Reliability estimation in Pareto-I distribution based on progressively type II censored sample with binomial removals

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1 Journal of Scentfc esearch Developent (): Avalable onlne at wwwjsradorg ISSN JSAD elablty estaton n Pareto-I dstrbuton based on progressvely type II censored saple wth bnoal reovals Ilhan USTA * Hanef Gezer Doctor of Phlosophy Assocate Professor Departent of Statstcs Faculty of Scence Anadolu Unversty Esksehr Turkey Master of Statstcs Faculty of Scence Anadolu Unversty Esksehr Turkey Abstract: In ths study we deal wth the estaton proble for the paraeters relablty characterstcs; relablty functon hazard rate functon ean te syste to falure of Pareto-I dstrbuton based on progressvely type-ii censored saple wth ro reovals The nuber of unts reoved at each falure te s assued to follow a bnoal dstrbuton The axu lkelhood ethod s used to obtan the estators of the paraeters relablty characterstcs functons of Pareto-I dstrbuton Monte Carlo sulaton s perfored to copare the perforance of axu lkelhood estates under progressvely type-ii censorng wth the dfferent ro schees Key words: Pareto-I dstrbuton; Maxu lkelhood ethod; Progressve type-ii censorng; Bnoal reovals; elablty characterstcs Introducton * The Pareto dstrbuton has been wdely used n the analyss of lfete data fro relablty survval nsurance econoy engneerng so on (Johnson et al 994) Also t s well known that n lfete testng experents the falure tes of all unts placed on the test are not always observed by the experenter Saples that result fro such cases are called censored saples There are several types of censorng schees However progressve censorng schees n the last few years have been studed rather extensvely by any authors Because these schees allow the experenter to reove unts before the ternaton of the experent Therefore progressve censorng schees are coonly used n relablty experents clncal trals lfe-testng experents etc For a coprehensve recent revew of progressve censorng see Balakrshnan Aggarwala (000) One of the coon progressve censorng schees s progressve type-ii censorng was ntroduced by Cohen (963) The progressvely type- II censored lfe test s defned as follows The experenter places n dentcal unts on test at te zero copletely observe only falures When the frst falure s observed of the reanng n survvng unts are roly selected reoved Then after the second observed falure of the reanng n survvng unts are roly selected reoved so on Fnally the experent ternates untl the falure s t h observed reanng n survvng unts all reoved If 0 * Correspondng Author n then that corresponds Type-II censorng If 0 then n that corresponds the coplete saple Moreover are all prefxed n ths censorng schee However n soe practcal stuatons these nubers cannot be prefxed they occur at ro For exaple Yuen Tse (996) ponted out that the nuber of patents that wthdraw fro a clncal test at each stage s ro cannot be prefxed Therefore the statstcal nference on lfete dstrbutons under progressve type II censorng wth ro reovals has been studed n recent years by varous authors ncludng Yuen Tse (996) Wu etal (007) Yan et al (0) Dey Dey (04) Az et al (04) In ths study we consder the two-paraeter Pareto dstrbuton of the frst knd (Pareto I) as lfete dstrbuton Also for the case that the observed data are fro the Pareto I dstrbuton based on the progressve type II censorng wth bnoal reovals we deal wth the estaton probles of the paraeters relablty characterstcs such as relablty functon hazard rate functon ean te to falure In the lterature there are soe studes on the nference of the Pareto dstrbuton under progressve type II censorng wth ro reovals For nstance Wu Chang (003) studed the estaton proble for Pareto I dstrbuton wth one paraeter based on progressve censorng wth unfor reovals They used the axu lkelhood (ML) ethod to obtan the estator of paraeter Wu (003) provded the nference for the estaton of the two-paraeter Pareto I 08

2 Ilhan USTA Hanef Gezer/ Journal of Scentfc esearch Developent () 05 Pages: 08-3 dstrbuton under progressve censorng wth unfor reovals The ML ethod was also used for the estaton procedure of paraeters An (008) consdered the estaton predcton probles usng the Bayesan approach for the Pareto I dstrbuton based on type-ii progressve censorng wth bnoal reovals Shanubhogue Jan (0) obtaned the unforly nu varance unbased estator for powers of the shape paraeter ts functons of Pareto I dstrbuton wth known scale paraeter under progressve type II censored data wth bnoal reovals However these authors were studed the proble for paraeter(s) estaton of Pareto I dstrbuton In ths paper we consder the estaton proble for not only two paraeters but also relablty characterstcs of Pareto I dstrbuton under progressve type II censored saple wth bnoal reovals The ML ethod s used to obtan the estators of the paraeters relablty characterstcs functons of Pareto-I dstrbuton Moreover Monte Carlo sulaton s perfored to copare the perforance of ML estates under progressvely type-ii censorng wth the dfferent ro schees The reer of ths paper s organzed as follows In Secton the propertes of Pareto-I dstrbuton relablty functon hazard-rate functon ean to syste falure are brefly presented The ML estators are derved under type II progressve censorng wth bnoal reovals n Sectons 3 4 The results of the sulaton study are presented n Secton 5 Secton 6 suarzes the conclusons of the study The odel The Pareto dstrbuton was frstly proposed Pareto (897) as a odel for the dstrbuton of ncoe but s now used as a odel n a wde range of felds such as nsurance busness econocs engneerng survval relablty Let the lfete of a unt X have a Pareto-I dstrbuton wth the shape scale paraeters The probablty densty functon (pdf) of Pareto -I dstrbuton s gven by ( ) f(x; ) x x 0 0 () are the shape scale paraeters respectvely The correspondng cuulatve dstrbuton functon (cdf) s gven by F(x; ) x x 0 0 () Soe relablty characterstcs of Pareto-I dstrbuton the relablty functon ( ( x) ) the h( x) hazard rate functon ( ) the ean te to syste falure (MTSF) are expressed respectvely as (x) F(x) x x 0 0 (3) f(x) h(x) x x 0 0 (x) (4) MTSF E[X] /( ) (5) E [ X ] s expected value of Pareto I dstrbuton More usefulness propertes of the Pareto I dstrbuton as a lfete odel were dscussed n Kus Kaya (007) Pars et al (00 Fua et al (0) 3 Estaton Let X X X be a progressvely type II censored saple fro Pareto-I dstrbuton n s pre-fxed before the test For progressve type II censorng wth a pre-deterned nuber of reovals ( r r) the condtonal lkelhood functon can be wrtten as(cohen963): L( ;x r) A f(x)( F(x) (6) A n(n r )(n r ) (7) Substtutng Eqs () () nto Eq(6) the lkelhood functon s derved as ( ) r L( ;x r) A(r) x( x) (8) Suppose that an ndvdual unt beng reoved th fro lfe test at the falure s ndependent of the others but wth sae probablty p Then the nuber of unts reoved at each falure te follows a bnoal dstrbuton wth paraeters p Thus n r nr P( r) p( p) 0 r n r n r l l (9) n rl r l l l P( r r r) n r p( p) (0) 0 r n r l l Moreover we presue that s ndependent of X for all Accordngly the lkelhood functon can be expressed as L( p;xr) L( ;x r)p( r) () P( r) s the jont probablty dstrbuton gven by P( r) P( r)p( r / r) P( r / r r) 3 3 P( r / r r) (n )! P( r) p( p) (n r) r! () r( )(n )( )r (3) 09

3 Ilhan USTA Hanef Gezer/ Journal of Scentfc esearch Developent () 05 Pages: 08-3 Now usng Eqs (8) () (3) we can wrte the full lkelhood functon as L( p;xr) BL( )L(p) (4) ( ) L( ) x( x) r r( )(n )( )r (5) p L(p) p( p)( p) (6) A(n )! B (n r) r! (7) It s clear that B does not depend on the paraeters L s ndependent of ; paraeters 4 Maxu lkelhood estaton In ths secton we obtan the axu lkelhood estators (MLEs) of the paraeters p (x) the relablty characterstcs h(x) MTSF based on progressvely type II censorng data wth bnoal reovals As entoned before L gven n Eq (5) does not nclude p Therefore the MLEs of can be derved by axzng Eq (5) drectly Snce ths lkelhood functon s an ncreasng functon of therefore the MLE of s gven by ˆ le X (8) The MLE of can be obtaned by solvng d l o g L( ) ˆ d ˆ le 0 Then t s found as (r )log(x) nlog() ˆ (9) L Slarly n Eq (6) does not nvolve p Therefore the MLE of can be derved by axzng Eq (6) drectly Solvng d l o g L( p) d p wth respect to p the MLE for p s gven by 0 r ˆp le r( )(n )( )r (0) Addtonally Once the MLEs of are obtaned as ˆ ˆ By usng nvarance property of the MLEs the MLEs of (x) h(x) MTSF are derved respectvely as ˆ ˆ ˆ(x) ˆ x x 0 () ˆ h(x) ˆ ˆ x x 0 () MTSF ˆ ˆ ˆ /( ˆ ) 5 Sulaton study (3) In ths secton a Monte Carlo sulaton study s conducted to copare the perforance of the ML estates derved n the prevous sectons for progressvely type II censorng wth the dfferent ro schees 5 Algorth for generatng progressvely type II censored saples fro Pareto I dstrbuton By usng the algorth gven n Balakrshnan Shu (995) frstly generate the nuber of progressve censorng r wth bnoal reovals between step step 6 Then the followng steps are used generate progressvely type II censored order statstcs fro Pareto-I dstrbuton The steps are: Specfy the value of n Specfy the value of 3 Specfy the values of paraeters p 4 Generate a ro nuber r fro B n o ( n p) r 5 Generate a ro nuber B n o ( n r l p) 6 Set r r l for each 3 fro accordng to the followng relaton n r n r k 0 k k k 0 o w 7 Generate ndependent unfor U( 0 ) W ro varables W W 8 For gven values of the progressve schee ( r r r) set V W for ( r) l l U V V V 9 Set then U U U are ranked progressve censored saple of sze fro U(0) wth bnoal reovals 0Fnally for gven values of paraeters U X F( U) we set Then ( X X X) s the progressve type II censored saple fro the Pareto-I dstrbuton wth bnoal reovals 5 Sulaton desgn The desgn of sulaton study s outlned n the followng ) Gven the values of paraeters p the sson te x saple sze n nuber of falures generate a progressvely type II censored saple of sze n wth falures usng the algorth gven n Secton 5 For each value of n= the values of are taken as (/n) 00 = 40% 60% 80% the value of p s consdered as / 0

4 Ilhan USTA Hanef Gezer/ Journal of Scentfc esearch Developent () 05 Pages: 08-3 ) Copute the ML estates of paraeters p relablty characterstcs (x) h(x) MTSF by usng MLEs gven n Secton 4 ) Gven =3 = p=03 07 x=035 =4 =3 p=03 07 x=3039 repeat frst second steps N tes N s taken as 0000 v) Copare the estates of paraeters relablty characterstcs wth the true values of the by coputng the bas MSE defned as (Karshna Kuar 0): N Bas()() ˆ N (4) N MSE()(()()) ˆ ˆ N (5) ˆ() = N are N estates of () N s the nuber of sulaton replcatons It s note that all calculatons are perfored on the MATLAB 53 Sulaton results The obtaned results fro the sulaton study are reported as the bas MSE of the ML estators n Tables - for =3 = =4 =3 under progressvely type II censorng wth bnoal reovals accordng to p=03 07 The further results are suarzed n Tables 3-4 (x) 095 h(x) 474 for x=035 MTSF 3 x=3039 (x) 095 h(x) 36 MTSF 4 based on progressvely type II censorng wth bnoal reovals accordng to p=03 07 Table : Sulaton results for =3 = p=03 07 p 03 ˆ ˆ ˆp n Bas MSE Bas MSE Bas MSE p 07 ˆ ˆ ˆp n Bas MSE Bas MSE Bas MSE Table : Sulaton results for =4 =3 p=03 07 p 03 ˆ ˆ ˆp n Bas MSE Bas MSE Bas MSE p 07 ˆ ˆ ˆp n Bas MSE Bas MSE Bas MSE It can be seen fro Tables that for the bases MSEs decrease as long as the saple sze n the falure nforaton ncrease under dfferent choces of censorng ro schees accordng to p=03 07 Furtherore for p=03 the bas MSE values of paraeters are slar to the results obtaned for p=07 On the other h for p the bases MSEs ncrease as n ncrease Table 3: Sulaton results for x=035 (x) 095 h(x) 474 MTSF 3 p=03 07 p 03 ˆ(t) ĥ(t) MTSF ˆ n Bas MSE Bas MSE Bas MSE p 07 ˆ(t) ĥ(t) MTSF ˆ n Bas MSE Bas MSE Bas MSE Table 4: Sulaton results for x=3039 h(x) 36 MTSF 4 p=03 07 p 03 ˆ(t) ĥ(t) ( x) ˆ MTSF n Bas MSE Bas MSE Bas MSE

5 Ilhan USTA Hanef Gezer/ Journal of Scentfc esearch Developent () 05 Pages: p 07 ˆ(t) ĥ(t) MTSF ˆ n Bas MSE Bas MSE Bas MSE The results wth regard to the bas MSE n Tables 3 4 then pont out that for the relablty characterstcs; ( x) h( x) M T S F the bases MSEs decrease once n ncrease under dfferent choces of censorng ro schees accordng to p=03 07 Addtonally consderng the bases MSEs of the relablty characterstcs for p=03 07 the slar results are observed for both p values 6 Concluson In ths paper we consder the estaton probles of not only two paraeters but also relablty characterstcs of two paraeters Pareto I dstrbuton under progressve type II censored saple wth bnoal reovals Moreover Monte Carlo sulaton s conducted to copare the perforance of axu estates under progressvely type-ii censorng wth the dfferent ro schees As a consequence the overall sulaton results reveal that () the MLEs of shape paraeter are very good n ters of the bas MSE for all censorng schees () the MLEs are strongly suggested to estate scale paraeter wth regard to the bas MSE for all censorng schees () the pont estates wth the ML ethod of the paraeter p are so good n ters of the bas MSE for all censorng schees but the estates are worse as n ncrease (v) the MLEs of relablty characterstcs relablty functon hazard rate functon ean te syste to falure gve the satsfactory results n ters of the bas MSE for all censorng schees eferences A Shanubhogue N Jan (0) Mnu varance unbased estaton n the Pareto dstrbuton of frst knd under progressve Type II censored data wth bnoal reovals ProbStat Foru vol5 pp-3 AC Cohen (963) Progressvely censored saples n the lfe testng Technoetrcs vol 5 pp C Kus MF Kaya (007) Estaton for the paraeters of the Pareto dstrbuton under progressve censorng Coun Stat Theory vol 36(7) pp H Karshna K Kuar (0) elablty estaton n Lndley dstrbuton wth progressvely type II rght censored saple Math Coput Sulat vol 8 pp 8-94 HK Yuen S K Tse (996) Paraeters estaton for Webull dstrbuted lfetes under progressve censorng wth ro reovals J Stat Coput S vol 55() pp 57-7 J Fua A Xub Y Tanga (0) Objectve Bayesan analyss of Pareto dstrbuton under progressve Type-II censorng Stat Probabl Lett vol 8 pp N Balakrshnan Aggarwala (000) Progressve censorng: Theory Methods Applcaton Brkhauser Boston N Balakrshnan A Shu (995) A sple sulaton algorth for generatng progressvely type- censored saple A Stat vol 49() pp 9-30 NL Johnson S Kotz N Balakrshnan (994) Contnuous Unvarare Dstrbutons Vol nd ed John Wley & Sons New York Az B Fash FA Sarkhanoglu ( 04) Statstcal nference for generalzed Pareto dstrbuton based on progressve type-ii censored data wth ro reovals Int J Sc World vol() pp -9 S Dey T Dey (04) Statstcal Inference for the aylegh dstrbuton under progressvely Type-II censorng wth bnoal reoval App Math Model vol 38(3) pp S Pars M Ganjal NS Farspour (00) Sultaneous confdence ntervals for the paraeters of Pareto dstrbuton under progressve censorng Coun Stat Theory vol 39 pp SJ Wu (003) Estaton for the two -paraeter Pareto dstrbuton under progressve censorng wth unfor reovals J Stat Coput S vol 73() pp 5-34 SJ Wu YJ Chen CT Chang (007) St atstcal nference based on progressvely censored saples wth ro reovals fro the Burr type XII dstrbuton J Stat Coput S vol 77 pp 9-7 SJ Wu CT Chang (003) Inference n the pareto dstrbuton based on progressve type censorng wth ro reovals J Appl Stat vol 30 pp 63-7

6 Ilhan USTA Hanef Gezer/ Journal of Scentfc esearch Developent () 05 Pages: 08-3 V Pareto 897 Cours d econoe Poltque Vol II F ouge Lausanne WA Yan YM Sh BW Song ZY Zao (0) Statstcal analyss of generalzed exponental dstrbuton under progressve censorng wth bnoal reovals J Syst Eng Electron Vol (4) pp ZH An (008) Bayesan nference for the Pareto lfete odel under progressve censorng wth bnoal reovals J Appl Stat vol 35 pp