Inference for the Rayleigh Distribution Based on Progressive Type-II Fuzzy Censored Data

Transcription

1 Journal of Modern Appled Statstcal Methods Volue 13 Issue 1 Artcle Inference for the Raylegh Dstrbuton Based on Progressve Type-II Fuzzy Censored Data Abbas Pak Shahd Charan Unversty, Ahvaz, Iran, a-pak@scu.ac.r Ghola Al Parha Shahd Charan Unversty, Ahvaz, Iran, Parha_g@scu.ac.r Mansour Saraj Shahd Charan Unversty, Ahvaz, Iran, seraj.a@scu.ac.r Follow ths and addtonal works at: Part of the Appled Statstcs Coons, Socal and Behavoral Scences Coons, and the Statstcal Theory Coons Recoended Ctaton Pak, Abbas; Parha, Ghola Al; and Saraj, Mansour (014) "Inference for the Raylegh Dstrbuton Based on Progressve Type-II Fuzzy Censored Data," Journal of Modern Appled Statstcal Methods: Vol. 13 : Iss. 1, Artcle 19. DOI: 10.37/jas/ Avalable at: Ths Regular Artcle s brought to you for free and open access by the Open Access Journals at DgtalCoons@WayneState. It has been accepted for ncluson n Journal of Modern Appled Statstcal Methods by an authorzed edtor of DgtalCoons@WayneState.

2 Journal of Modern Appled Statstcal Methods May 014, Vol. 13, No. 1, Copyrght 014 JMASM, Inc. ISSN Inference for the Raylegh Dstrbuton Based on Progressve Type-II Fuzzy Censored Data Abbas Pak Shahd Charan Unversty Ahvaz, Iran Ghola Al Parha Shahd Charan Unversty Ahvaz, Iran Mansour Saraj Shahd Charan Unversty Ahvaz, Iran Classcal statstcal analyss of the Raylegh dstrbuton deals wth precse nforaton. However, n real world stuatons, eperental perforance results cannot always be recorded or easured precsely, but each observable event ay only be dentfed wth a fuzzy subset of the saple space. Therefore, the conventonal procedures used for estatng the Raylegh dstrbuton paraeter wll need to be adapted to the new stuaton. Ths artcle dscusses dfferent estaton ethods for the paraeters of the Raylegh dstrbuton on the bass of a progressvely type-ii censorng schee when the avalable observatons are descrbed by eans of fuzzy nforaton. They nclude the au lkelhood estaton, hghest posteror densty estaton and ethod of oents. The estaton procedures are dscussed n detal and copared va Monte Carlo sulatons n ters of ther average bases and ean squared errors. Fnally, one real data set s analyzed for llustratve purposes. Keywords: Progressve type-ii censorng, fuzzy nforaton, au lkelhood prncple, hghest posteror densty estaton Introducton The Raylegh dstrbuton was orgnally ntroduced by Lord Raylegh (1880) n the feld of acoustcs; snce ts ntroducton, any researchers have used the dstrbuton n dfferent felds of scence and technology. The Raylegh dstrbuton s frequently used to odel wave heghts n oceanography, n councaton engneerng and t also has a wde applcaton n lfete data analyss, especally n relablty theory and survval analyss. An portant characterstc of the Raylegh dstrbuton s that ts hazard rate s a lnearly ncreasng functon of te at constant rate, whch akes t a sutable odel for Abbas Pak s n the Departent of Statstcs. Eal at: a-pak@scu.ac.r. Ghola Al Parha s n the Departent of Matheatcs. Eal at: parha_g@scu.ac.r. Mansour Saraj s n the Departent of Matheatcs. Eal at: seraj.a@scu.ac.r. 87

3 INFERENCE FOR THE RAYLEIGH DISTRIBUTION the lfete of coponents/tes that age rapdly wth te. Thus, as te ncreases, the relablty functon of the Raylegh dstrbuton decreases at a uch hgher rate than the eponental relablty functon does. The probablty densty functon (pdf) and the cuulatve dstrbuton functon (cdf) of a Raylegh rando varable X can be wrtten as: f ( ) e ; 0, 0, (1) and F( ) 1 e ; 0, () respectvely. Inferences for the Raylegh dstrbuton have been dscussed by several authors. Dyer and Whsenand (1973) deonstrated the portance of ths dstrbuton n councaton engneerng. Bhattacharya and Tyag (1990) entoned that n soe clncal studes dealng wth cancer patents, the survval pattern follows the Raylegh dstrbuton. Chung (1995) obtaned the best nvarant estator and the Bayes estator of the paraeter of Raylegh dstrbuton under entropy loss. Fernandez (010) addressed the probles of estatng the paraeter, hazard rate and relablty functon of the Raylegh dstrbuton on the bass of saple quantles. Dey and Mat (01) derved Bayes estator of the Raylegh paraeter and ts assocated rsk based on etended Jeffrey s pror. In any lfe testng and relablty eperents, a saple of n tes s tested, and the eperent s ternated when all of the fal. Ths procedure ay take a long te when the lfete dstrbuton of tes has a thck tal. Moreover, f the tes are epensve, such as edcal equpent, t s costly to gather nforaton fro the whole saple. There are any stuatons where eperental unts are lost or reoved fro the test before coplete falure. For eaple, ndvduals n a clncal tral ay drop out of the study, the study ay have to be ternated early for lack of funds or the test unts ay accdentally break. In other scenaros, the eperent ay have to be ternated n order to free up testng facltes for other purposes. In vew of above, censorng s used n lfe testng to save te and cost of testng unts. The reoval of unts n a test ay be unntentonal or pre-planned. Data obtaned fro such eperents are called censored saple. There are any types of censorng schees used n lfete analyss. The two ost coon 88

4 PAK ET AL censorng schees are tered type-i and type-ii censorng schees. In the conventonal type-i censorng schee, the eperent contnues up to a prespecfed te T; the conventonal type-ii censorng schee requres the eperent to contnue untl a pre-specfed nuber of falures occur. These schees, however, do not allow reoval of unts before the ternaton of the eperent; thus, a ore general knd of censorng schee called progressve type-ii censorng s consdered, whch s as follows: Suppose that n unts are placed on a lfe test and the eperenter decdes beforehand a quantty, the nuber of unts to be faled. Now at the te of the frst falure, R1 of the reanng n 1 survvng unts are randoly reoved fro the eperent. Contnung on, at the te of the second falure, R of the reanng n R1 unts are randoly reoved fro the eperent. Fnally, at the te of the th falure, all the reanng n R1 R 1(=R) survvng unts are reoved fro the eperent. The work on progressve censorng has becoe popular n lfe-testng and relablty studes. K and Han (009) studed the proble of estatng the scale paraeter of the Raylegh dstrbuton under general progressve censorng. Krshna and Kuar (011) dscussed relablty estaton for the Lndley dstrbuton wth progressve type-ii censored data. Lee et al. (011) obtaned a Bayes estator under the Raylegh dstrbuton wth a progressve type-ii rght censored saple. Raqab and Mad (011) addressed nference for the generalzed Raylegh dstrbuton based on progressvely censored data. Az et al. (01) consdered the Bayesan estaton of the paraeter and relablty functon of Raylegh dstrbuton based on a progressvely type-ii censored saple. Rastog and Trpath (01) studed paraeter estaton of the Burr type XII dstrbuton on the bass of a progressvely type-ii censored saple. Pradhan and Kundu (009) consdered the statstcal nference of the unknown paraeters of the generalzed eponental dstrbuton n presence of progressve censorng. A recent account on progressve censorng schees can be obtaned n the onograph by Balakrshnan and Aggarwala (000) or n the ecellent revew artcle by Balakrshnan (007). The above referenced studes for estatng paraeters of dfferent lfete dstrbutons under progressve type-ii censorng are lted to precse data. However, n real world stuatons, eperents do not provde eact nforaton. For eaple, the reacton te of a person to a certan stulus n a psychologcal eperence cannot be eactly deterned, but the psychologst s able to deterne t by eans of the followng precse nforaton, such as: The te of reacton s approately 5 to 35 seconds. To deal wth the lack of precson of the data, t s necessary to ncorporate the fuzzy concept to statstcal technques. Recently, 89

5 INFERENCE FOR THE RAYLEIGH DISTRIBUTION Pak et al. (013) proposed a new ethod to deterne the au lkelhood estate of the scale paraeter of a Raylegh dstrbuton under doubly type-ii censored saple fro fuzzy data. Further, n a lfe testng eperent, soe test unts ay need to be reoved at dfferent stages n the study for varous reasons. Ths would lead to progressve censorng. The purpose of ths artcle s to develop the nferental procedures for the Raylegh dstrbuton under a progressve type-ii censorng schee when the avalable observatons are reported by eans of fuzzy nforaton. The au lkelhood estate (MLE) of the paraeter s obtaned by usng EM algorth and the hghest posteror densty (HPD) estate of the unknown paraeter s coputed. The estaton va ethod of oents s dscussed, a Monte Carlo sulaton study s presented, and a coparson of all estaton procedures developed and one real data set s analyzed for llustratve purposes. Frst, the fundaental notaton and basc defntons of fuzzy set theory used heren s revewed. Consder an eperent characterzed by a probablty space S, F, P where, F s a easurable space and P belongs to a specfed faly of probablty easures { P, } on, F. Assue that the observer cannot dstngush or transt wth eactness the outcoe n the perforance of S, but that rather the avalable observaton ay be descrbed n ters of fuzzy nforaton, whch s defned as: Defnton 1 A fuzzy event on, characterzed by a Borel easurable ebershp functon ( ) fro to [0,1], where ( ) represents the grade of ebershp of to, s called fuzzy nforaton assocated wth the eperent. The set consstng of all observable events fro the eperent fuzzy nforaton syste assocated wth t, whch s defned as: deternes a Defnton A fuzzy nforaton syste (f..s.) assocated wth the eperent s a fuzzy partton { 1,..., K} of,.e., a set of K fuzzy events on satsfyng the orthogonalty condton (see Tanaka et al., 1979): 90

6 PAK ET AL K ( ) 1, k k1 where k denotes the ebershp functon of k. Accordng to Zadeh (1968), gven the eperent, FP,,, and a f..s. assocated wth t, each probablty easure P on, F nduces a probablty easure on Defnton 3 defned as: The probablty dstrbuton on nduced by P s the appng P fro to [0,1] such that ( ) ( ) dp( ),. (3) In partcular, the condtonal densty of a contnuous rando varable Y wth p.d.f. g( y ) gven the fuzzy event can be defned as ( y) g( y) g( y ) ( u) g( u) du. (4) In order to odel precse lfetes, a generalzaton of real nubers s necessary. These lfetes can be represented by fuzzy nubers. A fuzzy nuber s a subset, denoted by, of the set of real nubers (denoted by ) and s characterzed by the so called ebershp functon (.). Fuzzy nubers satsfy the constrants (Dubos and Prade, 1980): 1. : 0,1 s Borel-easurable;. : 1; and

7 INFERENCE FOR THE RAYLEIGH DISTRIBUTION 3. The so-called cuts B :, 0 1, defned as are all closed nterval,.e., B ( ) [ a, b ], (0,1]. Aong the varous types of fuzzy nubers, the trangular and trapezodal fuzzy nubers are ost convenent and useful n descrbng fuzzy lfete data. The trangular fuzzy nuber can be defned as ( a, b, c) and ts ebershp functon s defned by the followng epresson: a a b, b a c ( ) b c, c b 0 otherwse. The trapezodal fuzzy nuber can be defned as ( a, b, c, d) wth the ebershp functon: a a b, b a 1 b c ( ) d c d, d c 0 otherwse. Mau lkelhood estaton Suppose that n dentcal unts are put on a lfe testng eperent and that the lfete dstrbuton of each unt s gven by (1). Pror to the eperent, a nuber < n s deterned and the censorng schee ( R1,..., R ) wth R 0 and R n s specfed. Let 1 1 (,..., ) denote the correspondng progressvely type-ii censored saple. The lkelhood functon for the paraeter becoes proportonal to 9

8 PAK ET AL e (1 R) 1 L( ; ) (5) Now consder the proble where s not observed precsely and only partal nforaton about s avalable n the for of fuzzy nubers ( a, c, b ), 1,...,, wth the correspondng ebershp functons ( ),..., ( ). Let c(1) c()... c( ) denote the ordered values of the 1 1 eans of these fuzzy nubers. The lfete of R survvng unts, whch are reoved fro the test after the th falure, can be encoded as fuzzy nubers z,..., 1 z R wth the ebershp functons 0 z c() z ( z), j 1,..., R. j 1 z c() The fuzzy data w ( 1,...,, z1,..., z) where z s a 1 R vector wth z ( z,..., z ) for 1,...,, s thus the set of observed lfetes. The 1 R correspondng observed data log-lkelhood functon can be obtaned by usng the epresson (3) as follows: R z O w z j 1 1 j1 L ( ; ) log e ( ) d log ze ( z) dz () 1 1 log log e ( ) d R c. (6) The au lkelhood estate of the paraeter can be obtaned by azng the log-lkelhood L ( w ; ). Equatng the partal dervatve of the O log-lkelhood (6) wth respect to to zero, the resultng equaton s: 3 e ( ) LO ( w; ) d Rc () 0. 1 e ( ) d 1 Because there s no closed for of the soluton to equaton (7), an teratve nuercal search can be used to obtan the MLE. The Epectaton Mazaton (EM) algorth s a broadly applcable approach to the teratve coputaton of au lkelhood estates and useful n a varety of ncoplete-data (7) 93

9 INFERENCE FOR THE RAYLEIGH DISTRIBUTION probles. Because the observed fuzzy data w can be seen as an ncoplete specfcaton of a coplete data, the EM algorth s applcable to obtan the au lkelhood estate of the paraeter. In the followng, the fuzzy EM algorth (see Denoeu, 011) s used to deterne the MLE of. Frst, denote the lfetes of the faled and censored unts by X ( X1,..., X ) and Z ( Z1,..., Z ), respectvely, where Z s a 1 R vector wth Z ( Z 1,..., ZR ), for 1,...,. The cobnaton of W ( X,Z ) fors the coplete lfetes and the correspondng log-lkelhood functon s denoted by L (, ) c W, then, gnorng the addtve constant, R L ( W c, ) nlog zj. (8) 1 1 j1 For the E-step, t s necessary to copute the pseudo log-lkelhood functon. It can be obtaned fro (8) as follows: n E X E Z z R log ( ) ( j j ) 1 1 j1 (9) By usng (4), the condtonal epectatons coputed as: E ( X ) and E Z z can be ( j j ) 3 e ( ) d E ( X ), 1,...,, e ( ) d 1 E ( Zj zj ) c(), 1,...,, j 1,..., R. Net, the M-step nvolves the azaton of the pseudo functon (9). Therefore, f at the h th ( h) ( h 1) stage, the estate of s, then can be obtaned by azng R * L ( W c, ) nlog E ( h) ( X ) E ( h) ( Z zj ) (10) 1 1 j1 94

10 PAK ET AL wth respect to. Fro L * c( W, ) 0, ˆ ( h1) 1 [ E ( X ) R ( c 1/ )] ( h) n ( h) () (11) The teraton process contnues untl convergence,.e., untl ( h 1) ( h) L ( w; ) - L ( w ; ) for soe pre-fed 0. O O HPD estaton Consder the hghest posteror densty (HPD) estaton of the Raylegh paraeter based on observed fuzzy saple w. As a conjugate pror for, take the Gaa ( a, b) densty wth pdf gven by b ( a) a a1 b ( ) e, 0, (1) where a 0 and b 0. Based on ths pror, the posteror densty functon of gven the data can be wrtten as follows: ( ) ( brc( ) ) a1 1 ( ) 1 w e e d (13) The ethod of HPD estaton then estates as the ode of the posteror densty w ( ); therefore, the HPD estate of can be obtaned by solvng the equaton log ( w) 0 (14) where 95

11 INFERENCE FOR THE RAYLEIGH DISTRIBUTION 3 e Rc() 1 1 e log ( w) a1 ( ) d b. (15) ( ) d However, the soluton cannot be obtaned eplctly. Theore 1 dscusses the estence and unqueness of the HPD estate of. Theore 1 Let g( ) denote the functon on the rght-hand sde of the epresson n (15). Then the root of the equaton g( ) 0 ests and s unque. Proof. Fro (15), t s seen that a1 g( ), (0, ), and consequently l g( ). Also, note that 0 a1 l g( ) l 0, (0, ). 0 0 Thus, the equaton g( ) 0 has at least one root n (0, ). To prove that the root s unque, consder the frst dervatve of g, g ( ) gven by a1 g( ) log e ( ) d 1 (16) Because the ntegrand of the second ter n (16) s a log-concave functon of, and g( ) 0. It follows that g s a strctly decreasng functon w.r.t. and hence the equaton g( ) 0 has eactly one soluton. The HPD estate of ust be derved nuercally. In the followng, the Newton-Raphson algorth to deterne the HPD estate s descrbed. The Newton-Raphson algorth s a drect approach for estatng the relevant paraeters n a lkelhood functon. In ths algorth, the soluton of the ( ) lkelhood equaton s obtaned through an teratve procedure. Let ˆ h be the paraeter value fro the h th step. Then, at the ( h 1) th step of teraton process, the updated paraeter s obtaned as 96

12 PAK ET AL ˆ ˆ ( h1) ( h) log ( w) log ( w) h h (17) where the notaton evaluated at A h, for any partal dervatve A, eans the partal dervatve ˆ( h ). The second-order dervatve of log w ( ) wth respect to the paraeter, requred for proceedng wth the Newton-Raphson ethod, s obtaned as: [ ]. (18) 5 3 n log ( w) a1 e ( ) d e ( ) d 1 e ( ) d e ( ) d The teraton process then contnues untl convergence,.e., untl for soe pre-fed 0. ˆ ˆ, ( h1) ( h) Method of oents Let X be a rando varable whch has the Raylegh dstrbuton wth pdf gven by (1). It s known that the k th oent of the Raylegh odel wth paraeter s k (19) k ( k ) (1 ). EX Equatng the frst saple oent to the correspondng populaton oent, the followng equaton can be used to fnd the estate of oent ethod: n j1 R E ( X ) E( Z zj ). (0) Because the closed for of the soluton to (0) could not be obtaned, an teratve nuercal process to obtan the paraeter estate s descrbed as: 97

13 INFERENCE FOR THE RAYLEIGH DISTRIBUTION (0) 1. Let the ntal estate of be, wth h 0.. In the ( h 1) th teraton, frst copute ( h) e ( ) d E1 E ( h) ( X ) ( h), 1,...,, e ( ) d and ( h) z z e z () z dz j ( h) j ( h) z ze z () z dz j E E ( Z z ), 1,...,, j 1,..., R. The new estate of, for eaple ( h 1), can be obtaned fro: ( h1) n 4 ( E1 R E ) Checkng convergence, f the convergence occurs then the current ( h 1) s the estate of by the ethod of oents; otherwse, set hh 1 and go to Step. Fgure 1. Fuzzy nforaton syste used to encode the sulated data 98

14 PAK ET AL Nuercal Study Eperental results llustrate how the dfferent ethods behave for varyng saple szes. Frst, for fed 1 and dfferent choces of n, and censorng schee ( R1,..., R ), progressvely censored saples fro the Raylegh dstrbuton were generated, usng the ethod proposed by Balakrshnan and Sandhu (1995), as follows: 1. Generate Z fro U(0,1) for 1,...,.. For gven values of the progressve censorng schee ( R1,..., R ), set V 1/ a Z, a R, 1,...,. j1 j 3. Set U 1 V 1V... V, 1,...,. 4. Thus, X F 1 ( U ), 1,...,, s the desred progressve type-ii censored saple fro the Raylegh dstrbuton. Each realzaton of was then fuzzfed usng the f..s. shown n Fgure 1, and the ML, HPD and oent estates (MME) of for the fuzzy saple were coputed usng the ethods provded n the precedng sectons. For coputng the HPD estate of the unknown paraeter, assue that has Gaa ( a, b ) pror. To ake the coparson eanngful, t s assued that the prors are nonnforatve, and they are ab 0. Note that n ths case the prors are nonproper also. Press (001) suggested usng very sall non-negatve values of the hyperparaeters n ths case, and t wll ake the prors proper. Ths study uses ab The results are not sgnfcantly dfferent than the correspondng results obtaned usng non-proper prors, and are not reported due to space. The average values and ean squared errors of the estates, coputed over 1,000 replcaton, are presented n Tables

15 INFERENCE FOR THE RAYLEIGH DISTRIBUTION Table 1. Average value (AV) and ean squared error (MSE) of the estates of λ for dfferent censorng schees. (n = 0) Censorng schee MLE HPD MME AV MSE AV MSE AV MSE (0,,0,1) (1,0, 0) (0,1,0,,0) (0,,0,10) (10,0, 0) (0,10,0,,0) (0,,0,5) (5,0, 0) (0,5,0,,0) Table. Average value (AV) and ean squared error (MSE) of the estates of λ for dfferent censorng schees. (n = 30) Censorng schee MLE HPD MME AV MSE AV MSE AV MSE (0,,0,1) (1,0, 0) (0,1,0,,0) (0,,0,10) (10,0, 0) (0,10,0,,0) (0,,0,5) (5,0, 0) (0,5,0,,0) Table 3. Average value (AV) and ean squared error (MSE) of the estates of λ for dfferent censorng schees. (n = 50) Censorng schee MLE HPD MME AV MSE AV MSE AV MSE (0,,0,1) (1,0, 0) 1, (0,1,0,,0) (0,,0,10) (10,0, 0) (0,10,0,,0) (0,,0,5) (5,0, 0) (0,5,0,,0)

16 PAK ET AL Several ponts are clear fro the eperent: Even for sall saple szes, the perforances of the estates are satsfactory n ters of AVs and MSEs. For all the ethods, t s observed that for fed n as ncreases, the MSEs of the estates decrease. Aong the three estaton procedures developed n the paper, the HPD procedure gves the ost precse paraeter estates as shown by MSEs n Tables 1-3. Applcaton eaple To deonstrate the applcaton of proposed ethods to real data, consder the data collected durng the eperent reported by Pak et al. (013). In ths eperent, a saple of 5 ball bearngs s placed on a lfe test. A ball bearng ay work perfectly over a certan perod but be breakng for soe te and fnally be unusable at a certan te. So, the observed falure tes of the ball bearngs are reported by fuzzy nubers ( a,, b ), where a 0.05 and b 0.03 wth the ebershp functons ( a) a, a ( ) b b, b 1,...,5. Progressvely censored saples of sze 10 were consdered fro these fuzzy data usng three dfferent saplng schees, naely: Schee 1: R1... R 1 0 and R 15. Schee : R1 15 and R... R 0. Schee 3: R1... R 1 1 and R 6. the estate of the paraeter was then coputed usng the ML, HPD and oent ethods. For coputng the HPD estates, t was assued that has Gaa ( a, b) pror, ncludng the non-nforatve gaa pror,.e. ab 0 and nforatve gaa pror,.e. ab. All the results are suarzed n Table

17 INFERENCE FOR THE RAYLEIGH DISTRIBUTION Table 4. ML, HPD and oent estates of the paraeter for Eaple Schee MLE MME HPD (a = b = 0) HPD (a = b = ) Conclusons Soe work has been done n the past on the estaton of the paraeter of Raylegh dstrbuton based on coplete and censored saples, but tradtonally t s assued that the data avalable are perfored n eact nubers. In real world stuatons, however, soe collected lfete data ght be precse and are represented n the for of fuzzy nubers. Therefore, sutable statstcal ethodology s needed to handle these data. Ths artcle proposed dfferent procedures for estatng the paraeter of Raylegh dstrbuton under progressve type-ii censorng when the avalable observatons are descrbed by eans of fuzzy nforaton. They are au lkelhood estaton (MLE), hghest posteror densty (HPD) estaton and ethod of oents (MME). A sulaton study was conducted to assess the perforance of these procedures. Based on the results of the sulaton study, t ay be observed that, the perforance of the HPD estates s generally best followed by the MLE and MME. Thus, t would see reasonable to recoend the use of the HPD procedure for estatng the unknown paraeter fro the Raylegh dstrbuton. References Az, R., Yaghae, F., & Az, D. (01). Coparson of Bayesan estaton ethods for Raylegh progressve censored data under the dfferent asyetrc loss functon. Internatonal Journal of Appled Matheatcal Research, 1(4): Balakrshnan, N. (007). Progressve censorng ethodology: an apprasal. Test, 16(): Balakrshnan, N. & Aggarwala, R. (000). Progressve censorng, theory, ethods and applcatons. Boston, MA: Brkhauser. 30

18 PAK ET AL Balakrshnan, N. & Sandhu, R. A. (1995). A sple algorth for generatng progressvely Type-II censored saples. The Aercan Statstcan, 49(): Bhattacharya, S. K., & Tyag, R. K. (1990). Bayesan survval analyss based on the Raylegh odel. Trabajos de Estadstca, 5: Chung, Y. (1995). Estaton of scale paraeter fro Raylegh dstrbuton under entropy loss. Journal of Appled Matheatcs and Coputng, (1): Denoeu, T. (011). Mau lkelhood estaton fro fuzzy data usng the EM algorth. Fuzzy Sets and Systes, 183(1): Dey, S., & Mat, S. S. (01). Bayesan estaton of the paraeter of Raylegh dstrbuton under the etended Jeffrey's pror. Electronc Journal of Appled Statstcal Analyss, 5(1), Dubos, D., & Prade, H. (1980). Fuzzy Sets and Systes: Theory and Applcatons. Acadec Press, New York. Dyer, D. D. & Whsenand, C. W. (1973). Best lnear unbased estator of the paraeter of the Raylegh dstrbuton. IEEE Transactons on Relablty R- : 7-34, Fernandez, A. J. (010). Bayesan estaton and predcton based on Raylegh saple quantles. Qualty & Quantty, 44: K, C., and Han, K. (009). Estaton of the scale paraeter of the Raylegh dstrbuton under general progressve censorng. Journal of the Korean Statstcal Socety, 38: Lee, W., Wu, J., Hong, M., Ln, L., & Chan, R. (011). Assessng the lfete perforance nde of Raylegh products based on the Bayesan estaton under progressve type II rght censored saples. Journal of Coputatonal and Appled Matheatcs, 35: Krshna, H. & Kuar, K. (011). Relablty estaton n Lndley dstrbuton wth progressvely type-ii rght censored saple. Matheatcs and Coputers n Sulaton, 8: Pak, A., Parha, G. A., & Saraj, M. (013). On estaton of Raylegh scale paraeter under doubly type-ii censorng fro precse data. Journal of Data Scence, 11: Pradhan, B., & Kundu, D. (009). On progressvely censored generalzed eponental dstrbuton. Test, 18:

19 INFERENCE FOR THE RAYLEIGH DISTRIBUTION Raqab, M. Z., & Mad, T. M. (011). Inference for the generalzed Rayleg dstrbuton based on progressvely censored data. Journal of Statstcal Plannng and Inference, 141: Rastog, M. K., & Trpath, Y. M. (01). Estatng the paraeters of Burr dstrbuton under progressve type II censorng. Statstcak Methodology, 9: Raylegh, J. W. S. (1880). On the resultant of a large nuber of vbraton of the soepth and of arbtrary phase. Phlosophcal Magazne, 5th seres, 10: Press, S. J. (001). The subjectvty of scentsts and the Bayesan approach. New York: Wley. Tanaka, H. Okuda, T. & Asa, K. (1979). Fuzzy nforaton and decson n statstcal odel. In M. M. Gupta et al., Eds. Advances n Fuzzy Sets Theory and Applcatons, pp Asterda: North-Holland Publshng Co. Zadeh, L. A. (1968). Probablty easures of fuzzy events. Journal of Matheatcal Analyss and Applcatons 10: