[Saxena, 2(9): September, 2013] ISSN: Impact Factor: INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY

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1 [Saena, (9): September, 0] ISSN: Impact Factor:.85 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY Contant Stre Accelerated Life Teting Uing Rayleigh Geometric Proce Model S. Saena *, Arif-Ul-Ilam Department of Statitic & Operation Reearch, Aligarh Mulim Univerity, Aligarh, India Abtract The current tudy introduce the Rayleigh geometric proce model for the analyi of accelerated life teting under contant tre. The geometric proce decribe a imple monotone proce and ha been applied to a variety of ituation uch a the maintenance problem in engineering. By auming that the lifetime under increaing tre level form a geometric proce, we derive the maimum lielihood etimator of the Rayleigh parameter in cae of complete and cenored data. For each type, we alo derive confidence interval for the parameter uing aymptotic ditribution. The performance of the etimator i evaluated by a imulation tudy with different pre-fied parameter. Keyword: Geometric proce, Maimum Lielihood Etimator, Fiher Information Matri, Aymptotic Confidence Interval, Simulation Study. Introduction Development of highly ophiticated product, intene global competition, and increaing cutomer epectation ha put new preure on manufacturer to produce high-quality product. In order to acertain the ervice life and performance of a product, life tet under normal operating condition i clearly the mot reliable. The tandard life teting method would require a long period of time to obtain enough failure data neceary to mae inference. Hence, they are not uitable in above ituation. For ome product, it i poible to accelerate failure, and hence obtain failure information quicly by uing the product more intenively than in uual cae. According to uch propertie, the deign and analyi of the Accelerated Life Tet (ALT) are very important from a practical viewpoint. ALT, generally deal with three type of tre loading: contant tre, tep tre and Progreive tre. Contant tre i the mot common type of tre loading, in which every item i teted under a contant level of the tre, which i higher than normal level. In thi ind of teting, we may have everal tre level, which are applied for different group of the teted item. Thi mean that every item i ubjected to only one tre level until the item fail or the tet i topped for other reaon. If the tre level of the tet i not high enough, many of the teted item will not fail during the available time and one ha to be prepared to handle a lot of cenored data. To avoid thi problem, tep-tre teting can be applied, in which, all item are firt ubjected to a pecified contant tre for a pecified period of time. Item that do not fail will be ubjected to a higher level of tre for another pecified time. The level of tre i increaed tep by tep until all item have failed or the tet top for other reaon. Progreive-tre loading i quite lie the tep tre teting with the difference that the tre level increae continuouly. Failure data obtained from ALT can be divided into two categorie: complete (all failure data are available) or cenored (ome of failure data are miing). Complete data conit of the eact failure time of tet unit, which mean that the failure time of each ample unit i oberved or nown. In many cae when life data are analyzed, all unit in the ample may not fail. Thi type of data i called cenored or incomplete data. See for more detail, Bagdonaviciu and Niulin [5], Meeer and Ecobar [7], Nelon [9, 8], Mann and Singpurwalla [6]. with different type of data and planning ha been tudied by many author. Pan et al. [0] propoed a bivariate contant tre accelerated degradation tet model by auming the parameter, a function of the tre level. Yang [4] propoed an optimal deign of 4-level contant-tre ALT. Fan and Yu [8] dicu the reliability analyi of the contant tre accelerated life tet in cae of generalized gamma lifetime ditribution. Chen et al. [6] dicu the optimal deign of multiple CSALT plan on non-rectangle tet region. Saena et al. [] dicued the cae of Rayleigh lifetime ditribution for tep tre accelerated life teting (SSALT). Watin and John [] conider contant tre accelerated life http: // (C) International Journal of Engineering Science & Reearch Technology [-9]

2 [Saena, (9): September, 0] ISSN: Impact Factor:.85 tet baed on Weibull ditribution with contant hape and a log-linear lin between cale and the tre factor. Ding et al. [7] dealt with Weibull ditribution to obtain ALT ampling plan under type I progreive interval cenoring with random removal. Ahmad et al. [], Ilam and Ahmad [0], Ahmad and Ilam [], Ahmad, et al. [4] and Ahmad [] dicu the optimal contant tre accelerated life tet deign under periodic inpection and Type-I cenoring. Definition A tochatic proce { X n, n =,,...} i a geometric proce (GP), if there eit a real > 0 uch that n { X n, n =,,...} form a renewal proce (RP). The number i called the ratio of the GP. The concept of GP i introduced by Lam [] in the tudy of repair replacement problem. Large amount of tudie in maintenance problem and ytem reliability have been hown that a geometric proce model (GPM) i a good and imple model for analyi of data with a ingle trend or multiple trend, for eample, Lam and Zhang [5], Lam [4] and Zhang [5]. So far, there are only three tudie that utilize the GP in the analyi of ALT. Huang [9] introduced the GPM for the analyi of ALT with complete and cenored eponential ample under the contant tre. Kamal et al. [] etended the GPM for the analyi of ALT with complete Weibull failure data under contant tre. Zhou et al. [6] conidered the GP implementation of the CSALT model baed on the progreive Type-I hybrid cenored Rayleigh failure data. Kamal et al. [] ued the GP for the analyi of CSALT for Pareto Ditribution with complete data. Saena et al. [] tudied the cae of loglogitic GPM in cae of cenored data. In thi paper, the analyi of CSALT for Rayleigh ditribution with complete and cenored data by uing the GPM i conidered. Etimation of parameter i carried out by maimum lielihood (ML) technique. Aymptotic confidence interval for parameter are alo obtained. Statitical propertie of etimate and confidence interval are eamined through a imulation tudy. The Model The Geometric Proce Let u define the GP. Suppoe that X, X,..., X n i a equence of random variable. If n there eit > 0 uch that { X n, n =,,...} form a renewal proce (RP) with a contant mean µ, then X, X,..., X n i called a GP and the real number i called the ratio of the GP. It can eaily be noted that a GP i tochatically increaing for 0 < < and tochatically decreaing in cae of >. GPM can identify trend effect by two parameter: the mean µ of the underlying RP and the ratio which meaure the direction and trength of a trend. With the inherent geometric tructure, forecat uing the GPM i imple and traightforward. Mean and Variance of a Geometric Proce: It can be hown that if { X n, n =,,,...} i a GP and the pdf of X i f () with mean µ and variance then the pdf of X will be n n n f ( X n ) = f ( ), µ with E( X n ) = n σ and Var( X n ) = ( n) n =,,,... σ Thu, µ and σ are three important parameter of a GP. The Rayleigh Ditribution The Rayleigh ditribution ha played an important role in modeling the lifetime of random phenomena. It arie in many area of application, including reliability, life teting and urvival analyi. The life time of a product at any level of tre i aumed to follow Rayleigh ditribution with cale parameter. The probability denity function (p.d.f.) of Rayleigh ditribution i given by f ( ) = e, 0 <, > 0 () Cumulative ditribution function (c.d.f) i given by F( ) = e, 0 <, > 0 The correponding urvival function i S( ) = e, 0 <, > 0 () The hazard function of, denoted a h ( ) = f ( ) / S( ) i obtained a h( ) =, 0 <, > 0 Aumption The geometric model for ALT i baed on the following aumption: () Suppoe that an ALT under z, =,,...,, arithmetically increaing tre level i () http: // (C) International Journal of Engineering Science & Reearch Technology [-9]

3 [Saena, (9): September, 0] ISSN: Impact Factor:.85 performed. A random ample of N i, i =,,..., n, identical item i placed under each tre level and tart to operate at the ame time. Whenever an item fail, it i removed from the tet and it oberved failure time i i recorded. () At any contant tre level, the product lifetime ha a ingle parameter Rayleigh ditribution. () Let the equence of random variable X 0, X, X,..., X, denote the lifetime under each tre level, where X denote item lifetime under the deign 0, = i a GP with ratio > 0. Baed on the definition given in ubection., if denity function of X 0 i f ( ), then the pdf of X will be given by f ( X ) = f ( ), = 0,,, L, Therefore the pdf of a product lifetime (following th Rayleigh ditribution) at the tre level i f (, ) = ep (4) tre. We aume { X 0,,,..., } Maimum Lielihood Etimation Cae (i) Complete Data Here the ML method of etimation i ued becaue ML method i very robut and give the etimate of parameter with good tatitical propertie. In thi method, the etimate of parameter are thoe value which maimize the ampling ditribution of data. The ML etimation method i very appropriate for one parameter ditribution and alo it implementation in ALT i mathematically more intene. Generally, etimate of parameter do not eit in cloed form, therefore, numerical technique uch a Newton Method and ome computer program are ued to compute them. The lielihood function for CSALT for the Rayleigh ditribution uing GPM can be written a n L(, ) = f (, ) = i= n = i ep i (5) = i= It i uually eaier to maimize the logarithm of the lielihood function rather than the lielihood function itelf. Therefore, by taing the logarithm of the lielihood function, (5) become n i l = log L(, ) = log + log i log (6) = i= l l The maimum lielihood etimator of, and can be obtained by olving = 0, and = 0 repectively l l for, and where the value of, and are given a n ( ) l i = (7) = i= n l = + i (8) = i= From equation (7) and (8), it i oberved that thee equation are non linear. Therefore, the cloed form of MLE of, and do not eit. So, Newton-Raphon method mut be ued to olve thee equation imultaneouly to obtain the MLE of, and. Cae (ii) Cenored Data For Type I cenoring cheme, the tet at each tre level terminate at time t. An item eact failure time i th oberved only if it lifetime i i t. It i aumed that at the tre level r ( n) failure are oberved before the http: // (C) International Journal of Engineering Science & Reearch Technology [-9]

4 [Saena, (9): September, 0] ISSN: Impact Factor:.85 tet i upended. Correpondingly, ( n r ) unit urvive the entire tet without failing. The oberved ordered failure time th under the tre level can be written a ( ) () L ( r ). Here, t i fied in advance and r i random and therefore the lielihood function uing GP for the Rayleigh ditribution under CSALT for Type I cenored data i given by r n! ( nr ) L (, ) = f X ( ( i) ) [ S X ( t) ] (9) ( n r )! i= where, S X (t) i the probability that an item i cenored at time t and ( t) S X ( t) = ep (0) Uing eq. (0), the lielihood function for one of the tre level correponding to eq. () for obtaining the ML etimate of and i given by ( nr ) r! n ( ) ( t) Li (, ) ep ep, ( n r )! i= = () It follow that the lielihood function of oberved data in a total tre level i: L (, ) = L L L L r! ( ) ( ( ) ) n i i ( t) ep ep = ( n r )! i= = ( nr ) 0 ( ) () L ( r ) t ; () The log-lielihood function correponding to () tae the form l = log L(, ) r n! = log + log = ( n r )! = i The firt order derivative of log L (, ) are given by l = r r ( i ) = i= i= ( ( i) ) ( n r )( t) + log ( i) log () = ( n r )( t) r l ( i ) ( n r )( t) = + (5) = i= = The equation (4) and (5) are quite comple in form to be olved. So, the Newton-Raphon method i ued to olve thee equation imultaneouly to obtain ˆ and ˆ. Aymptotic Confidence Interval According to large ample theory, the ML etimator, under ome appropriate regularity condition, are conitent and normally ditributed. Since ML etimate of parameter are not in cloed form, therefore, it i impoible to obtain the eact confidence interval, o aymptotic confidence interval baed on the aymptotic normal ditribution of ML etimator intead of eact confidence interval are obtained here. The Fiher-information matri compoed of the negative econd order partial derivative of log lielihood function can be written a (4) http: // (C) International Journal of Engineering Science & Reearch Technology [-9]

5 [Saena, (9): September, 0] ISSN: Impact Factor:.85 F = ˆ ˆ I I Where in cae (i) = = = = n = i= = = and in cae (ii) = = = = ( ) + n i 4 = i= = n = i= ( ) ( ) i i r r ( ) ( i ) ( ) + + = i= i= r ( i ) + 4 = i= = = = = Now, the variance covariance matri can be written a ( n r r ( i ) = i= = 4 ( )( n r )( t) ( n r )( t) )( t) ˆ ˆ ˆ I AVar( ˆ) ACov( Σ = = I ˆ ˆ ˆ I ACov( ˆ) AVar( ˆ) The 00( γ )% aymptotic confidence interval for and, and are then given repectively a ˆ ± Z γ AVar( ˆ), and ˆ ± Z γ AVar( ˆ), Simulation Study To ae the performance of the method decribed in preent tudy, a number of data et with ample ize n = 50, 00,...,50 are generated from Rayleigh ditribution. The value for parameter and tre level are choen to be =.750, = and = and. For different given ample and tre level, the ML etimate, Mean quared error (MSE), abolute relative biae (RBia), Relative Error (RE), and the 95% aymptotic confidence interval for, and are obtained by uing the preent GP model uing the Newton-Raphon iteration procedure. The reult of the etimate for, and baed on 000 replication are ummarized in Table and for cae (i) and in Table and 4 for cae (ii) repectively. http: // (C) International Journal of Engineering Science & Reearch Technology [-9]

6 [Saena, (9): September, 0] ISSN: Impact Factor:.85 Table : Simulation tudy reult with =. 750, = , and = for complete data Sample Size Parameter Etimate MSE RAB RE LCL UCL http: // Table : Simulation tudy reult with =. 750, = , and = for complete data Sample Size Parameter etimate MSE RAB RE LCL UCL Table : Simulation tudy reult with =. 750, = , and = for cenored data Sample Size Parameter etimate MSE RAB RE LCL UCL Table 4: Simulation tudy reult with =. 750, = , and = for cenored data Sample Size Parameter etimate MSE RAB RE LCL UCL (C) International Journal of Engineering Science & Reearch Technology [-9]

7 [Saena, (9): September, 0] ISSN: Impact Factor: Dicuion and Concluion Thi paper deal with ue of GP model in the analyi of contant tre ALT plan for Rayleigh ditribution with complete data a well a cenored data. The MLE, MSE, RBia, and RE of the model parameter were obtained. Baed on the aymptotic normality, the 95% aymptotic confidence interval of the model parameter were alo obtained in both the cae. It i oberved that the etimate obtained in the imulation tudy are very cloe to the true value of the parameter and are alo quite well with relatively mall mean quared error. In the whole tudy, the parameter are etimated for different cae and it i found that a the ample ize increae, the MSE get maller. It implie that a larger ample ize reult in a better large ample approimation. Hence, it can be aid that the propoed GPM can be ued in the analyi of ALT. Reference [] Ahmad, N. (00), Deigning Accelerated Life Tet for Generalized Eponential Ditribution with Log-linear Model, International Journal of Reliability and Safety, vol. 4, no. /, pp [] Ahmad, N. and Ilam, A. (996), Optimal accelerated life tet deign for Burr type XII ditribution under periodic inpection and type I cenoring, Naval Reearch Logitic, vol. 4, pp [] Ahmad, N. Ilam, A., Kumar, R. and Tuteja, R. K. (994), Optimal Deign of Accelerated Life Tet Plan Under Periodic Inpection and Type I Cenoring: The Cae of Rayleigh Failure Law, South African Statitical Journal, 8, pp [4] Ahmad, N., Ilam, A. and Salam, A. (006), Analyi of optimal accelerated life tet plan for periodic inpection: The cae of Eponentiated Weibull failure model, International Journal of Quality & Reliability Management, vol., no. 8, pp [5] Bagdonaviciu, V. and Niulin, M. (00) Accelerated life model: Modeling and tatitical analyi Boca Raton: Chapman & Hall. [6] Chen, W., Gao, L., Liu, J., Qian, P. and Pan, J. (0), Optimal deign of multiple tre contant accelerated life tet plan on non- http: // [7] rectangle tet region, Chinee Journal of Mechanical Engineering, vol. 5, no. 6, pp. -7. [8] Ding, C., Yang, C. and Te, S. K. (00), Accelerated life tet ampling plan for the Weibull ditribution under type I progreive interval cenoring with random removal, Journal of Statitical Computation and Simulation, vol. 80, no. 8, pp [9] Fan, T. H. and Yu, C. H. (0), Statitical Inference on Contant Stre Accelerated Life Tet under Generalized Gamma Lifetime Ditribution, Quality and Reliability Engineering International. [0] Huang, S. (0), Statitical inference in accelerated life teting with geometric proce model, Mater thei, San Diego State Univerity. [] Ilam, A. and Ahmad, N. (994), Optimal deign of accelerated life tet for the Weibull ditribution under periodic inpection and type I cenoring, Microelectronic Reliability, vol. 4, no. 9, pp [] Kamal, M., Zarrin, S. and Ilam, A. (0): Accelerated life teting deign uing geometric proce for Pareto ditribution, International Journal of Advanced Statitic and Probability, vol., no., pp. 5-. [] Kamal, M., Zarrin, S., Saena, S. and Ilam, A. (0), Weibull Geometric Proce Model for the Analyi of Accelerated Life Teting with Complete Data, International Journal of Statitic and Application, vol., no. 5, pp [4] Lam, Y. (988): Geometric procee and replacement problem, Acta Mathematicae Applicatae Sinica, 4, pp [5] Lam, Y. (005), A monotone proce maintenance model for a multitate ytem, Journal of Applied Probability, vol. 4, no., pp. -4. [6] Lam, Y. and Zhang, Y. L. (996), Analyi of a two-component erie ytem with a geometric proce model, Naval Reearch Logitic, vol. 4, no. 4, pp (C) International Journal of Engineering Science & Reearch Technology [-9]

8 [Saena, (9): September, 0] ISSN: Impact Factor:.85 [7] Mann, N. R. and Singpurwalla, N.D. (98) Life Teting Encyclopedia of tatitical cience: USA: Wiley, Vol. 4, pp [8] Meeer, W. Q. and Ecobar, L. A. (998) Statitical method for reliability data New Yor: Wiley. [9] Nelon, W. (990): Accelerated Teting: Statitical Model, Tet Plan, and Data Analye, New Yor, John Wiley & Son. [0] Nelon, W. B. (980). Accelerated life teting: tep-tre model and data analye. IEEE Tranaction on Reliability, 9, [] Pan, Z., N. Balarihnan, and Quan Sun (0), Bivariate contant-tre accelerated degradation model and inference, Communication in Statitic-Simulation and Computation, vol. 40, no., [] Saena, S., Kamal, M., Zarrin, S., and Ilam, A. (0): Analyi of Accelerated Life Teting Uing Log-Logitic Geometric Proce Model in cae of Cenored Data, International Journal of Engineering Science and Technology, Vol.5, No. 7, pp [] Saena, S., Zarrin, S., and Ilam, A. (0): Optimum Step-Stre Accelerated Life Teting for Rayleigh ditribution, International Journal of Statitic and Application, Vol., No. 6, pp 0-5. [4] Watin, A.J. and John, A.M. (008), On contant tre accelerated life tet terminated by Type II cenoring at one of the tre level, Journal of Statitical Planning and Inference, vol. 8, no., pp [5] Yang, G. B. (994), Optimum contant-tre accelerated life-tet plan, IEEE Tranaction on Reliability, vol. 4, no. 4, pp [6] Zhang, Y. L. (008): A geometrical proce repair model for a repairable ytem with delayed repair, Computer and Mathematic with Application, vol. 55, no. 8, (008), pp [7] Zhou, K., Shi, Y. M. and Sun, T. Y. (0), Reliability Analyi for Accelerated Life-Tet with Progreive Hybrid Cenored Data Uing Geometric Proce, Journal of Phyical Science, vol. 6, -4. http: // (C) International Journal of Engineering Science & Reearch Technology [-9]