ESTIMATION OF PARAMETERS OF THE MARSHALL-OLKIN EXTENDED LOG-LOGISTIC DISTRIBUTION FROM PROGRESSIVELY CENSORED SAMPLES

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1 ESTIMATION OF PARAMETERS OF THE MARSHALL-OLKIN EXTENDED LOG-LOGISTIC DISTRIBUTION FROM PROGRESSIVELY CENSORED SAMPLES Mahmoud Rad Mahmoud Isttute of Statstcs, Caro Uversty Suza Mahmoud Mohammed Faculty of scece, Helwa Uversty ABSTRACT: I ths paper, a ew class of Log-Logstc dstrbuto usg Marshall-Olk trasformato s troduced. Its characterzato ad statstcal propertes are obtaed. Ad The estmato of parameters of dstrbuto uder progressve cesorg s vestgated, maxmum lkelhood estmators of the ukow parameters are obtaed usg statstcal software (Mathematca, MLE performs for dfferet samplg schemes wth dfferet sample szes s observed, ad the asymptotc varace covarace matrx s computed also. KEYWORDS: Exteded Log-Logstc dstrbuto progressve type II cesorg Maxmum lkelhood estmators Asymptotc varace covarace matrx. INTRODUCTION The Log-Logstc dstrbuto (kow as the Fsk dstrbuto ecoomcs s the probablty dstrbuto of a radom varable whose logarthm has a logstc dstrbuto.it has attracted a wde applcablty survval ad relablty over the last few decades, partcularly for evets whose rate creases tally ad decrease later, for example mortalty from cacer followg dagoss or treatmet, see [10]. It has also bee used hydrology to model stream flow ad precptato, see [14] ad [6], for modelg good frequecy, see [1]. [4] Used ths model to descrbe the thermal actvato of Clostrdum botulum 13B at temperatures below 11:1 _C. Furthermore, t s appled ecoomcs as a smple model of the dstrbuto of wealth or come, see [8].The cumulatve dstrbuto fucto F(x ad desty fucto f(x of Log-Logstc dstrbuto are gve by F(x = x β α β + x β, x > o (1-1 Ad f(x = αβ β x β 1 (x β + α β, x > (1-1

2 Where α > 0 s the scale parameter ad s also the meda of the dstrbuto, β > 0 s the shape parameter.whe β > 1, the Log-Logstc dstrbuto s umodal. It s smlar shape to the logormal dstrbuto but has heaver tals. Its cumulatve dstrbuto fucto ca be wrtte closed form, ulke that of the log-ormal. [7] Used the rato of maxmzed lkelhood to cosder the dscrmato procedure betwee the two dstrbuto fuctos. O the other had, [1] troduced a ew famly of survval fuctos whch s obtaed by addg a ew parameter γ > 0 to a exstg dstrbuto. The ew parameter wll result flexblty the dstrbuto. Let F (x = 1 F (x be the survval fucto of a radom varable X. The G (x = γf(x 1 (1 γf (x (1-3 Is a proper survval fucto. G (xs called Marshall-Olk famly of dstrbutos. If γ= 1, we have that G = F. The desty fucto correspodg to (3 s gve by g (x = Ad the hazard rate fucto s gve by γf(x (1 (1 γf (x (1-4 h (x = h F (x 1 (1 γf (x (1-5 Where h F (x s the hazard rate fucto of the orgal model wth dstrbuto F.By usg the Marshall-Olk trasformato (1-3, several researchers have cosdered varous dstrbuto extesos the last few years. [1] geeralzed the expoetal ad Webull dstrbutos usg ths techque. [] troduced Marshall-Olk exteded sem Pareto model for Pareto type III ad establshed ts geometrc extreme stablty. Sem-Webull dstrbuto ad geeralzed Webull dstrbutos are studed by [3]. Ght ay et al. (005 coducted a detaled study of Marshall-Olk Webull dstrbuto, that ca be obtaed as a compoud dstrbuto mxg wth expoetal dstrbuto, ad apply t to model cesored data. Marshall-Olk Exteded Lomax Dstrbuto was troduced by [9]. [11] Ivestgated Marshall-Olk q-webull dstrbuto ad ts max-m processes.i ths paper, we use the Marshall-Olk trasformato to defe a ew model, socalled the Marshall-Olk Log-Logstc dstrbuto, whch geeralzes the Log-Logstc model. We am to reveal some statstcal propertes of the proposed model ad estmato of ts parameters.

3 Desty fucto Let t follows Log-Logstc dstrbuto, the ts survval fucto s gve by F (t = 1 t β α β + t β, α β α β + t β Substtutg t (1-3 we obta a survval fucto of Marshall-Olk Log-Logstc dstrbuto deoted by αβ γ G (t = t β + α β γ α, β, γ, t > 0 (1-6 The correspodg desty fucto s gve by αβ βγt β 1 g(t = (t β + α β γ α, β, γ, t > (1-7 If γ = 1, we obta the Log-Logstc dstrbuto wth parameter α, β>0.ths dstrbuto cotas the Log-Logstc dstrbuto as a partcular case.the followg theorem gves some codtos uder whch the desty fucto(1-7 s decreasg or umodal. Maxmum Lkelhood Estmato I case complete data Cosder a radom sample cosstg of observatos from the log logstc dstrbuto ad the lkelhood fucto of ths sample s L(t 1, t,, t ; β, α, γ = f(t (1-8 L(t, β, α, γ = (βγαβ t (t β + α β γ β 1 (1-9 O takg logarthms of (1-9 we get, 3

4 LL = (log(βγ + (log[α β ] + (β 1 log[t ] log [t β + (α β γ] (1-10 Dfferetatg (1-10 wth respect to β, αad γ ad equatg to zero are: LL β = β + log[α] + log[t ] αβ γ log[α] + log[t ] t α β β γt β (1-11 LL α = β α αβ 1 βγ α β β γt LL γ = γ αβ α β β γt (1.1 (1-13 The maxmum lkelhood estmate (MLE ( ^, ^, ^ of (,, s obtaed by solvg the olear lkelhood equatos (1-11, (1-1 ad (1-13. These equatos caot be solved aalytcally ad popular software ca be used to solve these equatos umercally. I case progressve type two rght cesored data. Let t1, t, tbe depedet ad detcally dstrbuted radom lfetmes of tems. A type ll progressvely rght cesored sample may be obtaed the followg way: the falure tme of the frst r1 tems to fal are ot observed; at the tme of the (r+1st falure, deoted wth tr+1, Rr+1 of the remag uts are wthdraw from the test radomly. At the tme of ext falure, deoted wth t, Rr+ survvg tems are removed at radom from the remag tems, ad so o. At the tme of the mth falure, where m s a predetermed umber, all the remag Rr+m tems are cesored. 4

5 Therefore, a progressvely type-ll rght cesorg scheme s specfed by a sequece tegers, m ad r1,,rm-1 wth the costrats - m- r1 -.rm-1 0, m 1. The lkelhood fucto to be maxmzed whe geeral progressve type- II cesored sample based o depedet from F (t s ( * r R r 1,, 1, (1-14 r1 L c F t f t F t Where,, t t... t, r1 r m! , r! r 1! * c R r R R m The lkelhood fucto gve Eq. (1.14 ca be expressed as: r 1 r 1 m 1 L t ;,, r m * t r r 1 c t r 1 1 ( t t ( ( t ( t R 1 (1-15 r m r 1 * t r1 ( t t r 1 1 t ( L t ;,, c ( ( t ( t The log-lkelhood fucto, deoted by l t,,,, takes the form R (1-16 5

6 l t t r 1 r 1 t ;,, costat m r log r log m r m r 1 1 ( m r log[ ] ( 1 log[ t ] log t R log 1 t (1-17 The three equatos obtaed by dfferetatg (1.17 ad equatg to zero are m r r log[ ] ( m rlog[ ] t 1 r 1 log[ ] log[ t ] t m r m r log[ t ] 1 1 t log[ ] ( log[ ] log[ t ] t R( t( ( ( t t 0, (1-18 m r ( m r r t t r t t R( t ( ( ( 0, (1-19 R( t ( m r m r r t t tr 1 1 t 1 ( ( 0, (1-0 6

7 The maxmum lkelhood estmate (MLE ˆ, ˆ, ˆ of,, s obtaed by solvg the olear lkelhood equatos (1-18, (1-19 ad (1-0. These equatos caot be solved aalytcally ad popular software ca be used to solve the equatos umercally. 4. Fsher formato matrx I ( l l l E E E I ( l l l E E E. l l l E E E The elemets of the sample formato matrx, for progressvely type II cesored wll be m r r C ( A D ( t B Q ( t F ( t r 1 1 ( G ( t G ( t 1 ( A ( A D ( t A ( A D ( t H R ( G ( t ( R ( ( ( A D t G ( t ( G ( t G ( t ( G ( t ( B A ( A D ( t ( A D ( t ( B Q ( t R( G ( t ( 3 G ( t G ( t G ( t G ( t (1-1 7

8 1 1 1 m r r N r r q q ( A D ( t ( ( F ( t F ( t F ( t F ( t F ( t 1 r 1 r 1 r 1 1 A ( A D ( t 1 A ( A D ( t R ( R ( ( G ( t ( G ( t G ( t ( G ( t ( 1 N ( q ( A D ( t ( B Q ( t RG ( t ( 3 ( Gt ( ( G ( t ( G ( t ( G ( t (1- r c r a a ( A D ( t ( ( F ( t F ( t G ( t ( G ( t 1 r 1 r 1 1 ( a ( A D ( t ( A D ( t 3 c R( G ( t ( ( G ( t G ( t ( G ( t ( G ( t A ( A D ( t A ( A D ( t R ( R ( ( ( G ( t ( G ( t G ( t ( G ( t (1-3 m r ( m r r b r d b d ( ( F ( t F ( t ( G ( t G ( t r 1 r R ( f g f g ( R ( G ( t ( ( ( ( ( ( ( 1 h b j d R ( G ( t ( 3 ( G ( t ( G ( t ( G ( t ( G ( t (1-4 1 r k r ( ( F ( t F ( t r1 r1 1 1 J 3k R ( G ( t ( 3 ( G ( t ( G ( t G ( t 8

9 1 ( 1 J 3k f g R 3 R ( G ( t ( ( ( ( ( ( ( ( ( ( f g R ( G ( t ( ( G ( t G ( t (1-5 m r m r r ( ( F ( t ( G ( t r R( G ( t ( 3 ( G ( t ( G ( t R ( R ( ( ( ( G ( t G ( t ( G ( t G ( t (1-6 Where, A log[ ] B log[ ] C log[ ] D ( t r 1 log[ t r 1] t r 1 Q ( t log[ t ] t F ( t t r 1 r 1 r 1 r 1 r 1 G ( t t H log[ ] J r1 r N log[ ] a log[ ] b c f 1 g 1 log[ ] d ( 1 h j (1 k q log[ ], the maxmum lkelhood estmators ( ^ ^ ^ For 0 ad ˆ, ˆ, ˆ s asymptotcally to ormal wth mea vector 0 ad varace- 1 covarace matrx I.,, of (,, are cosstet estmators, 9

10 Asymptotc varace covarace matrx l l l V ˆ Cov ˆ, ˆ Cov ˆ, ˆ va ˆ r ˆ, ˆ, ˆ Cov ˆ, ˆ V ˆ Cov ˆ, ˆ l l l Cov ˆ, ˆ Cov ˆ, ˆ V ˆ l l l A Numercal example ad data aalyss ˆ ˆ ˆ ˆ,,, ˆ, ˆ ˆ, ˆ, ˆ ˆ, ˆ, ˆ ˆ, ˆ, ˆ ˆ, ˆ, ˆ ˆ, ˆ, ˆ ˆ, ˆ, ˆ ˆ, ˆ, ˆ I ths secto, we preset some results from geeratg progressve cesored type II data from exteded logstc dstrbuto by usg Mathematca to observe how the MLEs perform for dfferet samplg schemes ad for dfferet sample szes, The asymptotc varace covarace matrx s computed also. We have take =50, 100, 150 ad00, m=0, 30,50ad70 Dfferet samplg schemes. Dfferet values of the parameters are takeβ=3, α=4 ad γ=, β=1.5, α=3 ad γ=1 each case, we have calculated the MLEs. We replcate the process 1000 tmes ad compute the average bases ad stadard devatos of the dfferet estmates. The scheme (, m, r,. Scheme [1] =50, m=0, r=0, R1= R=1, R3=, R4=1, R5=3, R6=, R7=R8=1, R9=3, R10= R11=1, R1=3, R13=, R14=0, R15= R16= R17=1, R18= R19=, R0=1 Scheme[]=100, m=30, r=0, R1= R=3, R3=6, R4= R5=1, R6=4,R7=R8=R9=R10=3R11=1,R1=,R13=1,R14=R15=R16=R17=, R18=6, R19=5, R0= R1=R=0, R3=1, R4=, R5=3, R6=4, R7= R8=1, R9=4, R30=1. 10

11 scheme[3] =150, m=50, r=0, R1=0, R=1, R3=0, R4= R5= R6=, R7=4, R8=,R9=3,R10=1, R11=3,R1=5,R13=4, R14=5, R15=0,R16=8,R17=4,R18=0,R19=1,R0=R1=,R=0,R3=3,R4=, R5=4,R6=R7=1,R8=,R9=0,R30=1,R31=,R3=4,R33=1,R34=3, R35=0,R36=1,R37=R38=0,R39=,R40=3,R41=,R4=6,R43=0,R44=1, R45=, R46=0, R47=R48=1, R49=4, R50=. Scheme[4] =00, m=70, r=0, R1=0, R=4, R3=, R4=0,R5=3,R6=R7=,R8=R9=1,R10=4,R11=R1=,R13=4,R14=R15=1 R16=R17=0,R18=,R19=R0=1,R1=,R=0,R3=R4=,R5=1, R6=0R7=R8=,R9=4, R30=R31=1, R3=5,R33=R34=1, R35=, R36=3,R37=0,R38=,R39=3,R40=,R41=1,R4=3,R43=4,R44=1, R45=,R46=3,R47=,R48=0,R51=R5=1,R53=R54=0,R55=,R56=0, R57=R58=1, R59=4, R60=R61=1, R6=, R63=R64=0,R65=3,R66= R67=, R68=6, R69=5, R70=. Table 1: Estmato of parameters whe β=3, α=4, γ=. Scheme ˆ ˆ ˆ va r ˆ, ˆ, ˆ (50, 0, (100,30,

12 Table :Estmato of parameters usg Scheme[3] Β Α γ ˆ ˆ ˆ va r ˆ, ˆ, ˆ Table 3:Estmato of parameters usg Scheme[4] Β Α γ ˆ ˆ ˆ va r ˆ, ˆ, ˆ REFERENCES [1] MI Ahmad, CD Sclar, ad A. Werrtty, Log-logstc ood frequecy Aalyss, Joural of Hydrology 98 (1988, o. 3, 05{4. [] T. Alce ad KK Jose, Marshall-olkpareto processes, Far East Joural of Theoretcal Statstcs 9 (003, o., 117{13. [3], Marshall{olk sem-webull mcato processes, Recet Adv Stat Theory Appl I (005, 6{17. [4] WA Aderso, PJ McClure, AC Bard-Parker, ad MB Cole, theap- plcato of a log-logstc model to descrbe the thermal actvato of Clostrdumbotulum 13b at temperatures below 11.1 c, Joural of Appled Mcrobology 80 (1996, o. 3, 83{90. 1

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