Change Point Estimation for Pareto Type-II Model

Transcription

1 Journal of Modern Appled Statstcal Methods Volume 3 Issue Artcle Change Pont Estmaton for Pareto Type-II Model Gyan Prakash S. N. Medcal College, Agra, U. P., Inda, ggyanj@yahoo.com Follow ths and addtonal works at: Part of the Appled Statstcs Commons, Socal and Behavoral Scences Commons, and the Statstcal Theory Commons Recommended Ctaton Prakash, Gyan (04) "Change Pont Estmaton for Pareto Type-II Model," Journal of Modern Appled Statstcal Methods: Vol. 3 : Iss., Artcle. DOI: 0.37/jmasm/ Avalable at: Ths Regular Artcle s brought to you for free and open access by the Open Access Journals at DgtalCommons@WayneState. It has been accepted for ncluson n Journal of Modern Appled Statstcal Methods by an authorzed edtor of DgtalCommons@WayneState.

2 Journal of Modern Appled Statstcal Methods May 04, Vol. 3, No., Copyrght 04 JMASM, Inc. ISSN Change Pont Estmaton for Pareto Type-II Model Gyan Prakash S. N. Medcal College Agra, U.P., Inda Some Bayes estmators of the change pont for the Pareto Type-II model under rght tem falure-censorng scheme are proposed. The Bayes estmators are obtaned here n two cases, the frst s when one parameter s known and second when both parameters are consdered as the random varable. The performances of the procedures are llustrated by smulaton technque. Keywords: Change pont, Pareto Type-II model, Bayes estmaton Introducton The Pareto dstrbuton and ts close relatves provde a flexble famly of fattaled dstrbutons, whch may be used as a model for ncome dstrbuton of hgher ncome group and n soco-economc studes. Ths dstrbuton has played mportant role n varety of other problems such as sze of ctes and frms, busness mortalty, servce tme n queung system. It s often used as a model for analyzng areas ncludng cty populaton dstrbuton, stock prce fluctuaton, ol feld locatons and mltary areas. It has been found to be sutable for approxmatng the rght tals of dstrbuton wth postve skewness. Pareto dstrbuton has a decreasng falure rate, so t has often been used for model survval after some medcal procedures (the ablty to survve for a longer tme appears to ncrease, the longer one survves after certan medcal procedures). Harres (968) used ths dstrbuton n determnng tmes of mantenance servce whle Dyer (98) found that two-parameter Pareto dstrbuton transformaton s equvalent to the two-parameter exponental dstrbuton. Mad & Raqab (004) dscussed about the forecastng of the temperatures records by Gyan Prakash s an Assstant Professor n the Department of Communty Medcne. Emal hm at: ggyanj@yahoo.com. 339

3 CHANGE POINT ESTIMATION FOR PARETO TYPE-II MODEL Pareto dstrbuton. Sngh et al (007) dscussed about dfferent types of testestmaton for the Pareto Model. The length of Bayes predcton lmts have been obtaned recently by Prakash & Sngh (03) for the Pareto model. Panah & Asad (0) presented stress-strength model for a Lomax dstrbuton. Some nferences regardng the Lomax dstrbuton under the generalzed order statstcs has dscussed by Moghadam et al. (0). Nasr & Hossen (0) presented Bayesan and classcal statstcal nferences for Lomax model based on record values. Recently, Al-Zahran & Al-Sobh (03) presents some parameter estmaton for Lomax dstrbuton under general progressve censorng crteron. The probablty densty functon of the consdered Pareto Type-II model s gven as ( ) f ( x;, ) ( x ) ; x 0, 0, 0 () Here, s the shape parameter and s the scale parameter. The proposed Pareto Type-II model s the result of mxture of the Exponental dstrbuton wth the parameter α, and the exponental scale parameter α s dstrbuted as a Gamma wth parameters and. Ths artcle dscusses the Bayes estmaton of change pont for Pareto Type- II model. The Bayes estmator has been obtaned under the rght tem falure censorng crtera n two cases: the frst s when the scale parameter s known and second when both parameters are consdered as the random varable. A numercal study was carred out for llustraton of the procedures n next secton by MCMC technque. The Change Pont In order to obtan nformaton on ther endurance, manufactured tems such as mechancal or electronc components are often put to lfe tests and lfe tmes are observed perodcally. Physcal systems manufacturng the tems are often subject to random fluctuatons. It may happen that at some pont of tme, there s a change n the parameter. The objectve of study s to fnd out when and where ths change has started occurrng, whch s called the change pont nference problem. Bayesan model may play an mportant role n the study of such change pont estmaton problem and have been studed by Broemelng & Tsurum (987), Jan & Pandya (999), Ebrahm & Ghose (00), Goldenshluger, et al. (006). Pandya & Jadav (00) presents Bayesan estmaton of change pont n mxture of left truncated exponental and degenerate dstrbuton. Some Bayes estmaton 340

4 GYAN PRAKASH of shft pont n Posson model was presented by Srvastava (0). Recently, Pandya (03) presented Bayes estmaton of auto regressve model wth change pont. n 3 such as Consder a sequence of ndependent random sample of sze x, x,..., xm, xm, xm,..., xn from the consdered model wth survval functon ( t) at tme t ( 0) but later t s found that there s a change n the system at some pont of tme m and t s reflected n the sequence after the observaton x m by the change n the survval functon. The probablty densty functon and survval functon of the frst m observatons x, x,..., x m are gven from model () as: f ( x ;, ) ( x ) ; x 0, 0, 0,,,..., m () ( ) and ( t) ( t ) ; t 0, 0, 0. Smlarly, the probablty densty functon and survval functon of remanng ( n m) components xm, xm,..., xn are and f x ( ( ;, ) ) ( ) ; x 0, 0, 0, m, m,..., n x (3) ( t) ( t ) ;t 0, 0, 0. In lfe testng, the observatons usually occur n ordered manner such that the weakest tems fal frst and then second one and so on. Suppose that n tems are put to test under the consdered model wthout replacement and only k ( n) tems are fully measured, whle the remanng ( n k) tems are censored. These ( n k) censored tems wll be ordered separately. Ths censorng scheme s known as the rght tem falure-censorng crtera. The change pont crtera was ntroduced nsde the rght tem-censorng scheme; assume a sequence of ordered ndependent random sample of sze n such as x(), x(),..., x( k ), x ( k ), x ( k ),..., x ( n ) from the model (), wth the 34

5 CHANGE POINT ESTIMATION FOR PARETO TYPE-II MODEL parameters and. All n tems are tested wthout replacement and frst k ordered tems are fully measured whle remanng ( n k) tems are censored. From the frst fully measured k ( x(), x(),..., x( k), x( k) ) tems, t s found that there s a change n the system at some pont of tme m and t s reflected n the sequence after x ( ) ( ) m k by the change n the survval functon. m The probablty densty functon of frst m ( m, k n) random samples x (), x (),..., x ( m) wth parameters and, are f ( x ;, ) ( x ) ; ( ) ( ) ( ) x,, 0,,,..., m( m k, k n ). ( ) (4) The frst remanng group of random samples x( m), x ( m ),..., x( k) wth sze ( k m) usng a consdered Pareto model has a probablty densty functon wth parameters and f ( x ;, ) ( x ) ; ( ) ( ) ( ) x,, 0, m, m,..., k ( k m, k n ). ( ) (5) The last remanng group of random samples x( k), x ( k ),..., x( n) of sze ( n k) dstrbuted agan a Pareto model wth parameters and has the probablty densty functon f ( x ;, ) ( x ) ; ( ) ( ) ( ) x,, 0, k, k,..., n ( k ). ( ) (6) Under the above scenaro the lkelhood functon for the random sample x ( x(), x (),..., x( n )) s defned as m k L x,,, m f ( x ( ) ;, ) f ( x( ) ;, ) m n x( ) f ( x ;, ) dx k 0 ( ) ( ) 34

6 GYAN PRAKASH L x m T e e (7) m km ( nkm ) ( km ) T T (,,, ) 0 ; where T k m k m n 0 ( ( ) ), T x( ) x( ) k T x log ( x() ). Remark. Substtute n (7) and log ( ) log ( ) L x T e k n T 3 (, ) 0 ; T log ( x ). 3 ( ) n Here, L ( x, ) shows the lkelhood functon under the rght temfalure censorng crteron wthout consderaton of change pont.. Substtute and k n n (7) to obtan the lkelhood functon for complete sample case wthout consderaton of change pont. L x T e * n n T 3 (, ) 0 ; n * 0 ( ) T ( x ). Change Pont Estmaton (Scale Parameter Is Known) From a Bayesan vewpont; there s clearly no way n whch one can say that one pror s better than other. It s more frequently the case that, that a pror s selected to restrct attenton to a gven natural famly of prors, and one s chosen from that famly, whch seems to match best wth one s personal belefs. A natural famly of conjugate pror for shape parameter θ s consdered here as a Gamma dstrbuton (when scale parameter s known) wth probablty densty functon a g ( ) e ; 0, a 0, 0. (8) Based on change pont crteron the pror densty (8) s re-parameterzed as 343

7 CHANGE POINT ESTIMATION FOR PARETO TYPE-II MODEL a j j j j j j j j j g ( ) e ; 0, a 0, 0, j,. (9) A dscrete unform over the set (,,..., k ), s consdered as the pror dstrbuton of change pont m and defned as g 3 ( m). (0) k The jont pror dstrbuton when scale parameter s consdered to be known, s defned as h (,, m) g ( ) g ( ) g ( m). 3 The jont posteror densty functon s now obtaned as (,, m x) m L ( x,,,m ) h (,, m) L ( x,,,m ) h (,, m) d d * * m k m T T m x e e,, ; () k ( m a ) ( k m a) where,, * m a * km a m T T * log and T T ( k m)log. T T n k m * ( ) Hence, the margnal posteror densty for change pont m s,, m x m x d d. () * The choce of the loss functon may be crucal n Bayesan analyss. It has always been recognzed that the most commonly used loss functon, squared error loss functon (SELF), s napproprate n many stuatons. The Bayes estmator of a parameter under SELF s the posteror mean. If SELF s taken as a measure of naccuracy then the resultng rsk s often too senstve to the assumptons about the behavor of the tal of the probablty dstrbuton. To overcome ths dffculty, 344

8 GYAN PRAKASH a useful asymmetrc loss functon based on the squared error loss functon (ISELF) s defned for any estmate ˆ correspondng to the parameter as L ( ˆ, ) ; ˆ. The Bayes estmator of the change pont m under ISELF s obtaned as ˆ I P( ) P( ) m E m E m k k mˆ ( m ) ( m ). I m m (3) Here, the suffx P ndcates the expectaton taken under the posteror densty. When postve and negatve errors have dfferent consequences, the use of squared error loss functon (SELF) n Bayesan estmaton may not be approprate. In addton, n some estmaton problems overestmaton s more serous than the underestmaton, or vce-versa. To deal wth such cases, a useful and flexble class of asymmetrc loss functon (LINEX loss functon (LLF)) s gven as a L ( ) e a. The shape parameter of LLF s denoted by ' a '. Negatve (postve) value of shape parameter ' a ', gves more weght to overestmaton (underestmaton) and ts magntude reflect the degree of asymmetry. It s also observed that, for a, the functon s very asymmetrc wth overestmaton beng more costly than underestmaton. For small values of a, the LLF s almost symmetrc and s not far from the SELF. Bayes estmator of m under the LLF s obtaned as m ˆ L log E P e a log a k m a m a m e. (4) A close form of both the estmators does not exst. A numercal method s appled for obtanng the values of ther estmates. 345

9 CHANGE POINT ESTIMATION FOR PARETO TYPE-II MODEL Change Pont Estmaton (Both Parameter Unknown) In the case of when both the parameters and are unknown for the consdered Pareto model, there does not exsts any jont conjugate pror. Assume that the pror belefs about the parameters and are ndependent. The natural famly of conjugate prors for parameters and non-nformatve pror for parameter are consdered ndependently here. The non-nformatve pror of the parameter s the lmtng form of the approprate natural conjugate pror. The jont pror dstrbuton when both parameters are unknown s defned as g, g ( ) h ( ) ; h ( ), 0. (5) The pror dstrbuton g ( ) s gven n equaton (8). The lkelhood functon n present case s redefned as m L x,,, m f ( x ( ) ;, ) k n x( ) f ( x( ) ;, ) f ( x ( ) ;, ) dx( ) m k 0 L x m T e e (6) m km ( nkm ) log ( km ) log T T (,,, ) 0 e e. The jont pror densty for the parameters,, and m s wrtten as h (,,, m) g ( ) g ( ) h ( ) g ( m). 3 Hence, the jont posteror densty s obtaned as (,,, m x) m L ( x,,, m ) h (,,, m) L ( x,,, m) h (,,, m) d d d T * * 0,,, m a k m a m x T T e e ; (7) 346

10 GYAN PRAKASH where k T 0 and d. m Hence, the margnal posteror densty for the change pont m s (8) m x,,, m x d d d, ** and, the Bayes estmator under ISELF and LLF for the change pont m are obtaned as and Numercal Analyss One Parameter Known Case k k m ˆ m m I (9) m m k ˆ a m m L log e. (0) a m To assess and study the propertes of the Bayes estmator for the change pont m, a smulaton study was performed. The random samples were generated as: Generate ;, ; through pror densty g ;, ; for the gven as values of pror parameters, ;, ;, (0.5, 0.50), (4, ), (9,3);,. The selectons of pror parametrc values meet the crteron that the pror varance should be unty. Usng generated values of ;, ; and 0.50,.00,.50, 3.00; generate 0,000 random samples of sze n 5 by usng the model () and (3). The values of the Bayes estmate mˆ I under the ISELF have been obtaned and presented them n the Table, for selected set of censored sample sze k = 04, 06, 08,

11 CHANGE POINT ESTIMATION FOR PARETO TYPE-II MODEL Table. Bayes Estmate of m under ISELF (Scale Parameter Known) n=5 k (, ) , , , , , , , , , , , , Table shows that when censored sample sze k ncreases, the magntude of the estmate ncreases, but ncrement n magntude s nomnal (robust). A smlar trend also noted when scale parameter ncreases, however for large value of (.5) the magntude of the estmate decreases. The opposte trend has been seen when set of pror parameter ncreases. Usng above consdered set of parametrc values wth a 0.5, 0.50,.00,.00; (shape parameter of LLF) the magntude of the Bayes estmate under LLF have been obtaned and present n the Table, only for a 0.5,

12 GYAN PRAKASH Table. Bayes Estmate of m under LLF (Scale Parameter Known) n5, a0.5 k (, ) , , , , , , , , , , , , n5, a.00 k (, ) , , , , , , , , , , , ,

13 CHANGE POINT ESTIMATION FOR PARETO TYPE-II MODEL Table shows that when shape parameter ncreases, the magntude of the estmator decreases (except for large pror parametrc value). An ncreasng trend n the magntude of the estmate also s also noted when ' a ' ncreases but the ncrement n magntude s robust. Others propertes are smlar to ISELF. When Both Parameters Unknown When both parameters are consdered as a random varable, a smulaton study was carred out to study the propertes of Bayes estmators of change pont. Smlarly, a 0,000 random sample of sze n 5 was generated. The Bayes estmate of m under the ISELF and LLF were obtaned and are presented n Tables 3-4 respectvely for dfferent selected set of values. Table 3. Bayes Estmate of m under ISELF (Both Parameter Unknown) n 5 (, ) , , , k , , , , , , , , ,

14 GYAN PRAKASH Table 4. Bayes Estmate of m under LLF (Both Parameter Known) n5, a0.5 k (, ) , , , , , , , , , , , , n5, a.00 k (, ) , , , , , , , , , , , ,

15 CHANGE POINT ESTIMATION FOR PARETO TYPE-II MODEL The behavor of m ˆ was shown to be smlar as compare to ˆ I m I under ISELF. It s also noted that the magntude of the estmator m ˆ I ncreases as ncreases for all selected parametrc set of values. Further, the magntude of the estmate of m ˆ I s closer than the estmate of m ˆ I except for large value of. All propertes of estmator m ˆ were smlar as compared to ˆ L ml under LLF. For small values of ' a ', the magntude of estmate of m ˆ L s wder than m ˆ L for all consdered values of (except for 0.50 ). For large values of ' a ', the magntude of estmate of m ˆ L becomes narrower than m ˆ L for all consdered values of (except for.00 ). Other propertes are the same, as n the case of a known shape parameter. Remark In the case when the censored sample sze r 5, the censorng crteron reduces to the complete sample sze crteron and, hence, all results are vald for the complete sample case. References Al-Zahran, B. & Al-Sobh, M. (03). On parameters estmaton of Lomax dstrbuton under general progressve censorng. Journal of Qualty and Relablty Engneerng, 03: -7. Broemelng, L. D. & Tsurum, H. (987). Econometrcs and structural change. New York: Marcel Dekker. Dyer, D. (98). Structural probablty bounds for the strong Pareto law. Canadan Journal Statstcs, 9: Ebrahm, N. & Ghosh, S. K. (00). Bayesan and frequentst methods n change-pont problems. In N. Balkrshna & C. R. Rao (Eds). Handbook of statstcs, Vol. 0: Advances n Relablty, (Elsever), Goldenshluger, A., Tsybakov, A. & Zeev, A. (006). Optmal change pont estmaton from ndrect observatons. The Annals of Statstcs, 34(): Harrs, C. M. (968). The Pareto dstrbuton as a queue dscplne. Operatons Research, 6():

16 GYAN PRAKASH Jan, P. N. & Pandya, M. (999). Bayes estmaton of shft pont n left truncated exponental sequence. Communcatons n Statstcs-Theory and Methods, 8(): Mad, M. T. & Raqab, M. Z. (004). Bayesan predcton of temperature records usng the Pareto model. Envronmetrcs, 5(7): Moghadam, M. S., Farhad, Y. & Manoochehr, B. (0). Inference for Lomax dstrbuton under generalzed order statstcs. Appled Mathematcal Scences, 6(05): Nasr, P. & Hossen, S. (0). Statstcal nferences for Lomax dstrbuton based on record values (Bayesan and Classcal). Journal of Modern Appled Statstcal Methods, (): Panah, H. & Asad, S. (0). Inference of stress-strength model for a Lomax dstrbuton. World Academy of Scence, Engneerng and Technology, 55: Pandya, M. (03). Bayesan estmaton of AR () wth change pont under asymmetrc loss functons. Statstcal Research Letters, (): Pandya, M. & Jadav, P. (00). Bayesan estmaton of change pont n mxture of left truncated exponental and degenerate dstrbuton. Communcaton n statstcs-theory and Methods, 39(5): Prakash, G. & Sngh, D. C. (03). Bayes predcton ntervals for the Pareto model. Journal of Probablty and Statstcal Scence, (): 09-. Sngh, D. C., Prakash, G. & Sngh, P. (007). Shrnkage testmators for the shape parameter of Pareto dstrbuton usng LINEX loss functon. Communcatons n Statstcs-Theory and Methods, 36(4): Srvastava, U. (0). Bayesan estmaton of shft pont n Posson model under asymmetrc loss functons. Pakstan Journal of Statstcs and Operaton Research, 8():