GENERAL INTRODUCTION AND SURVEY OF LITERATURE

Transcription

1 CHAPTER 1 GENERAL INTRODUCTION AND SURVEY OF LITERATURE 1.1 Inroducion In reliabiliy and survival sudies, many life disribuions are characerized by monoonic failure rae. Trayer (1964) inroduced he inverse Rayleigh disribuion. The probabiliy densiy funcion (pdf) and cumulaive disribuion funcion (cdf) of sandard inverse Rayleigh disribuion (IRD), respecively, are given by 1 f( z) exp( ), 3 z z z > 0 (1.1.1) = 0, z 0 1 Fz ( ) exp( ), z > 0 (1.1.) z = 0, z 0 If a scale parameer Rayleigh disribuion are respecively, given by 0 is inroduced, he pdf and cdf of inverse f (; ) exp( / ), > 0, 0 (1.1.3) 3 = 0, 0 F (; ) exp( / ), > 0, 0 (1.1.4) = 0, 0

2 The inverse Rayliegh disribuion (IRD) can be considered as a life ime disribuion. The reliabiliy funcion R() and hazard funcion h() of IRD are R () 1 exp( / ), > 0, 0 (1.1.5) h (), > 0, 0 (1.1.6) 3 [exp( / ) 1] The failure rae of a single parameer IRD is increasing for < and decreasing for > as shown by Mukherjee and Saran (1984). The graphs of pdf, reliabiliy funcion and hazard funcion (failure rae) of a sandard IRD are given below : Pdf curve f()

3 Reliabiliy funcion curve R() Failure rae curve h() I can be seen ha, he hazard funcion shows a rend of firs increasing and hen decreasing naure of failure occurrence, which is dual o a ypical bahub hazard model. Therefore, his model can be used in siuaions of producs which have iniially more probable failures and ge sabilized evenually resuling in a decreasing failure rae quie common in sofware producs (Gokhale and Trivedi (1998); Srinivasa Rao (001)). 3

4 1. Research lieraure based on inverse Rayleigh disribuion The maximum likelihood (ML) mehod of esimaion, o esimae he parameers of cerain disribuions does no yield explici soluions from complee/censored samples. In such siuaions, some auhors sugges cerain modificaions or approximaions o he log-likelihood equaions, which yield explici soluions o he parameers. Several auhors like Tiku (1967), Mehrora and Nanda (1974), Persson and Roozen (1977), Cohen and Whien (1980), Tiku e al. (1986), Balakrishnan (1990) and Tiku and Suresh (199) obained modified maximum likelihood (MML) esimaes by making linear approximaions o cerain funcions in esimaing equaions of parameers of normal, lognormal, logisic, exponenial and Rayleigh disribuions. The modificaions or approximaions suggesed o some erms in he loglikelihood equaion by he auhors menioned above are for esimaion in he respecive densiies from complee or censored samples. The above lieraure shows ha no much is known abou he esimaion of scale parameer in inverse Rayleigh disribuion from complee and censored samples. Rosaiah e al. (1993a and 1993b) considered he modified ML (MML) esimaion in gamma disribuion wih and wihou prior relaions beween is parameers. Oher similar works include Kanam and Dharma Rao (1993) in half logisic disribuion from complee samples, Kanam and Srinivasa Rao (1993) in Rayleigh disribuion from lef censored samples, Kanam and Srinivasa Rao (00) exended MML esimaion o log-logisic disribuion. Skewed disribuions o develop conrol chars are aemped by many auhors. Some of hem are Edgeman (1989) inverse Gaussian disribuion, Gonzalez and Viles (000) Gamma disribuion, Kanam 4

5 and Sriram (001) Gamma disribuion, Beul and Yaziki (006) Burr disribuion, Kanam e al. (006a) Log logisic disribuion and references herein. Since inverse Rayleigh disribuion is a skewed disribuion, we developed he conrol char consans for mean, median, mid range and range chars. Chan and Cui (003) have developed conrol char consans for x and R chars in a unified way for a skewed disribuions where he consans are dependen on he coefficien of skewness of he disribuion. We make an aemp o develop he ANOM procedure of O (1967) when he daa variae is assumed o follow inverse Rayleigh disribuion. In he 1930 s and 1940 s, accepance sampling was one of he major componens of he field of saisical qualiy conrol and was used primarily for incoming or receiving inspecion. Sobel and Tischendrof (1959) have developed accepance sampling plans based on life ess in exponenial model. Similar works are Goode and Kao (1961) for Weibull disribuion; by Gupa and Groll (1961) for Gamma disribuion; by Kanam and Rosaiah (1998) for half logisic disribuion; by Kanam e al. (001) for log-logisic disribuion; by Rosaiah and Kanam (005) for inverse Rayleigh disribuion and Rosaiah e al. (005) for exponeniaed log-logisic disribuion; Srinivasarao e al. (009b) for Marshall-Olkin exended Lomax disribuion. Balakrishnan e al. (007) have proposed he accepance sampling plans for he quaniles and derived he formulae. Sampling plans in a new economic approach for log-logisic disribuion are suggesed by Kanam e al. (006b) and Srinivasarao e al. (009a) for Marshall-Olkin exended Lomax disribuion. Srinivasa Rao and Kanam (010) have developed accepance sampling plans from runcaed life ess based on he loglogisic disribuion for perceniles and Srinivasa Rao (010a) developed 5

6 accepance sampling plans from runcaed life ess based on he Marshall-Olkin exended exponenial disribuion for perceniles. Lio e al. (010) developed for he accepance sampling plans for perceniles of he Birnbaum-Saunders (BS) model. The concep of sress-srengh in engineering devices has been one of he deciding facors of failure of he devices. I has been cusomary o define safey facors for longer lives of sysems in erms of he inheren srengh hey have and he exernal sress being experienced by he sysems. Church and Harris (1970) have compued he confidence inervals for P(Y<X) under he assumpion ha X and Y are independenly normally disribued and he disribuion of Y is known. The procedure developed by hem is compared wih he disribuion free confidence bounds of P(Y<X) suggesed by Govindarajulu (1966). Enis and Geisser (1971) considered he problem of esimaing P Y X in boh disribuion free and parameric frame works. Downown (1973) sudied he esimaion of P(X>Y) in he normal case. Bhaacharya and Johnson (1974) have formulaed a sysem consising of k idenical componens and defined a muli-componen s ou of k sress-srengh model. Tong (1977) inroduced he esimaion of P(Y<X) for exponenial families. Awad e al. (1981) have worked ou some inference resuls in P(Y<X) in he bivariae exponenial model. Sahe and Shah (1981) sudied on esimaion of P(Y<X) for he exponenial disribuion. Pandey and Borhan (1985) sudied he reliabiliy in a muli-componen sresssrengh sysem when boh sress and srengh follow Burr disribuion. Awad and Gharraf (1986) provided a simulaion sudy which conains hree esimaors for R P Y X when Y and X are wo independen bu no idenically disribued Burr random variables. Benjamin and 6

7 Guman (1986) examined saisical inference for P(Y<X), where X and Y are independen normal variaes wih unknown means and variances. Consanile and Karson (1986) considered esimaion of P(Y<X) where X ( M, ), Y ( N, ) are independen when M and N are known. Kakai and Srivasav (1986) considered an acceleraed life esing problem for he sress-srengh model. Pandey and Upadhyay (1986) have aken up he problem of comparing he sysem reliabiliy esimaes in muli-componen sress-srengh sysems when sress, srengh disribuions are Weibull models wih equal scale parameers. Dua and Srivasav (1989) sudied an n-uni warm sandby redundan sysem for sress-srengh model. Gupa and Gupa (1990) considered esimaion of P(aX > by) in he mulivariae normal case. Travadi and Raani (1990) sudied esimaion of reliabiliy funcion for inverse Guassian disribuion wih known coefficien of variaion. McCool (1991) examined inference on P(Y<X) in he case of Weibull disribuion. Handi & Johnson (199) considered he sress-srengh problems in which a componen of srengh X is subjeced o environmenal sress Y, where X and Y have independen normal disribuions. Nandi and Aich (1994) have invesigaed he problem of esimaing he reliabiliy R=P(Y<X) ha arises in sress-srengh relaionship where X follows an exponenial disribuion while Y has an inverse Gaussian / half normal / half Cauchy disribuion. Pora, e al. (1994) presened a procedure for he poin and inerval esimaion for a reliabiliy model of P(X>Y) or P(X Y). Pham and Almhana (1995) have reviewed basic properies of hree parameer generalized gamma disribuion and presened resuls on he hazard rae and sress-srengh model of he generalized gamma disribuion. Surles and Padge (1998, 001) sudied inference for P(Y<X) in he Burr ype X 7

8 model. Gupa e al. (1999) presened a procedure for he poin and inerval esimaion of P(X>Y) in he normal case wih common coefficien of variaion. Gupa and Brown (001) have aken up reliabiliy sudies of he skew-normal disribuion and is applicaion o sress-srengh models. Raqab and Kundu (005) sudied he comparison of differen esimaors of P(Y<X) for a scaled Burr ype X disribuion. Mokhlis (005) considered he reliabiliy of a sress-srengh model wih Burr ype III disribuions. Kundu and Gupa (005, 006) sudied on he esimaion of P(Y<X) for he generalized exponenial disribuion and Weibull disribuion respecively. Kanam e al.(007) sudied on sress-srengh reliabiliy model in log-logisic disribuion. Raqab e al. (008) inroduced he esimaion of P(Y<X) for he hree-parameer generalized exponenial disribuion. Kundu and Raqab (009) sudied on he esimaion of R=P(Y<X) for hree-parameer Weibull disribuion. Gupa e al. (010) derived he esimaion of reliabiliy from Marshall-Olkin exended Lomax disribuion. Kanam e al. (007) sudied on sresssrengh reliabiliy model in log-logisic disribuion. Srinivasa Rao and Kanam (010b) sudied esimaion of reliabiliy in muli-componen sress-srengh for log-logisic disribuion Problems of invesigaion The conens of he research aricles/ lieraure described in Secion 1. reveal ha he following problems are no paid much aenion nor available in published form in respec of inverse Rayleigh disribuion. I is also a basic probabiliy model in saisical inference. Therefore an aemp is made o sudy he following problems and he findings of our invesigaion are presened in he form of his hesis. 8

9 1. Reliabiliy esimaion hrough esimaion of he parameers in inverse Rayleigh disribuion by ML and modified ML mehods from complee and censored samples and sudy of sofware failure phenomenon hrough a non homogeneous Poisson process model wih he help of inverse Rayleigh disribuion as mean value funcion.. Consrucion of variable conrol chars for process mean, process dispersion where he probabiliy model of he underlying measurable qualiy characerisic is an inverse Rayleigh model, including he analysis of means (ANOM) mehod. 3. Accepance sampling plans based on average and perceniles for he inverse Rayleigh disribuion under a runcaed life es along wih operaing characerisic curves and comparaive sudy of wo approaches. 4. Esimaion of reliabiliy of sress-srengh model of inverse Rayleigh disribuion by ML mehod and momen mehod along wih asympoic, exac and boosrap inervals in sress-srengh model and also reliabiliy of muli-componen sress-srengh model using ML mehod and asympoic confidence inervals of muli-componen sress-srengh model of inverse Rayleigh disribuion. 9