ISSN 76-338 J P S S A comprehesve joural of probablty ad statstcs for theorsts methodologsts practtoers teachers ad others JOURNAL OF PROBABILITY AND STATISTICAL SCIENCE Volume 8 Number August 00
Joural of Probablty ad Statstcal Scece 8( 99-3 Aug 00 Asymptotc Propertes of MLE s for Dstrbutos Geerated from a -Parameter ebull Dstrbuto by a Geeralzed Log-Logstc Trasformato James U Gleato ad M Mahbubur Rahma Uversty of North Florda ABSTRACT A geeralzed log-logstc (gll famly of lfetme dstrbutos s oe whch ay par of dstrbutos are related through a gll trasformato for some (oegatve value of the trasformato parameter (the odds fucto of the secod dstrbuto s the -th power of the odds fucto of the frst dstrbuto e cosder gll famles geerated from a -parameter ebull dstrbuto It s show that the Maxmum Lkelhood Estmators (MLE s for the parameters of the geerated or composte dstrbuto have the propertes of strog cosstecy ad asymptotc ormalty ad effcecy Keywords Geeralzed log-logstc famles; -parameter ebull dstrbuto Itroducto Gleato ad Lych [] examed the relablty of a partcular physcal system a homogeeous budle (varyg cross-sectoal areas of brttle elastc fbers subjected to tesle stress It was show that for equal load sharg the Maxmum Etropy Prcple mples that the fber survval dstrbutos are related to each other through a geeralzed log-logstc (gll trasformato defed below More geerally ths type of relatoshp betwee the survval dstrbutos for the compoets of a system wll hold for systems havg the followg characterstcs: there s a parallel system of compoets each subjected equally to the same load or put; each compoet s respose s proportoal to the compo- et load up to the pot of compoet falure; there are dffereces amog the costats of proportoalty of the compoet resposes; ad v compoet resposes are codtoally - Receved Jue 009 revsed December 009 fal form Jauary 00 James U Gleato ad M Mahbubur Rahma are afflated wth the Departmet of Mathematcs ad Statstcs at the Uversty of North Florda Jacksovlle FL 34 USA; emal address of James U Gleato: jgleato@ufedu 00 Susa Rvers Cultural Isttute Hschu Tawa Republc of Cha ISSN 76-338
00 JPSS Vol 8 No August 00 pp 99-3 depedet gve the system load I a later paper Gleato ad Lych [3] dscussed propertes of lfetme dstrbutos belogg to famles geerated by a geeralzed log-logstc trasformato: [ ( x] ( x l ( x [ ( x] [ ( x] for x > 0 relatg two lfetme dstrbuto fuctos ad Here the dstrbuto may also be a fucto of a m-dmesoal ( ths paper m = o-egatve parameter vector φ ( m The trasformato s defed for each > 0 by l ( u [ ( u/ u ] for 0 u where u u It was show that the trasformato parttos the set of all lfetme dstrbutos so that wth each equvalece class every par of dstrbuto fuctos are related through a gll trasformato I each equvalece class oe arbtrarly chose dstrbuto called the embedded dstrbuto may be cosdered the geerator of the class The other dstrbutos the class are called composte dstrbutos A equvalece class may be characterzed oe of two ways: a ay par of dstrbuto fuctos the class are related by a gll trasformato or b the log-odds rates d d [l( / ] [l( / ] dx dx for ay two dstrbutos a equvalece class are the same apart from a multplcatve costat whch s the trasformato parameter It was show [3] that: ether every member of a equvalece class has a momet geeratg fucto or else oe does; every member of a equvalece class has exactly the same umber of momets; each equvalece class s learly ordered accordg to the trasformato parameter wth larger values of ths parameter correspodg to smaller dsperso of the dstrbuto about the commo class meda; ad v wth a equvalece class the Kullback-Lebler formato s a creasg fucto of the rato of the trasformato parameters I ths paper we establsh suffcet codtos for the asymptotc propertes of the MLE s of the parameters of dstrbutos a equvalece class whch the embedded dstrbuto s a two-parameter ebull dstrbuto I secto certa propertes of the ebull dstrbuto are preseted for later use ad some results are establshed for use provg the lemmas secto 3 I secto 3 we prove a theorem that shows that for a certa rages of values of the trasformato parameter ad the ebull parameters the lkelhood equatos have a sequece of solutos satsfyg codtos of strog cosstecy ad asymptotc ormalty ad effcecy
Asymptotc Propertes of MLE s for Dstrbutos James U Gleato ad M Mahbubur Rahma 0 The Two-Parameter ebull Dstrbuto wth cdf e wll be cocered wth famles geerated by a two-parameter ebull dstrbuto ( x ( x; e for 0 < x < + wth odds fucto ( x ( x ( x Hereafter for coveece we wll wrte the cdf as ad the embedded desty as wth the parameter vector mplct here there s o chace for cofuso the argumet x wll also be omtted It s also kow that all momets of the two-parameter ebull dstrbuto exst [5] the th momet beg ( E X ( / (A expectato operator wthout a subscrpt wll deote expectato wth respect to the -parameter ebull measure I the followg sectos partal dervatves wth respect to the parameters or wth respect to wll be deoted by subscrpts paretheses such as ( Ths should cause o cofuso sce all of the partal dervatves are cotuous o (0 e wll eed the frst- ad secod-partal dervatves of both ad wth respect to the parameters lsted below (The fuctos d through d 8 are gve explctly the appedx: ( (/ d ( ( / d ( (/ d 5 ( (/ d3 ( (/ d4 3 (/ d ( 6 ( (/ d7 ( (/ d8 ad d ( ( (/ 9 ( ( / d0 ( (/ d (/ d ( (/ d 4 ( ( / d ( 3 3 ( ( / d5 ( (/ d6 ad (/ d ( (/ d 7 ( 8 The asymptotc propertes of the MLE s of the parameters of the three-parameter ebull dstrbuto wth double-cesorg have bee vestgated by Harter ad Moore [4] It was foud that the regularty codtos [6] ecessary for the MLE s to be strogly cosstet asymptotcally ormal ad asymptotcally effcet are satsfed f ad oly f at least oe of the followg codtos holds:
0 JPSS Vol 8 No August 00 pp 99-3 a the shape parameter s greater tha b the locato parameter s kow c a proporto of the sample s cesored from below Sce we are assumg that the geeratg dstrbuto for the equvalece class s a two-parameter ebull dstrbuto wthout cesorg the locato parameter s 0 ad the regularty codtos hold for ay o-egatve values of ad of the scale parameter The lkelhood equatos ad asymptotc stadard errors for the MLE s for the parameters of the two-parameter ebull dstrbuto may be foud Johso Kotz ad Balakrsha [5] The followg two lemmas wll be used the ext secto the proof of Lemma 3 Lemma For a two-parameter ebull dstrbuto wth > ad for ay postve tegers ad m let 8 The the fucto m G ( x d m s bouded absolute value by a tegrable fucto Proof The fucto s a sum of terms of four types Each term s a product of ( m wth G m oe of the followg: a a costat b a product of a costat ad a power of j j x o less tha c a product of a costat ad a postve power of l( x or d a product of a costat a power of x o less tha ad a postve power of l(x The frst type of term s tegrable due to the fact that s bouded The boudedess of together wth the exstece of all momets of the -parameter ebull dstrbuto mply that the secod type of term s tegrable Due to the boudedess of the thrd type of term s bouded absolute value by a fucto of the form C [l( 3 x ]q for some postve costat C3 ad postve teger q Such terms are tegrable sce all dervatves of the gamma fucto exst o (0 + Due to the boudedess of the fourth type of term s bouded absolute value by a fucto of the form C4 x [l(x] q for some postve costat C 4 costat o less tha ad postve teger q Such fuctos are tegrable sce all dervatves of the gamma fucto exst o (0 + Lemma For a two-parameter ebull dstrbuto For > 0 [l( ] s bouded absolute value by a tegrable fucto for ay postve teger For > 0 ( s bouded absolute value by a tegrable fucto For > ( s bouded absolute value by a tegrable fucto 3 v For > 0 ( x l( s bouded absolute value by a tegrable fucto 3 v For > 0 {l[( x ]} l( s bouded absolute value by a tegrable fucto Proof Let (l / the meda of the dstrbuto For x [l] 0 O ths terval we also have by mootocty of the log-odds fucto
Asymptotc Propertes of MLE s for Dstrbutos James U Gleato ad M Mahbubur Rahma 03 x ( x x x e [l( e ] x e But ( ( ( x gx x e s tegrable o the terval ( + sce For x we have gxdx EX! 0 l dx l y dy! 0 0 Hece [l( ] s bouded absolute value by a tegrable fucto ( x ( x e have ( x e ( e 0 for all x > 0 For x we have ( x x e whch s tegrable o ( + sce t s proportoal to a ( fucto whch whe tegrated over (0 + yelds x whch s tegrable o (0 e also have ( x ( x ( l 0 x x e e EX ( For x < ( ( x for x > / or ( 0 x max / ( x e whch s tegrable o (max{ / } sce t s proportoal to a fucto whch whe tegrated over (0 + yelds EX ( For x m{ / } we have otherwse For whch s tegrable o m / v For x x l 0 for > x 0l x The 3 4 x dx x dx ( l 3 sce all momets exst for the -parameter ebull dstrbuto For x 0 ( x 3 so that x l l( whch s bouded by a tegrable fucto by part ( above 3 3 3 v For x 0 { l[( x ]} ( x whch mples that {l[( x ]} l dx s fte by the precedg argumet For 3 x 0 ( x l ( x 3 l l 3 x l ( x whch s bouded by a tegrable fucto by Lemma
04 JPSS Vol 8 No August 00 pp 99-3 3 Asymptotc Propertes of MLE s for Dstrbutos Geerated by GLL Trasformatos of a -Parameter ebull Dstrbuto Let the embedded dstrbuto be a two-parameter ebull dstrbuto The the pdf for the composte dstrbuto s l [ ] (3 Throughout we wll use prmes to deote dfferetato wth respect to ether x or wth respect to u Sce all momets of the embedded dstrbuto exst the all momets of the composte dstrbuto exst [3] Ths fact wll be used the proof of Lemma 33 The ext three lemmas show that the regularty codtos hold for the gll-trasformed two-parameter ebull dstrbuto (hereafter called the GLL dstrbuto for certa rages of values of the parameters The frst lemma proves that each frst partal dervatve of the atural log of the composte desty wth respect to each parameter compoet has for each x a Taylor expaso the parameters [6 7] Lemma 3 Let l be a dstrbuto related to through a gll trasformato The all frst- secod- ad thrd-order partal dervatves of l( wth respect to the parameter compoets exst for all x 0 Proof Sce s oe-to-oe ad has a desty we may let u ad exame the exstece of the partal dervatves of l( wth respect to terms of the partal dervatves of l[ l ( u] l l( uu l( u u wth respect to If we let l( u l ( u we have l ( u l ( u ( ad l[ l ( u] (/ [ l ( u]l( u/ u (3a l[ l ( u] (/ l ( u l ( u[l( u/ u] ( (3b 3 3 l[ l ( u] ( / l ( u l ( u[ l ( u][l( u/ u] (3c It s clear that for x > 0 e u(0 each of the dervatves (3 exsts for all > 0 Thus all frst partal dervatves of l( wth respect to the trasformato parameter exst for all > 0 e also eed the followg dervatves of l ( u: l( u l ( u s( u( uu (33a l ( u l ( u g( u( uu (33b l ( u l( u{[ 3u l ( u] g( u uug( u}( uu (33c ( 3
Asymptotc Propertes of MLE s for Dstrbutos James U Gleato ad M Mahbubur Rahma 05 where ad g u u u l u l u u l u uul u (33d 6 4 6 ( su u l ( u (33e of e have l( l[ l ] The usg (33 we fd that the frst partal dervatve l( wth respect to a parameter of the embedded dstrbuto s gve by [l] s l( (34a Now the embedded dstrbuto satsfes the regularty codtos I partcular ( exsts for x > 0 But x x due to cotuty of the partal dervatves Ths mples that exsts for x > 0 Sce all frst partal dervatves of l( wth respect to ts parameter compoets exst the t s clear that for x > 0 all frst partal dervatves of l ( wth respect to the embedded parameter compoets also exst for all > 0 The secod partal dervatves wth respect to parameters of the embedded dstrbuto are gve by (where the argumets of g ad s have bee omtted [l] ( [l] j gs s j j ( j (34b Usg the same ratoale as gve for the exstece of the prevous dervatves we see that for x > 0 all secod-order partal dervatves of l( wth respect to the embedded parameter compoets exst for all > 0 The thrd partal dervatves wth respect to parameters of the embedded dstrbuto are gve by (the argumet of the trasformato operator s mplct v { } [l( ] ( l / l [ l l /( l ] ( l / l l( l ( j k ( ( j ( k [l( l ] [ ] ( ( k ( j ( j k ( k ( j [l( l] [l] (34c ( ( j j k ( k j Usg (33 (34b ad the fact that all frst- secod- ad thrd-order partal dervatves of the embedded dstrbuto pdf wth respect to ts parameter compoets exst t s clear that for all x > 0 all thrd-order partal dervatves of l( wth respect to the compoets of the parameter of the embedded dstrbuto also exst for all > 0 The mxed partal dervatves wth respect to both trasformato parameters ad para- meters of the embedded dstrbuto are as follows: [l] [( l l l( ] (35a ( [l] [( l l l( ] ( j ( j ( ( ( [4 l ( l l( l] j (35b
06 JPSS Vol 8 No August 00 pp 99-3 [l( ] ( [ l ( l l l l l( ] (35c ( ( Usg (3 (33 ad the fact that all frst- secod- ad thrd-order partal dervatves of the embedded dstrbuto pdf wth respect to ts parameter compoets exst t s clear that for all x > 0 all frst- secod- ad thrd-order partal dervatves of l( wth respect to all parameters also exst for all > 0 The ext lemma proves that f the embedded dstrbuto obeys certa regularty codtos the each of the frst ad secod partal dervatves of the composte desty wth respect to the trasformato parameters s bouded absolute value by a tegrable fucto ad that the expectatos of each thrd partal dervatve s bouded absolute value by a fucto wth fte expectato As a result of ths lemma ad the Lebesgue Domated Covergece Theorem we fd that for each of the tegrals ( x dx ad [l( ] ( dx 0 tegrato wth respect to x ad dfferetato wth respect to ay compoet of χ are terchageable [6 7] 3 Lemma 3 Let ad be defed as Lemma 3 Let χ ( be the 3 parameter vector of the composte dstrbuto wth > The for each χ 0 wth 0 3 there exst fuctos g( x hj( x H jk ( x for j k = 3 (possbly depedg o χ such that for χ a eghborhood 0 N( 0 the relatos for j k = 3 hold for all x where for j k = 3 (Here 0 g( x (36a ( h ( x (36b j j ( H ( x (36c j k jk g ( x dx h j( x dx ad E[ Hjk ( x] E [] deotes expectato wth respect to the GLL measure Proof Let / the odds fucto for the embedded dstrbuto The we have l l [ l] (37a Now ( /( so that (/ l( The frst term ths equato s a tegrable fucto For the factor (37b
Asymptotc Propertes of MLE s for Dstrbutos James U Gleato ad M Mahbubur Rahma 07 Hece for ( s bouded by a tegrable fucto f l( s bouded by a tegrable fucto whch s true by Lemma Let (l / the meda of the -parameter ebull dstrbuto ad of the com- poste GLL dstrbuto [3] e assume that For x > we have ( x x e whch s tegrable o the terval ( + sce t s proportoal to a fucto whch tegrated o (0 + yelds EX For x we have ( x [ e ]l[( x ] O the terval (0 ths fucto has a maxmum value of ( M where ( x [ e ]l[( x ] M for x x y / 0 0 y ad y 0 s the soluto of the trascedetal equato y l( y e o the terval (0 Thus ( s bouded absolute value by a tegrable fucto v l 3 [l( 3 ( 39 [l] 75 l ] (37c (37d Now from (37b ad Lemma t s clear that l( l( ad 3 l( are all bouded absolute value by tegrable fuctos Thus ( ad ( are bouded by tegrable fuctos Also f the fucto boudg ( s tegrable the the expectato of that fucto s fte The frst partal dervatve wth respect to may be foud by multplyg (34a by l : If > the by (37b l[ ( l ] l (38 (39 l Also ( l By assumpto ( s bouded absolute value by a tegrable fucto By Lemma for > ( s bouded absolute value by a tegrable fucto Hece for > ( ( s the bouded by a tegrable fucto e let s l (30a so that s ( l (30b ad s l l s (30c The the secod partal dervatves of the composte desty wth respect to the parameters of the embedded dstrbuto have the form
08 JPSS Vol 8 No August 00 pp 99-3 ls( s ls [ ] ( j ( j ( j ( j ls l( l l (3 ( j ( j ( j Now for > 3 we have 0 ( l 6 Also sce s for > the frst ad fourth terms are bouded absolute value by tegrable fuctos f (a ( ( j s bouded absolute value by a tegrable fucto for all ad j th > 3 the secod ad thrd terms are bouded absolute value by tegrable fuctos f (b ( j ( ad (c ( j respectvely are bouded absolute value by tegrable fuctos for all ad j Smlarly for > 3 the ffth term s bouded absolute value by a tegrable fucto by assumpto The codtos (a (b ad (c above hold due to Lemma The thrd part of the lemma holds f each thrd partal dervatve of the composte desty s bouded absolute value by a fucto wth fte expectato The product of the composte desty ad a thrd partal dervatve of the composte desty wth respect to the parameters of the embedded dstrbuto ( s a sum of 5 terms each of whch s a product of a j k costat ad oe or more of: powers of l s powers of λ egatve powers of the frst secod ad thrd partal dervatves of wth respect to the parameters ad the frst ad secod partal dervatves of wth respect to the parameters If ths sum s bouded absolute value by a tegrable fucto the the expectato of the thrd partal dervatve of the composte desty wth respect to embedded parameters s bouded absolute value by a fucto wth fte expectato Now s 3( s 3( ad Hece sce l for > the thrd partal dervatve of the composte desty wll be bouded absolute value by a fucto wth fte expectato f each of the followg terms s bouded absolute value by a tegrable fucto for all j k: ( ( j k ( ( j ( k j ( k ( ( j ( k ad j ( k (3 Each of the above terms s bouded absolute value by a tegrable fucto as a result of Lemma Hece the thrd partal dervatve s bouded absolute value by a fucto wth fte expectato It remas to exame the mxed partal dervatves of the composte desty wth respect to both parameters of the trasformato ad the embedded dstrbuto There are sx remag dervatves: ( ( ad for = e wll eed the followg fact: From (37a ad (38 we have l l( (33 l [ l] ( (6 4l l 4
Asymptotc Propertes of MLE s for Dstrbutos James U Gleato ad M Mahbubur Rahma 09 It has already bee proved that the frst term s tegrable The secod ad thrd terms are tegrable by Lemma The last term s tegrable by Lemma Now usg (37c we have { [ ( l ( 6 6 l ]} ( ( l 3[l] 4 l ( 6 [l] 6 l j j The usg (38 ad the fact that for > j l 4 we have 8( l l 5 ( [l] ( ( ( ( 3 [l] 4 l 8 6 [l] 04 l( ( ( ( By Lemmas ad each of the above terms s bouded by a tegrable ucto ad thus has fte expectato wth respect to the GLL dstrbuto e also have from (3 that l {9( ( ( 6( ( 3( } l ( ( ( { ( } 4 ( l l ( ( Multplyg equato (3a by l we have l l [ l( ] Hece 44( ( [ l ] ( 4( [ l ] ( ( [ l ] [ l( ] ( ( 8[ l ] ( 48 ( 3 l( By Lemmas ad each of the above terms s tegrable Hece ( s bouded absolute value by a fucto wth fte expectato wth respect to the GLL dstrbuto Lemma 33 Let ad be defed as Lemmas 3 ad 3 The for > ad > 3 0 E ({[l] for = 3 Proof } ( j j From (3a ad usg the fact that for j l 4 we have E ( ( dx dx 0 0 0 {[l] } l [l]
0 JPSS Vol 8 No August 00 pp 99-3 sce by Lemma l( s bouded by a tegrable fucto for ay postve teger e also have ] s [l( [l ] by (33a The ({[l( χ ]} E dx E( 0 44( {[l] } 4( dx 0 The frst ad thrd tegrals are fte by Lemma The secod term s fte sce the embedded dstrbuto satsfes the regularty codtos Theorem 34 Let X X X be a radom sample from a GLL dstrbuto wth > ad > 3 The wth GLL-probablty the lkelhood equatos admt a sequece of solutos ˆ χ satsfyg a strog cosstecy: ˆ χ as ad b asymptotc ormalty ad effcecy: χˆ s AN( I where χ I E {[(l ][(l ]} j 33 Proof Ths result s a mmedate cosequece of the precedg three lemmas [4 5] 4 Cocluso As a result of Theorem 34 ferece about the parameters of the GLL dstrbuto may be doe usg asymptotc ormal dstrbuto theory As see the proofs of the lemmas the prevous secto the regularty of the GLL dstrbuto s due largely to: a the regularty of the -parameter ebull dstrbuto (cludg Lemmas ad ad b the boudedess j of the trasformato ( partcular ( l 4 j for greater tha some postve teger j For a radom sample x x x x from a GLL dstrbuto wth > 0 > ad > 3 the log-lkelhood fucto s x ( ; x l ( l[ ( x ( x ] l{[ ( x ] [ ] } l[ ( x ] The usg equatos ( ad ( we fd that the lkelhood equatos are ( x l x d x x d6 x 0 x l x dx x d7x 0 ad [ l ( x ]l[ ( x ] 0 (
Asymptotc Propertes of MLE s for Dstrbutos James U Gleato ad M Mahbubur Rahma These equatos may be solved smultaeously usg a approprate umercal procedure such as Newto s method to obta the MLE s for the parameters The elemets of the formato matrx dvded by are gve below Usg equatos (33d (33e ad (34b we have I ( E ( I E [( s g ( s ] E {[l( ] } ( E ( E [( s g s ] E {[l( ] } ( I E ( E {[(l ( X l l l( ] d ( X} 3 I ( E ( E {( s g [ ] s } E {[l( ] } ad I E ( E {[(l ( X l l l( ] d ( X} 3 I E ( E { l l [l] } 33 The expectatos would eed to be calculated by umerc tegrato after the MLE s of the parameters are obtaed The asymptotc stadard errors of the estmators may be foud by vertg the formato matrx ad takg square roots of the dagoal elemets gvg SE ˆ I ˆ [ ] SE ˆ I [ ] ˆ ad SE ˆ I ˆ [ ] 33 These results may the be used to obta approxmate large-sample cofdece terval estmates of the parameters Theorem 34 establshes the asymptotc propertes of the MLE s for a subset of the parameter space wth > 0 > ad > 3 It would be useful to vestgate whether these restrctos o the values of ad mght be relaxed I addto the results hold for a system whose lfetme dstrbuto s GLL It would also be useful to vestgate the lfetme dstrbuto of the type of parallel system descrbed secto whch the compoets have GLL lfetme dstrbutos Fally the asymptotc results were establshed for famles geerated from a -parameter ebull dstrbuto Other geeratg dstrbutos mght be of terest for applcatos to systems such as the oe metoed the precedg paragraph
JPSS Vol 8 No August 00 pp 99-3 Appedx: Fuctoal Deftos d ( x x d ( x xl( x d ( x x[ ( x ] 3 d x xd x x 4 6[l( ] d ( x xd ( x 5 7 d x x x x 6 3 ( 3 d 3 ( x x [l( x ] ( x 3 x 7 d ( x x [l( x ][ x l( x 3 x l( x x l( x ] 8 d x x x x x x x 9 [ l ( 3 l( ] x[ ( l( x ] d ( x ( x 0 d ( x l( x ( x l( x d x x x x 3 d x x x x x x x x 3 l( [ l( ( ( l( 3 ( l( ] d x x x x x x x 4 [ ] l( 3 ( l( ( l( d ( x 6 ( x ( x 7 ( x 9 ( x 3 ( x 5 3 3 3 ( x d x x x x x x x x x 3 3 6 6 l 3 [l] [l] 7 3 [l] x 3 9( x [l( x] [l( x] 3[l( ] d x x x x x x x x 3 7 9 3 3 6 l l( 7 ( x l( x 3 ( x l( x ( x l( x l( x l( x d x x x x x x x x x 8 ( l( ( 4( l( 4l( 6 ( [l( ] 3 ( x [l( x] 7 x [l( x] [l( x] Refereces [] Dubey S D (965 Asymptotc propertes of several estmators of ebull parameters Techometrcs 7 43-434 [] Gleato J U ad Lych J D (004 O the dstrbuto of the breakg stra of a budle of brttle elastc fbers Advaces Appled Probablty 36 98-5
Asymptotc Propertes of MLE s for Dstrbutos James U Gleato ad M Mahbubur Rahma 3 [3] Gleato J U ad Lych J D (006 Propertes of geeralzed log-logstc famles of lfetme dstrbutos Joural of Probablty ad Statstcal Scece 4( 5-64 [4] Harter H L ad Moore A H (967 Asymptotc varaces ad covaraces maxmumlkelhood estmators from cesored samples of the parameters of the ebull ad gamma populatos The Aals of Mathematcal Statstcs 38( 557-570 [5] Johso N L Kotz S ad Balakrsha N (994 Uvarate Cotuous Dstrbutos Volume d Edto Joh ley & Sos Ic New York [6] Serflg R J (980 Approxmato Theorems of Mathematcal Statstcs Joh ley & Sos Ic New York [7] lks S S (96 Mathematcal Statstcs Joh ley & Sos Ic New York [8] Schervsh M J (995 Theory of Statstcs Sprger-Verlag New York [9] Self S G ad Lag K-Y (987 Asymptotc propertes of maxmum lkelhood estmators ad lkelhood rato tests uder ostadard codtos Joural of the Amerca Statstcal Assocato 8 605-60