Confidence intervals for functions of coefficients of variation with bounded parameter spaces in two gamma distributions

Transcription

1 Sogklaakar J. Sc. Techol. 39 (), 7-39, Ja. - Feb. 7 Orgal Artcle Cofdece terval for fucto of coeffcet of varato th bouded parameter pace to gamma dtrbuto Pataraa Sagaakj ad Sa-Aat Ntpog * Departmet of Appled Stattc, Faculty of Appled Scece, Kg Mogkut Uverty of Techology North Bagkok, Bag Sue, Bagkok, 8 Thalad Receved: 9 December 5; Reved: March 6; Accepted: 9 March 6 Abtract The problem of etmatg parameter a gamma dtrbuto ha bee dely tuded th repect to both theore ad applcato. I pecal cae, he the parameter pace bouded, the cotructo of the cofdece terval baed o the clacal Neyma procedure uatfactory becaue the formato regardg the retrcto of the parameter dregarded. I order to develop the etmator for th ue, the cofdece terval for the coeffcet of varato for the cae of a gamma dtrbuto ere propoed. Extedg to to populato, the cofdece terval for the dfferece ad the rato of coeffcet of varato th retrcted parameter ere preeted. Mote Carlo mulato ere ued to vetgate the performace of the propoed etmator. The reult hoed that the propoed cofdece terval performed better tha the compared etmator term of expected legth, epecally he the coeffcet of varato ere cloe to the boudary. Addtoally, to example ug real data ere aalyzed to llutrate the fdg of the paper. Keyor: bouded parameter pace, coeffcet of varato, cofdece terval, gamma dtrbuto, mulato. Itroducto Iterval etmato a commo approach for etmatg the parameter of teret. Sce t guarateed by the cofdece level that the uko parameter cotaed the cofdece terval th commo probablty, the terval etmator more meagful, ad t provde more formato th repect to the parameter tha the pot etmator (Caella & Berger, ). I frequett theory, he lackg a pror koledge of the parameter, the cofdece terval uually derved from the clacal Neyma procedure. That mea tattcal ferece baed o that tradtoal approach avalable for the atural parameter pace (Madelker, ). Hoever, fact, the bouded parameter foud may practcal applcato, uch a egeerg proce cotrol, health cece, ad phycal expermet. I th cae, * Correpodg author. Emal addre: a-aat.@c.kmutb.ac.th the crucal problem of the cofdece terval obtaed by the Neyma procedure that he the cofdece terval partly or completely depart from the permble rage for the parameter, t valdate the aerto of the ( )% cofdece terval (Fraer et al., 4). Therefore, alteratve approache have bee dcued for obtag accurate cofdece terval. Recet ork related to tattcal ferece for bouded parameter a follo. I the paper of Wag (8), the cofdece terval for the ormal mea cae here the parameter pace bouded ere derved ug the rp terval, Bayea terval, ad lkelhood terval. Thee cofdece terval ere compared th the tadard cofdece terval ad the mmax terval by mulato. It a foud that, although the coverage probablte of the tadard cofdece terval ere loer tha thoe of the rp terval ad Bayea terval, they ere greater tha the omal coverage level ad mpler to ue practce. The tadard cofdece terval alo provded the hort legth terval. Furthermore, the tadard cofdece terval th bouded parameter the ormal

2 8 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 dtrbuto ere tuded by Ntpog () ad Sappaktkamjor ad Ntpog (3). The reult va Mote Carlo mulato hoed that ther cofdece terval performed ell term of coverage probablty ad expected legth. I a keed dtrbuto, Eede (995) troduced mmax etmato for cale varat quare error lo he the cale parameter bouded belo. Chag () preeted the admble etmator of the retrcted cale parameter the gamma dtrbuto. Ug the rp terval, Wag () tuded the cofdece terval for the mea th bouded parameter pace the expoetal famle. Moreover, Ntpog (3a, 3b) propoed tadard cofdece terval for the mea ad the coeffcet of varato a logormal dtrbuto th retrcted parameter pace, ad the developed to the tadard cofdece terval for the dfferece ad the rato of to logormal mea (Ntpog, 5). The reult hoed that the cofdece terval of Ntpog (5) performed ell term of coverage probablty ad expected legth. A the reveed lterature dcate, the retrcto parameter ha bee tuded th both ormal ad keed dtrbuto. Thu, t alo lkely that the parameter pace may be bouded the gamma dtrbuto. It probablty dety fucto gve (). Th dtrbuto appled actuaral cece ad may fel of appled tattc a the atg tme utl th evet occur. I th tudy, e focu o the cofdece terval for fucto of coeffcet of varato to gamma dtrbuto he the parameter pace are retrcted. The coeffcet of varato a tattcal meauremet ued to report the dpero of varable, ad t ca be appled to compare everal varable expreed dfferet ut. I the gamma dtrbuto, the coeffcet of varato the fucto of oly oe parameter, the hape parameter, hle t varace depe o both the hape ad cale parameter. Th the reao for coderg the tattcal ferece of the coeffcet of varato. I th ork, the coverage probablte ad expected legth of the propoed ad the extg cofdece terval are tuded through Mote Carlo mulato. Moreover, e ue realorld example to llutrate the cofdece terval propoed th paper.. Cofdece Iterval for the Coeffcet of Varato of the Gamma Dtrbuto I th ecto, e expla the metho for cotructg the cofdece terval for the gle coeffcet of varato. The crtero of the tudy a follo. Let X = ( X, X,, X ) be a radom ample from the gamma dtrbuto th the hape parameter ad cale parameter, deoted a X ~ Gamma(, ). The probablty dety fucto of X gve by x exp{ x / } ; x f ( x ;, ) ( ), X () ; x here ad. { : } ad { : } are the atural parameter pace. The mea ad varace of X are E( X ) ad Var( X ), repectvely. Thu, the coeffcet of varato of X gve by /. Sce the uko parameter, t requred to be etmated. We frt coder the maxmum lkelhood etmator for ad. From the dety ho (), the log-lkelhood fucto of ad gve by X l L (, ) ( ) l X l ( ) l. Takg partal dervatve of the above equato th repect to ad, repectvely, the core fucto derved a U (, ) = l X l / ( ) l X / /. The, e yeld the maxmum lkelhood etmator for ad, repectvely, here ˆ (l X l X / ) ˆ X, ˆ X X / the ample mea of X. Alo, the ample coeffcet of varato for gve by ˆ / ˆ. Next, the cofdece terval for ug to metho, the core ad the Wald terval are vetgated. Thee approache are codered later.. Cofdece terval baed o the core method Let ad be the parameter of teret ad the uace parameter, repectvely. I geeral, the core or Rao tattc deoted a T ( ˆ W U, ) I (, ˆ ) U (, ˆ ), u here ˆ the maxmum lkelhood etmator for uder the ull hypothe H :, U (, ˆ ) the vector of the core fucto, ad I (, ˆ ) the matrx of the Fher formato. Here, t eay to derve that the core fucto uder H ˆ l X / ( ) l X U (, ) =. The vere of the Fher formato ca be derved a / X / I (, ˆ ). 3 X / X ( ) / ( )

3 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 9 Ug the property of the core fucto, e ca ee that the pvotal () Z l X l X core coverge dtrbuto to the tadard ormal dtrbuto. Sce the varace of ˆ /, e approxmate t by ubttutg ˆ t varace. Uder H, tattc () gve a ˆ Z l X l X. core From the probablty tatemet, P( Z Z Z ) / core /, t ca be mply rtte a P( l u ). Therefore, the ( )% cofdece terval for baed o the core method,, gve by [ l,u ], z Z z Z, / / ˆ ˆ (3) here z l X l X ad Z / the ( / ) th percetle of the tadard ormal dtrbuto.. Cofdece terval baed o the Wald method The Wald tattc a aymptotc tattc derved from the property of the maxmum lkelhood etmator. The geeral form of the Wald tattc uder the ull hypothe H : defed a W ( ˆ ) [ I ( ˆ, ˆ )] ( ˆ ), e T here I ( ˆ, ˆ ) the etmated varace of ˆ obtaed from the frt ro ad the frt colum of I ( ˆ, ˆ ). Ug the formato of partal dervatve from the prevou ubecto, the vere matrx gve by ˆ / X / I ( ˆ, ˆ ) 3 X / X ( ˆ ) / ( ˆ ) th I ( ˆ, ˆ ) ˆ /. Therefore, uder H, e obta the Wald tattc Z ald ˆ ( ˆ ), hch ha the lmtg dtrbuto of tadard ormal dtrbuto. Therefore, the ( )% cofdece terval for baed o the Wald method,, gve by l,u / ˆ Z, / ˆ Z. / / ˆ ˆ (4) I addto, uppoe that Y ( Y,Y,,Y ) m be a radom ample, here Y ~ Gamma(, ). The coeffcet of varato of Y / th the pot etmator ˆ / ˆ. Alo, e have the cofdece terval for baed o the core method, [ l,u ], ad the Wald method, [ l,u ]. Note that thee cofdece terval are mlar to (3) ad (4), except that they ue the formato from Y. Extedg the problem of th ecto, the cofdece terval for the gle coeffcet of varato the gamma dtrbuto th bouded parameter pace are vetgated the ext ecto. 3. Cofdece Iterval for a Bouded Coeffcet of Varato the Gamma Dtrbuto The method for cotructg the cofdece terval for bouded parameter pace appled th paper the tadard approach. It derved ug the terecto of the geeral cofdece lmt ad the bou of parameter pace. For th method, the formato cocerg the retrcto of the parameter ued, cotrat to the clacal Neyma approach. Follog Wag (8), he a parameter bouded betee value c ad d, the tadard cofdece terval for [max( c,l ), m( d,u )], (5) here l ad u are the loer ad upper lmt of the geeral cofdece terval for, repectvely. Obvouly, e have four cae a follo: () If c l ad d u, the [ c,d] () If c l ad d u, the [ c,u ] () If c l ad d u, the [ l,d] (v) If c l ad d u, the [ l,u ]. Note that the cofdece terval obtaed from the above cae ha the hortet legth terval. The procedure for cotructg the cofdece terval for th parameter retrcto decrbed later. Aume that the parameter pace of ko to be retrcted ad bouded betee value a ad b, here a b. Sce a fucto of, he bouded, alo bouded. It ca be mply rtte a a b a b a / b / a b. R R Ug the cofdece terval for preeted the prevou ecto ad the formato of the retrcto, the ( )% cofdece terval ug the core ad the Wald terval for th bouded parameter pace are gve a R R [max( a,l ), m( b,u )] (6)

4 3 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 [max( a,l ), m( b,u )], (7) R R repectvely. We alo ote that l ad u, ad l ad u are the geeral cofdece lmt obtaed from (3) ad (4), repectvely. 4. Cofdece Iterval for the Dfferece of Coeffcet of Varato the Gamma Dtrbuto I th ecto, e follo the cofdece terval for the dfferece of coeffcet of varato preeted the paper of Sagaakj ad Ntpog (5). The otato are gve at the tart. Suppoe that X ad Y be to radom ample th X ~ Gamma(, ) ad Y ~ Gamma(, ). Alo, X ad Y are depedet. The dfferece of coeffcet of varato defed a. They troduced the cofdece terval for ug the method of varace of etmate recovery (MOVER) th the core terval,, ad Wald terval,. Thee cofdece terval are gve by [ l,u ] ˆ ( ˆ l ) ( u ˆ ), ˆ ( u ˆ ) ( ˆ l ) (8) [ l,u ] ˆ ( ˆ l ) ( u ˆ ), ˆ ( u ˆ ) ( ˆ l ) (9) here ˆ ˆ ˆ the ample dfferece of coeffcet of varato. I the mulato, t a foud that ad of Sagaakj ad Ntpog (5) performed ell term of coverage probablty almot all cae, ad the legth of ere lghtly horter tha thoe of. Thu, the ext ecto, e ue thee etmator to develop the cofdece terval for the dfferece of coeffcet of varato th bouded parameter pace. 5. Cofdece Iterval for the Dfferece of Coeffcet of Varato th Bouded Parameter the Gamma Dtrbuto Here, e are tereted the retrcto of parameter the gamma dtrbuto order to cotruct the cofdece terval for the dfferece of coeffcet of varato. Aume that the hape parameter ad are bouded, a b ad a b, here a b for =,. Thu, e have a b a / b / a b. R R Sce the dfferece of coeffcet of varato a fucto of parameter ad hch are bouded, t alo bouded a / b / a / a / b a a b b a b, DR R R R R DR here a / b, b / a, a / b, ad b R R R R / a. Ug the tadard approach, the ( )% cofdece terval for th bouded parameter baed o the core terval ad the Wald terval are [max( a,l ), m( b,u )] () DR DR DR [max( a,l ), m( b,u )], () DR DR DR repectvely. Note that l ad u, ad l ad u are the geeral cofdece lmt for obtaed from (8) ad (9), repectvely. 6. Cofdece Iterval for the Rato of Coeffcet of Varato the Gamma Dtrbuto Let X ad Y be to radom ample from the gamma dtrbuto a metoed Secto 4. Here, the rato of coeffcet of varato /. Recetly, Sagaakj et al. (5) troduced the cofdece terval for baed o the MOVER th the core terval,, ad the Wald r terval,, here [ l,u ] r r r ˆ ˆ ˆ ˆ l u ˆ u ˆ l / u ˆ u ˆ ˆ ( ˆ ˆ ) l u ( ˆ l )( ˆ u ) / l ( ˆ l ) ( ) ( )( ) ( ), () [ l,u ] r r ˆ ˆ ˆ ˆ l u ˆ u ˆ l / u ˆ u ˆ ˆ ( ˆ ˆ ) ( ˆ )( ˆ ) ( ˆ ) ( ) ( )( ) ( ), l u l u / l l, (3) repectvely. I the mulato of Sagaakj et al. (5), t a foud that the coverage probablte of ad r r atfed the omal coverage level ad performed ell term of expected legth all cae. Therefore, thee to extg etmator are codered to cotruct the e cofdece terval the ext ecto.

5 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, Cofdece Iterval for the Rato of Coeffcet of Varato th Bouded Parameter the Gamma Dtrbuto Suppoe that parameter ad are bouded a a b ad a b. Ug the formato regardg the retrcto of ad Secto 5, the rato of coeffcet of varato to gamma dtrbuto bouded a a / b / b / a a a / b b / a b. RR R R R R RR Hece, t eay to ee that the ( )% tadard cofdece terval for th bouded parameter baed o the core terval ad the Wald terval are [max( a,l ), m( b,u )] (4) R RR r RR r [max( a,l ), m( b,u )], (5) R RR r RR r repectvely, here l ad u, ad l ad u are the r r r r geeral cofdece lmt for obtaed from () ad (3), repectvely. 8. Smulato Stude I th tudy, the performace of the propoed cofdece terval vetgated ug Mote Carlo mulato. The mulato are doe ug the R tattcal program (Veable et al., 5) th M =, replcato each cae. The etmated coverage probablty (CP) ad the etmated expected legth (EL), repectvely, are gve by c L U CP ad EL M M h h h U L, M here c( L U ) the umber of mulato ru he parameter le th the cofdece terval. Here, e chooe a cofdece terval hch ha a coverage probablty greater tha or cloe to the omal coverage level, ad hort legth terval. For oe populato, the data are geerated from a gamma dtrbuto th ad adjuted to get the requred coeffcet of varato. For the retrcto at. 5. 5, e et =.5,.,.,.8,.3,.33,.35,.4,.45,.47,.49, ad.5. The ample ze are choe to be =, 3, 5,, ad. The, the performace of 95% cofdece terval for computed. For to populato, the data are geerated from to depedet gamma dtrbuto th (, ) here are fxed at, ad are adjuted to yeld the requred coeffcet of varato, hch computed by /, for =,. For the retrcto at ad. 5. 5, the coeffcet of varato are et at (, ) (.5,.5), (.,.49), (.5,.45), (.8,.8), (.45,.5), (.49,.), ad (.5,.5). Next, the coverage probablte ad expected legth of the 95% cofdece terval for the dfferece of coeffcet of varato ad the rato of coeffcet of varato are evaluated. The performace of all propoed cofdece terval alo compared th that of the extg cofdece terval. The mulato reult are decrbed the ext ecto. 9. Reult ad Dcuo We frt coder the performace of the cofdece terval for th bouded parameter pace. The reult are ho Table. For 5, provde coverage probablte le tha the omal coverage level at.95. Hoever, he the ample ze creae, the coverage probablte of creae, ad are greater tha.95. Table. The coverage probablte ad expected legth of the 95% cofdece terval for the coeffcet of varato, he Coverage probablty Expected legth,,

6 3 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 Table. Cotued Coverage probablty Expected legth,,

7 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 33 Meahle, the coverage probablte of atfy the omal coverage level. Sce the coverage probablte of ad, ad, repectvely, provde the ame reult, the performace of thoe cofdece terval appraed term of the expected legth. It a foud that the expected legth of ad are horter tha thoe of ad almot all cae. I addto, the expected legth of are lghtly maller tha thoe of for large. From Table, t ca be ee that the rato of expected legth betee ad, ad ad, repectvely, are greater tha oe, he cloe to the boudary of (.5,.5). That mea the cofdece terval obtaed from the method volvg a bouded terval more accurately cover the true parameter tha cofdece terval Table. The rato of expected legth of the 95% cofdece terval for the coeffcet of varato, he = = = 3 = =

8 34 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 derved by the clacal procedure. The reult of the cofdece terval for th bouded parameter pace are preeted Table 3. For,m 3, yel coverage probablte le tha DR.95. Hoever, he the ample ze creae, t ha coverage probablte greater tha or cloe to.95. The coverage probablte of atfy the omal coverage DR geeral, expect for ample ze equal to. The reult are mlar to thoe of Sagaakj ad Ntpog (5). Furthermore, the expected legth of ad are DR DR horter tha thoe of ad. A ca be ee from Table 4, the rato of expected legth of ad, ad DR ad, repectvely, are greater tha oe, epecally, DR he ad are cloe to the boudary of (.5,.5). Alo, the expected legth of are alo loger tha thoe DR of. DR Fally, the performace of the cofdece terval for th bouded parameter pace codered. The reult from Table 5 ho that ad provde R R coverage probablte greater tha or cloe to.95. I geeral, the coverage probablte of are hgher tha thoe of R. The reult are alo mlar to thoe of Sagaakj R Table 3. The coverage probablte ad expected legth of the 95% cofdece terval for the dfferece of coeffcet of varato, he. 5,. 5 (,m) (, ) Coverage probablty, DR, DR Expected legth DR DR (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (3,3) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (5,5) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5)

9 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 35 Table 4. The rato of expected legth of the 95% cofdece terval for the dfferece of coeffcet of varato, he (,m) (, ) DR DR DR DR (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (3,3) (.5,.5) (.,.49) (.5,.45) (.8,.8)...87 (.45,.5) (.49,.) (.5,.5) (5,5) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) et al. (5). From Table 6, the rato of expected legth betee ad, ad ad, repectvely, r R r R are greater tha oe, epecally he ad are cloe to the boudary of parameter pace. Moreover, the expected legth of are lghtly maller tha thoe of. R R. Real Data Example Example. The data of mothly rafall (mm) are ued to compute the cofdece terval for. From the report of the Hydrology Irrgato Ceter for the cetral rego of Thalad (5), 6 obervato September (betee 955-4), hch the moth th the loet rafall, are elected. The tattc are reported a follo. Before computg the cofdece terval, the Adero-Darlg tet ued to tet the dtrbuto of thee data. It a foud that the data of rafall September ft the gamma dtrbuto th the tet tattc of.45 ad the p- 6 value of.5. The average rafall x 38., l x = 34.4, ad the maxmum lkelhood etmator for ˆ.36. Suppoe that the bouded terval. 3. 5

10 36 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 Table 5. The coverage probablte ad expected legth of the 95% cofdece terval for the rato of coeffcet of varato, he. 5,. 5 (,m) (, ) Coverage probablty, r R, r R r Expected legth r R R (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (3,3) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (5,5) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) codered. The 95% cofdece terval are calculated ad preeted Table 7. Lke the mulato the prevou ecto, ha a horter legth tha. The legth of greater tha that of ad. Meahle, the legth of ot dfferet from that of, but greater tha that of. Example. We ue the data provded by Procha (963) to compute the cofdece terval for ad. Thee data preet the tme (hour) of ucceve falure of the ar codtog ytem of to jet arplae, umber 79 ad 799. Ug the Adero-Darlg tet, the data ft the gamma dtrbuto. For the frt plae, 3 ample are oberved th the ample mea x 59.6 ad the ample coeffcet of varato ˆ.. For 9 ample from the ecod plae, the ample mea ad ample coeffcet of varato are y 83.5 ad ˆ.8, repectvely. Suppoe that the coeffcet of varato are bouded,. 8,. 3. The 95% cofdece terval ad the legth for are preeted Table 7. It ca be ee that the legth of ad are horter tha thoe of ad DR DR. That mea the propoed cofdece terval are better tha the extg terval. I addto, thee data are

11 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 37 Table 6. The rato of expected legth of the 95% cofdece terval for the rato of coeffcet of varato, he. 5,. 5 (,m) (, ) r R r R R R (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (3,3) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (5,5) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) (.49,.) (.5,.5) (,) (.5,.5) (.,.49) (.5,.45) (.8,.8) (.45,.5) ued to coduct the 95% cofdece terval for. We alo ote that thee reult upport the mulato tude the prevou ecto.. Cocluo The am of th paper a tofold. The frt am a to propoe e cofdece terval for the coeffcet of varato th bouded parameter pace the gamma dtrbuto. Thee etmator ere exteded from the cofdece terval baed o the core ad Wald terval. The ecod am a to propoe e cofdece terval for the dfferece ad the rato of coeffcet of varato th bouded parameter pace to gamma dtrbuto. The reult obtaed from the mulato dcated that the propoed cofdece terval had the ame coverage probablte a ther extg cofdece terval. Hoever, all propoed cofdece terval performed better tha the compared etmator term of expected legth, epecally he the coeffcet of varato ere cloe to the boudary. That mea the propoed cofdece terval more accurately etmate the true parameter. Therefore, t hould be oted that he the coeffcet of varato ko to be bouded, the cofdece terval utlzed the formato about the retrcto of the parameter ued. Sce our etmator outperform ad are alo eay to

12 38 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 Table 7. The reult of the 95% cofdece terval ad the legth of terval for the data example Reult The extg The propoed cofdece terval cofdece terval for Iterval [.9,.4] [.3,.45] [.3,.4] [.3,.45] Legth for DR DR Iterval [-.,.77] [-.,.93] [-.,.45] [-.,.45] Legth for r r R R Iterval [.99,.5] [.98,.5] [.99,.63] [.98,.63] Legth compute ug the explct formula, thoe etmator ca be ued a the cofdece terval for fucto of coeffcet of varato th bouded parameter pace the gamma dtrbuto. Ackoledgemet We are grateful thak to the referee for the valuable commet ad uggeto. Moreover, the frt author grateful for recevg the facal upport from Thammaat Uverty, Thalad. Referece Caella, G., & Berger, R. L. (). Stattcal ferece. Moterey, CA: Duxbury Pre. Chag, Y. T. (). A ote o the etmato of retrcted cale parameter of gamma dtrbuto. I H. J. Lez, P. T. Wlrch, & W. Schmd (E.), Froter tattcal qualty cotrol 9 (pp ). Hedelberg, Germay: Phyca-Verlag. Eede, C. (995). Mmax etmato of a loer-bouded cale parameter of a gamma dtrbuto for calevarat quare-error lo. The Caada Joural of Stattc, 3(3), do:.37/ Fraer, D. A. S., Red, N., & Wog, A. C. M. (4). Iferece for bouded parameter. Phycal Reve, 69(3), -6. do:.3/phyevd Hydrology Irrgato Ceter for Cetral Rego of Thalad. (5, October 5). The Mothly Rafall. Retreved from =3 Madelker, M. (). Settg cofdece terval for bouded parameter. Stattcal Scece, 7(), Retreved from Ntpog, S. (). Coverage probablty of cofdece terval for the ormal mea ad varace th retrcted parameter pace. Iteratoal Joural of Mathematcal, Computatoal, Phycal, Electrcal ad Computer Egeerg, 5(8), Ntpog, S. (3a). Cofdece terval for the mea of logormal dtrbuto th retrcted parameter pace. Appled Mathematcal Scece, 7(4), Ntpog, S. (3b). Cofdece terval for the coeffcet of varato of logormal dtrbuto th retrcted parameter pace. Appled Mathematcal Scece, 7(77), Ntpog, S. (5). Cofdece terval for the dfferece ad the rato of log-ormal mea th bouded parameter. Sogklaakar Joural of Scece ad Techololgy, 37(), 3-4. Procha, F. (963). Theoretcal explaato of oberved decreag falure rate. Techometrc, 5(3), do:.37/6634 Sagaakj, P., & Ntpog, S. (5). Cofdece terval for the dfferece of coeffcet of varato to gamma dtrbuto. Proceedg of the Iteratoal Cogre o Ecoomy, Face, ad Bue, Sagaakj, P., Ntpog, S., & Ntpog, S. (5). Cofdece terval for the rato of coeffcet of varato of the gamma dtrbuto. I N. Huyh, M. Iuguch, & T. Deoeux (E.), Lecture ote computer cece: Vol Itegrated ucertaty koledge modelg ad deco makg (pp. 93-3). Cham, Stzerlad: Sprger Iteratoal. do:.7/ _9 Sappaktkamjor, J., & Ntpog, S. (3). Cofdece terval for the coeffcet of varato th bouded parameter. Iteratoal Joural of Mathematcal, Computatoal, Phycal, Electrcal ad Computer Egeerg, 7(9), 46-4.

13 P. Sagaakj & S. Ntpog / Sogklaakar J. Sc. Techol. 39 (), 7-39, 7 39 Wag, H. (8). Cofdece terval for the mea of a ormal dtrbuto th retrcted parameter pace. Joural of Stattcal Computato ad Smulato, 78(9), do:.8/ Wag, H. (). Cofdece terval for a bouded mea expoetal famle. Joural of Theoretcal ad Appled Stattc, 48(), do:.8/ Veable, W. N., Smth, D. M., & R Core Team. (5, July 3). A troducto to R: Note o R, A programmg evromet for data aaly ad graphc. Retreved from