg whe s creasg. 6. The rght tal error of BCI s the smallest. 7. Cocluso I ths artcle, several approaches for estmatg the cofdece tervals of Stress-Stregth elablty P(X < X ) are provded whe radom varables havg geeral verse expoetal form dstrbutos wth dfferet ukow parameters. A smulato study s performed to compare ts performace usg the verse Webull dstrbuto as a example of the geeral verse expoetal form dstrbuto. The comparso s carred out o bass of average legth, average coverage, ad tal errors. The results obtaed were very close to each other. efereces [] S. Kotz, Y. umelsk, M. Pesky, The Stress-Stregth Model ad ts Geeralzatos: Theory ad Applcatos, Sgapore: World Scetfc, 003. [] D. Al-Mutar, M. Ghtay, D. Kuddu, Ifereces o stressstregth relablty from dley dstrbutos, Commucatos Statstcs-Theory ad Methods, 8 (03) [3] N. Amr,. Azm, F. Yaghmae, M. Babaezhad, Estmato of stress-stregth parameter for two parameter Webull dstrbuto, Iteratoal Joural of Advaced Statstcs ad Probablty, (03) 4-8. [4] S. ezae,. Tahmasb, M. Mahmood, Estmato of P(Y < X) for geeralzed Pareto dstrbuto, Statstcal Plag ad Iferece, 40 (00) [5] N. Mokhls, E. Ibrahm, D. Ghareb, Stress-stregth relablty wth geeral form dstrbutos, Commucatos Statstcs- Theory ad Methods, 46 (07) [6] N. Mokhls, E. Ibrahm, D. Ghareb, Iterval estmato of a PX < X model wth geeral form dstrbutos for ukow parameters, Statstcs Applcatos ad Probablty, 6 (07) [7] B. Efro, A Itroducto to the Bootstrap, Chapma ad Hall, 994. [8] A. Asgharzadeh,. alollah, M. Z. aqab, Estmato of the stress-stregth relablty for the geeralzed logstc dstrbuto, Statstcal Methodology, 5 (03) [9] D.. Thoma,. J. Ba, C. E. Atle, Ifereces o the parameters of the Webull dstrbuto, Techometrcs, (969) [0] K. Krshamoorthy, Y., Y. Xa, Cofdece lmts ad predcto lmts for a Webull dstrbuto based o the geeralzed varable approach, Statstcal Plag ad Iferece, 39 (009) [] K. Krshamoorthy, Y., Cofdece lmts for stressstregth relablty volvg Webull models, Statstcal Plag ad Iferece, 40 (00)