Intenatonal Jounal of Innovatve Management, Infomaton & Poducton ISME Intenatonalc0 ISSN 85-5439 Volume, Numbe, June 0 PP. 78-8 INTERVAL ESTIMATION FOR THE QUANTILE OF A TWO-PARAMETER EXPONENTIAL DISTRIBUTION JUAN WANG AND XINMIN LI School of Scences, Shandong Unvesty of Technology Zbo, Shandong 55049, Chna xml@sdut.edu.cn ABSTRACT. In ths ae, we consde the nteval estmaton fo the quantles of two-aamete exonental dstbutons. Based on bootstang and fducal nfeences, two methods fo the nteval estmaton of quantles ae oosed. To evaluate the coveage obabltes and exected lengths of the two methods, a smulaton study s conducted. The esults ndcate that the fducal nfeence method efoms well unde all of the examned condtons. Keywods: Exonental Dstbuton; Quantle; Bootsta; Fducal Dstbuton; Coveage Pobablty. Intoducton. Two-aamete exonental dstbuton, whch occues an motant oston n obablty and statstcal aeas, has been wdely used n actce, esecally n the aea of elablty. Dung the ast few decades, thee ae many authos consdeed the statstcal nfeences fo the two-aamete exonental dstbuton, see Nelson (98), Lawless (98), Ban and Engelhadt (99), Balakshnan and Basu (995), and Meeke and Escoba (998). It s well known that the quantles of a andom samle ae the common used ndcatos to assess the elablty n statstcal analyss. Howeve, as the quantles do not deend on nusance aametes, thee s no exact fequency method fo nteval estmaton of quantles. Consequently, thee has not been much attemt to the study of the nfeences on quantles n ecent lteatues. In some comlcated stuatons, Efon (979) oosed the bootsta method fo statstcal nfeence. Fee of oulaton dstbutons and aametes, the bootsta theoy has been geatly develoed and exanded fo the last thee decades, and now ths technque s wldly used n vaous felds of statstcs, see Hall (988), DCcco and Efon (996), Chen and Tong (003), etc. Recently, the fducal nfeence has attacted a geat amount of attenton due to ts advantage of handlng the nfeence oblems unde cetan comlex stuatons. Fshe oosed and dscussed the fducal nfeence fstly n 935. Davd and Stone (998) deved a genealzed method to conduct the fducal nfeence based on the functon model. Moe ecently, L et al. (005) and Hanng et al. (006) futhe dscussed the fducal theoy and develoed a geneal method to constuct the fducal ntevals. The man wok of ths ae s to gve the nteval estmatons fo quantles of two-aamete exonental dstbuton based on the bootsta method and the fducal method. Numecal smulatons ae conducted to comae the two methods mentoned above.
INTERVAL ESTIMATION FOR THE QUANTILE 79. Inteval Estmaton of the Quantles... The Two-Paamete Exonental Dstbuton. Let X, X,, X be a andom n samle fom the two-aamete exonental dstbuton wth ts obablty densty functon (df) gven by x ex{ }, x x;, 0, else whee 0, 0. By Fx ( ) PX ( x), the th quantle x can be exessed as x u log( ) fo any (0,). Assume the obsevatons ae the tye II censoed data X X X () () (. It s ) obvous that W X and V () X ( n ) X nx ae the comlete ( ) ( ) () suffcent statstcs of the dstbuton and ndeendently dstbuted. Then, the unfomly mnmum-vaance unbased estmatos (UMVUE) fo and ae ˆ X { X ( n ) X nx }, () ( ) ( ) () n ( ) ˆ { ( ) } ( ) () X n X nx. () Hence, the UMVUE of the th quantle x s xˆ X { ( ) } () X n X nx ( ) ( ) () n ( ) { X ( ) ( n ) X nx () () }log( )... The Bootsta Method. The bootsta method, oosed by Efon (979), s often used to constuct the confdence ntevals fo aametes. The man thought of the bootstang s to adot the emcal obablty dstbuton as a elacement of the unknown dstbuton of undelyng oulaton fom whch the ognal samles ae dawn, and then constuct new andom vaables basng on the ndeendently dstbuted samles geneated fom emcal dstbuton, whch s a substtuton of the ognal samles, fo futhe statstcal nfeence. The bootsta ocedue fo the calculaton of confdence nteval of the th quantle s gven as follows: () Calculate the emcal dstbuton functon based on the tye II censoed data X () X () X ( ). () By Monte Calo smulaton method, geneate a samle x * x * x * fom the emcal dstbuton wth sze andomly. (3) Based on the andom samle, calculate the UMVUE of x by x ˆ ˆ log( ). ˆ
80 JUAN WANG AND XINMIN LI (4) Reeat ste ()-(3) fo N (=5000) tmes and get N coesondngx. ˆ (5) Fo the gven confdence coeffcent, sot the x s n an ascendng ode, that ˆ s, x x x ˆ () ˆ () ˆ ( N ). Fnd the / and / ecentles denoted by x ˆ, L and x, Then the bootsta confdence nteval of the quantle x s [ ˆ, L x, x ]. ˆU, ˆU, esectvely..3. The Fducal Method. Let P () denote the df of the andom vaable X wth ts samle sace, whee s the unknown aamete n the aamete sace. ( ) s a eal-valued aamete functon of nteest. Defnton.. Suose that thee exst a andom vaable E wth known dstbuton on sace and a functon h(,) e fom to such that X h(, E) fo all. Futhemoe, f fo any obsevaton x and e, the equaton x h(,) e has a unque soluton n, denoted by () e, then the dstbuton of ( ( E)) s called x x the fducal dstbuton of ( ). In the followng, we wll gve the confdence nteval of the quantle x usng the fducal method. Noted that W X (), V X ( n ) X nx ( ) ( ) () ae comlete suffcent statstcs, and ndeendently dstbuted. By nw ( ) V X ~ (), Y ~ ( ) we have X Y ( WV, ) (, ) n Fo a gven obsevaton ( wv, ) and (), xy, the equaton x y ( wv, ) (, ) n has a unque soluton vx v (, ) ( w, ). ny y Consequently, the fducal dstbuton of the th quantle s vx v F ( x ) P( w log( ) x ) x ny Y Hence, fo gven (0,), the fducal confdence nteval of x s [ x ( / ), x ( / )], whee x ( ) s the 00 quantle of x. Geneally seakng, thee exsts no exlct exesson fo the fducal dstbuton of x, and t s dffcult to fnd a numecal soluton. Howeve, the smulaton method would be
INTERVAL ESTIMATION FOR THE QUANTILE 8 helful to conduct the calculaton of the fducal ntevals. () Fo gven data, set the sze of the smulated samles N lage enough, say N 5000. () Fo,,, N, geneate X and Y fom () and (( )), esectvely. (3) Comute vx v x w, log( ) ny Y (End N loo) Denote x ( ) as the 00 ecentle of { x,, x,,, x, }. Then the fducal N confdence nteval of x s [ x ( / ), x ( / )]. 3. Smulaton Results. Ths secton s devoted to the comason of the bootsta method and fducal method usng numecal smulaton. In geneal, the mutual comason of the above two methods should take nto account the followng oetes: the coveage obabltes (CP) and the exected lengths (EL) of the ntevals. The nfeence ocedues wth lage CP ae desed fstly, and then a shote EL would be consdeed as the ndcaton of moe accuate nteval estmaton. In ode to evaluate the nteval estmaton of the above two methods, we hee aly Monte Calo smulaton to estmate CP and EL. Fo gven and, geneate M samles, comute the bootsta ntevals and fducal ntevals unde the nomnal level usng the elated algothms ut fowad n secton, and fnally calculate the ooton of the M ntevals contanng x and the aveage nteval lengths. In the smulaton ocedue, we set, 4, the samle sze n, the numbe of obseved censoed data 8, the confdence coeffcent 95% and M 3000. The smulaton esults ae shown n Table. TABLE. The smulated CP and EL P Bootsta ntevals Fducal ntevals CP EL CP EL 0.5 0.773.3698 0.947.839 0.5 0.8833.804 0.9407.3308 0.35 0.97.506 0.9477 3.95 0.45 0.9300 3.4505 0.9533 4.3433 0.50 0.940 3.9647 0.9553 5.475 0.55 0.9447 4.637 0.953 5.7954 0.65 0.9457 6.48 0.9480 7.4689 0.75 0.9477 8.573 0.957 0.085 0.85 0.9447.3563 0.9547 4.9056 0.95 0.9583 8.5748 0.9557 4.85
8 JUAN WANG AND XINMIN LI The numecal esults n Table ndcate that the CPs of the fducal ntevals ae close to, and aaently lage than that of the bootsta ntevals fo small. Unde ths condton, the fducal method efoms moe satsfactoly than the bootsta method. When the value of s modeately lage, the CPs of the two knds of ntevals ae close to each othe. When t comes to the ELs of the confdence ntevals, nevetheless, the bootsta method efoms bette. Theefoe, we can conclude that the fducal method would not be affected by the value of, and the bootsta method could be well acceted only wth modeate to lage values of. REFERENCES [] A. P. Dawd and M. Stone (98), The functonal model bass of fducal nfeence, Annals of Statstcs, vol.0,.054 067. [] B. Efon (979), Bootsta methods: Anothe look at the jackknfe, Annals of Statstcs, vol.7,. 6. [3] B. Efon and R. J. Tbshan (993), An Intoducton to the Bootsta, New Yok, Chaman & Hall. [4] J. P. Chen and L. I. Tong (003), Bootsta confdence nteval of the dffeence between two ocess caablty ndces, The Intenatonal Jounal of Advanced Manufactung Technology, vol.,.49-56. [5] J. Hanng, H. Iye and P. Patteson (006), Fducal genealzed confdence ntevals, Jounal of the Amecan Statstcal Assocaton, vol.0,.54 69. [6] J. F. Lawless (98), Statstcal Models and Methods fo Lfetme Data, John Wley & Sons, New Yok. [7] L. J. Ban and M. Engelhadt (99), Statstcal Analyss of Relablty and Lfe-testng Models, nd, Macel Dekke, New Yok. [8] N. Balakshnan and A.P. Basu (995), The Exonental Dstbuton: Theoy, Methods and Alcaton, Godon and Beach Scence Publshes, Langhone, PA. [9] P. Hall (988), Theoetcal comasons of bootsta confdence ntevals, Annals of Statstcs, vol.6,.97 953. [0] R. A. Fshe (930), Invese obablty, Poceedngs of the Cambdge Phlosohcal Socety, vol.6,.58 535. [] T. J. DCcco and B. Efon (996), Bootsta confdence ntevals, Statstcal Scence, vol.,.89 8. [] W. Q. Meeke and L. A. Escoba (998), Statstcal Methods fo Relablty Data, John Wley & Sons, New Yok. [3] W. Nelson (98), Aled Lfe Data Analyss, John Wley & Sons, New Yok. [4] X. L, G. L and X. Xu (005), Fducal ntevals of estcted aametes and the alcatons, Scence n Chna Sees A, vol.48,.567 583.