Inference on Reliability in the Gamma and Inverted Gamma Distributions

Transcription

1 Statstcs n the Twenty-Frst Century: Specal Volue In Honour of Dstngushed Professor Dr. Mr Masoo Al On the Occason of hs 75th Brthday Annversary PJSOR, Vol. 8, No. 3, pages , July Jungsoo Woo Departent of Statstcs Yeungna Unversty Gyongsan, South Korea jswoo@ynu.ac.kr Inference on Relablty n the Gaa and Inverted Gaa Dstrbutons Abstract We derve dstrbutons of rato for two ndependent gaa varables and two ndependent nverted gaa varables and then we observe the skewness of two rato denstes. We then consder nference on relablty n two ndependent gaa rando varables and two ndependent nverted gaa rando varables each havng known shape paraeters. Keywords: Beta functon, Hypergeoetrc functon, Inverted gaa, Lkelhood rato test, Relablty, Whttaker functon.. Introducton For two ndependent rando varables X and Y and a real nuber c, the probablty P( X <) cy nduces the followng facts: () the probablty P( X <) cy s the relablty when the real nuber c equals one, () the probablty P( X <) cy s the dstrbuton of the rato X /() X Y when c = t /() t for < t <, and () the probablty P( X <) cy s the densty of a skewed rando varable f X and Y are syetrc rando varables about orgn. The relablty wll ncrease the need for ndustry to perfor systeatc study for the dentfcatons and reducton of causes of falures. These relablty studes ust be perfored by persons who () can dentfy and quantfy the odes of falures, () know how to obtan and analyze the statstcs of falure occurrences, and () can co nstruct atheatcal odels of falure that depend on, for exaple, the paraeters of ateral strength or desgn qualty, fatgue or wear resstance, and the stochastc nature of the antcpated duty cycle (see Saunders (7)). Many authors have consdered propertes of gaa dstrbuton (see Johnson et al (994)). McCool (99) consdered the nference proble on relablty P( X <) Y n the Webull case. Al and Woo (5 a & b)studed nference on relablty P( Y <) X n power functon and Levy dstrbutons. Pal, Al, and Woo (5) studed estaton of testng P( Y >) X n two paraeter exponental dstrbuton. Al, Pal, and Woo () studed the rato of two ndependent exponentated Pareto varables. Saunders (7) Pak.j.stat.oper.res. Vol.VIII No.3 pp

2 Jungsoo Woo ntroduced relablty, lfe testng and predcton of servce lves for engneers and other scentsts. In ths paper, we derve dstrbutons of rato for two ndependent gaa and two ndependent nverted gaa rando varables and then we observe the skewness of the two rato denstes. We then consder nference on relablty n two ndependent gaa rando varables and two ndependence nverted gaa rando varables each havng known shape paraeter.. Dstrbuton of rato In ths Secton we consder densty of rato of two ndependent gaa rando varables each havng the followng densty functons n (.) and densty of the rato of two ndependent nverted gaa rando varables each havng the followng densty functons n (.4).. Gaa dstrbuton Let X and Y be ndependent gaa rando varables havng the followng denstes. x/ f X () x= x e, f < x< and, () / x fy () y= x e, f < y<, (.) () where 's and 's are postve. Let W = Y / X. Then fro the quotent densty n Rohatg (976, p.4) and forula 3.38(4) n Gradshteyn and Ryzhk (965, p.37), densty of W = Y / X s gven by In fact, () fw () w=(), < < w. w f w ()() f () = W w dw fro forula.9 n Oberhettnger (974, p.5). (.) Fro the densty (.) and forula 3.5 n Oberhettnger (974, p.6), oent generatng functon (gf) MW () t of W s gven by t () =(), () MW t t e W t ()() where Wa, b () x s the Whttaker functon, see Abraowtz and Stegun (97, p.55). It guarantees the exstence of the k th oent of W, and hence fro forula 3.94(3) n Gradshten and Ryzhk (965, p.85), k th oent of W = Y / X s gven by k ()() k k E W f k () =, > =,,3,. ()() k 636 Pak.j.stat.oper.res. Vol.VIII No.3 pp

3 Inference on Relablty n the Gaa and Inverted Gaa Dstrbutons We obtan the densty of the rato R = X /() X= /() Y W fro (.) as the followng. Proposton : Let X and Y be ndependent rando varables each havng the respectve densty as n (.). Let =. Then the densty of R = /() s gven by where B( a,) b fr () r =()(() /) r, < r<, r r r B(,) s the beta functon. Fro Proposton and forula.33 n Oberhettnger (974, p.9), k th oent of the rato R s obtaned as follows. For k =,, 3, and =, ()() k F (, ; k;) f < < ()() k k E() R= (.3) ()() k k F (, ; k;) < f ()() k where F ( a, b; c;) x s the hypergeoetrc functon, see Abraowtz and Stegun (97, p.555). Table : Means, Varances, and Coeffcents of Skewness for the densty of rato R n the Gaa Dstrbuton Mean Varance Skewness Mean Varance Skewness /3 /3 / / / / / / / /3 / / / / / / /3 / / / , / / / Pak.j.stat.oper.res. Vol.VIII No.3 pp

4 Jungsoo Woo Fro k th oent of the rato R n (.3) and recurson forulas of hypergeoetrc functon n Abraowtz and Stegun (97, p.558), Table provdes nuercal values of ean, varance, and coeffcent of skewness for rato densty n Proposton. Fro Table we observe the followng trends of the rato densty n Proposton. Fact : Let X and Y be ndependent gaa rando varables each havng the densty as gven n (.). Then for = /, and the rato R = X /() X Y, we observe the followng. () When =, the densty of the rato R s syetrc at r = / when =, the densty s left-skewed when <, t s rght-skewed when >. () It s left-skewed when = / 4()4 for (,) =(,/ 3) and (3, /3), but t s rghtskewed when = / 4()4 for (,) =(/ 3,) (/3, 3).. Inverted gaa dstrbuton We now consder the densty of the rato of two ndependent nverted gaa rando varables U and V, each havng the followng respectve densty. In the context of the gaa denstes n (.), U = / X and V =/ Y. u / fu () u= u e, u> and, () v / fv () v= v e, v>. (.4) () Let Q = V / U. Then fro Rohatg (976, p.4) and forula 3.38(4) n Gradshteyn and Ryzhk (965, p.37), the densty of Q = V / U s obtaned as () fq () w=( /)( / /), w< <. w w (.5) ()() In fact, f () = Q w dw by forula.9 n Oberhettnger (974, p.5). We obtan the densty of R = U /() U = /() V Q fro (.5) as follows. Proposton : Let U and V be ndependent nverted gaa rando varables each havng the respectve densty n (.4). Then the densty of the rato R s gven by fr () r =()(()) r, < <, r r r r B(,) where = /. Fro Proposton and forula.33 n Oberhettnger (974, p.9), k th oent of the rato R s gven below. For k =,,3, and = /, 638 Pak.j.stat.oper.res. Vol.VIII No.3 pp

5 Inference on Relablty n the Gaa and Inverted Gaa Dstrbutons E() R= k ()() k ()() k ()() k k F (, ; k;) f < < F (, ; k;) < f. ()() k Let us observe the followng relaton between R and R. (.6) / X The rato R = U /() U= V = /() = Y X /() Y= X X Y R. Hence, (/ X /) Y E() R= () E R, Var() R=() Var R, and skewness of R = 3 3/ 3 3/ E[(()) R / E[()] R = Var [(()) R / [()] E R E R Var R = - skewness of R. Thus the ean, varance, and skewness of R can be nuercally obtaned fro Table wthout provdng a separate Table. Proposton 3: Let U and V be two ndependent nverted gaa rando varables each havng the densty respectvely n (.4). For rato R = U /( U V ) and R = X /() X Y, skewness for R and R are n the opposte drecton, where ( X,) Y s a par of ndependent gaa varables each havng the respectve densty as n (.). 3. Relablty P( Y <) X n the Gaa Case In ths Secton we consder nference on relablty n two ndependent gaa rando varables each havng densty wth known respectve shape paraeter as n (.). Fro the densty (.) and forulas 3.38()&() n Gradshteyn and Ryzhk (965, p.37), we obtan the relablty as follows. Proposton 4 Let X and Y be two ndependent gaa rando varables each havng the respectve densty as n (.) wth known shape paraeter. Let = /. Then R() =( P<) Y=(, X ; ;). F ()()() () Fro Proposton 4 and forula 5.. n Abraowtz and Stegun (97, p.557), we obtan the followng. Proposton 5: Let X and Y be two ndependent gaa rando varables each havng the respectve densty as n (.). If ean of Y s greater than that of X,.e., f > n denstes (.), then relablty R() =( P<) Y X s a onotone decreasng fncton of. If ean of X s greater than ean of Y,.e., f < n denstes (.), then relablty R() =( P<) Y= ( X <) P X Y s a onotone ncreasng functon of. Pak.j.stat.oper.res. Vol.VIII No.3 pp

6 Jungsoo Woo We now consder nference on relablty P( Y <) X when the shape paraeters and are known. Because R() s a onotonc functon of, nference on relablty s equvalent to nference on (see McCool (99)). Hence, t s suffcent for us to consder nference on / when the shape paraeters and are known. We get the followng Lea easly fro forulas 3.38(4) n Gradshteyn and Ryzhk (965, p.37). Lea : Let X be a gaa rando varable havng ean and varance. Then (a) (/) =, > ( ) f. (b) (/) =, ( )( ) f >. Assue X, X,, X and Y, Y,, Yn be two ndependent saples fro each densty n (.) wth known shape paraeter, respectvely. Then MLE ˆ of, =, are gven by n ˆ ˆ = X, and = Y. n = = Therefore, MLE ˆ of s ˆ = ˆ ˆ /. Fro Lea, ean and varance of ˆ are gven by E() ˆ =, and ( n ) Var() ˆ =, f >. ( )( n ) (3.) (3.) Fro expectaton n (3.), an unbased estator of s defned by j j= =. n X = n Y Fro Lea, varance of s gven by n Var() =. n ( ) (3.3) 64 Pak.j.stat.oper.res. Vol.VIII No.3 pp

7 Inference on Relablty n the Gaa and Inverted Gaa Dstrbutons Fro the results n (3.), (3.), and (3.3), we fnd that MSE( ) < MSE( ˆ )and then we obtan the followng fro equvalence of nference between and R() (see McCool (99)). Proposton 6: Assue X, X,, X and Y, Y,, Yn be two ndependent rando saples fro the respectve denstes n (.) wth known shape paraeters. Then an estator R() perfors better than the MLE R() ˆ n the sense of MSE f >. 3. The shape paraeter s a postve nteger We consder nterval estator of, especally f the shape paraeters 's n the denstes (.) are known postve ntegers.,.e., the denstes (.) belong to Erlang dstrbutons. Then ˆ ˆ / s a pvot quantty havng the F -dstrbuton wth degrees of freedo (,n ). Therefore, an ()% confdence nterval for s ˆ ˆ,( F /, ) n, F ˆ ˆ /( n, ) where h () t dt = /,( c F,),() n h t c / degrees of freedo (,n ). Next, we consder the testng of the followng hypothess: H : = =( = ) aganst H:( ). s the densty of F -dstrbuton wth Applyng the lkelhood rato test, we reject H f and only f f ˆ ˆ X ( x, x,, x;)(, fy, y, y ;) yn ( x, x,, x; y, y,,) y= n, c f ( ˆ ˆ X x, x,, x;)(, fy, y, y ;) yn n where c s a constant and ˆ = X = Y j= j. n The above lkelhood rato results n the followng equvalent test: Reject H f for a gven test sze, where < <, ˆ ˆ < or >( F /,). n ˆ F (,) n ˆ / (3.4) 3. The shape paraeter s known postve We consder an asyptotc nterval estate for f the shape paraeters and are known postve. Then usng the asyptotc property of MLE and ean and varance n (3.) and (3.) of ˆ, for large and n, ˆ ( n ) /(( )( n n)) has an asyptotc standard noral dstrbuton. Pak.j.stat.oper.res. Vol.VIII No.3 pp

8 Jungsoo Woo Fro (3.) and (3.) we fnd that ˆ s a consstent estator of and then fro the lt Theore n Rohatg (976, p.53), we obtan the followng. For large and n, ˆ ˆ ( n ) /(( )( n n)) has an asyptotc standard noral dstrbuton. (3.5) Therefore, an asyptotc ()% confdence nterval for s gven by ˆ (( z ) /(( n)( ))), n n / where / =() t dt, where ()t z / s the standard noral densty. We also wsh to test the followng H : = =( = ) aganst H:( ). For large and n and gven a test sze, < <, reject H f ˆ / ˆ < z ˆ ˆ /( ) /(( n )( )) n n ˆ ˆ / ˆ /( ˆ ) /(( n )( )) n n > z / / or (3.6) 4. Relablty P( V < U ) n the nverted gaa case In ths Secton we consder nference on relablty n two ndependent nverted gaa rando varables each havng respectve densty as n (.4). Usng the sae ethod for the gaa case, we obtan the followng. Proposton 7: Let U and V be ndependent nverted gaa rando varables each havng the respectve densty n (.4) wth known shape paraeters. Then for = /, R() =( P<) V = U (/ < /) P V U = (, ; ;) ()()() () F s a onotone functon of. The proof coes fro Propostons 4 and 5. Assue U, U,, U and V, V,, Vn be two ndependent rando saples fro each respectve densty n (.4) wth known shape paraeters. Then / U,/ U,,/ U and / V,/ V,,/ Vn are two ndependent rando saples fro the respectve denstes n (.) wth known shape paraeters. 64 Pak.j.stat.oper.res. Vol.VIII No.3 pp

9 Inference on Relablty n the Gaa and Inverted Gaa Dstrbutons The MLE ˆ, of, =, are gven by n ˆ ˆ = and =. U n V = j= j Usng slar arguents as n Secton 3, we can consder nference on relablty R() n the nverted gaa dstrbuton usng the two rando saples / U,/ U,, / U and / V,/ V,,/ Vn. References. Abraowtz, M. and Stegun, I. A. (97). Handbook of Matheatcal Functons. Dover Publcatons Inc., New York.. Al, M. Masoo, Pal, M., and Woo, J. (). On the rato of two ndependent exponentated Pareto varables. Austran Journal of Statstcs 39, Al, M. Masoo, Pal, M., and Woo. J. (5a). Inferences on relablty P( Y <) X n a power functon dstrbuton. Journal of Statstcs & Manageent Systes 8, Al, M. Masoo, Pal, M., and Woo. J. (5b). Inference on relablty P( Y < X ) n the Levy dstrbuton. Matheatcs and Coputer Modellng 4, Gradshteyn, I. S. and Ryzhk, I. M. (965). Tables of Integrals, Seres, and Products. Acadec Press, New York. 6. Johnson, N. L., Kotz, S. and Balakrshnan, N. (994). Contnuous Unvarate Dstrbutons. Houghton Mffln Co., Boston. 7. McCool, J. I. (99). Inference on P( Y < X ) n the Webull case. Councatons n Statstcs - Sulaton & Coputaton, Oberhettnger, F. (974). Tables of Melln Transfor. Sprnger-Verlag, New York, New York. 9. Pal, M., Al, M. Masoo, and Woo, J. (5). Estaton and testng of P( Y > X ) n two paraeter exponental dstrbutons. Statstcs 39, Rohatg, V. K. (976). An Introducton to Probablty Theory and Matheatcal Statstcs. John Wley & Sons, New York.. Saunders, S. C. (7). Relablty, Lfe Testng, and Predcton of Servce Lves. Sprnger, New York. Pak.j.stat.oper.res. Vol.VIII No.3 pp