Estimation and Testing in Type-II Generalized Half Logistic Distribution

Size: px

Start display at page:

Download "Estimation and Testing in Type-II Generalized Half Logistic Distribution"

Brianne Newman
5 years ago
Views:

Joural of Moder Appled Statstcal Methods Volume 13 Issue 1 Artcle 17 5-1-014 Estmato ad Testg Type-II Geeralzed Half Logstc Dstrbuto R R. L. Katam Acharya Nagarjua Uversty, Ida, katam.rrl@gmal.

edu/jmasm Part of the Appled Statstcs Commos, Socal ad Behavoral Sceces Commos, ad the Statstcal Theory Commos Recommeded Ctato Katam, R R. L.; Ramakrsha, V; ad Ravkumar, M S.

1 Joural of Moder Appled Statstcal Methods Volume 13 Issue 1 Artcle Estmato ad Testg Type-II Geeralzed Half Logstc Dstrbuto R R. L. Katam Acharya Nagarjua Uversty, Ida, katam.rrl@gmal.com V Ramakrsha K. L. Uversty, Vaddeswaram, Ida, vramakrsha006@gmal.com M S. Ravkumar Acharya Nagarjua Uversty, Ida, msrk.raama@gmal.com Follow ths ad addtoal works at: Part of the Appled Statstcs Commos, Socal ad Behavoral Sceces Commos, ad the Statstcal Theory Commos Recommeded Ctato Katam, R R. L.; Ramakrsha, V; ad Ravkumar, M S. (014) "Estmato ad Testg Type-II Geeralzed Half Logstc Dstrbuto," Joural of Moder Appled Statstcal Methods: Vol. 13 : Iss. 1, Artcle 17. DOI: 10.37/jmasm/ Avalable at: Ths Regular Artcle s brought to you for free ad ope access by the Ope Access Jourals at DgtalCommos@WayeState. It has bee accepted for cluso Joural of Moder Appled Statstcal Methods by a authorzed edtor of DgtalCommos@WayeState.

2 Joural of Moder Appled Statstcal Methods May 014, Vol. 13, No. 1, Copyrght 014 JMASM, Ic. ISSN Estmato ad Testg Type-II Geeralzed Half Logstc Dstrbuto R. R. L. Katam Acharya Nagarjua Uversty Nagarjuaagar, Ida V. Ramakrsha K. L. Uversty Vaddeswaram, Ida M. S. Ravkumar Acharya Nagarjua Uversty Nagarjuaagar, Ida A geeralzato of the Half Logstc Dstrbuto s developed through expoetato of ts survval fucto ad amed the Type II Geeralzed Half Logstc Dstrbuto (GHLD). The dstrbutoal characterstcs are preseted ad estmato of ts parameters usg maxmum lkelhood ad modfed maxmum lkelhood methods s studed wth comparsos. Dscrmato betwee Type II GHLD ad expoetal dstrbuto pars s coducted va lkelhood rato crtero. Keywords: Geeralzed Half Logstc Dstrbuto (GHLD), maxmum lkelhood estmato (MLE), modfed maxmum lkelhood estmato (MMLE), mea square error (MSE), lkelhood rato type crtero, percetles, power of the test Itroducto I lfe testg ad relablty studes a combato of mootoe ad costat falure rates over varous segmets of the rage of lfetme of a radom varable s also kow as bath tub or o-mootoe falure rate. I bologcal ad egeerg sceces, stuatos of o-mootoe falure rates are commo (see Rajarsh & Rajarsh (1988) for a comprehesve arrato of these models). Mudholkar, et al. (1995) preseted a exteso of the Webull famly that cotas umodel dstrbutos wth bathtub falure rates ad also allows for a broader class of mootoe hazard rates. They amed ther exteded verso the Expoetated Webull Famly. Gupta ad Kudu (1999) proposed a ew model called the geeralzed expoetal dstrbuto. If s a postve real umber ad F(x) s the cumulatve Dr. R. R. L. Katam s a Professor the Departmet of Statstcs. Emal hm at katam.rrl@gmal.com. V. Ramakrsha s a Assocate Professor the Departmet of Computer Scece ad Egeerg. Emal hm at: vramakrsha006@gmal.com. M. S. Ravkumar s a UGC Research Fellow the Departmet of Statstcs. Emal hm at: msrk.raama@gmal.com. 67

3 ESTIMATION AND TESTING IN TYPE II GHLD dstrbuto fucto (cdf) of a cotuous postve radom varable, the [F(x)] θ ad the correspodg probablty dstrbuto may be termed a expoetated or geeralzed verso of F(x). A half logstc model obtaed as the dstrbuto of absolute stadard logstc varate s a probablty model of recet org (Balakrsha, 1985). Its stadard probablty desty fucto, cumulatve dstrbuto fucto ad hazard fuctos are gve by: x e f ( x), x 0 x (1 e ) (1) x 1 e F( x), x 0 x 1 e () x 1 e F( x), x 0 x 1 e. (3) Katam et al. (011) adopted ths geeralzato to the well-kow half logstc dstrbuto, ad amed t the Type-I Geeralzed Half Logstc Dstrbuto (GHLD). Cosder a seres system of compoets wth dvdually ad detcally dstrbuted (d) dvdual lfetmes, for example, F(x). The relablty fucto of such a system s gve by [1 F(x)] θ ; hece, the dstrbuto fucto of the lfetme radom varable of a seres system s 1 [1 F(x)] θ. Takg F(x) as the half logstc model gve by Equato (), the correspodg dstrbuto s termed the Type-II Geeralzed Half Logstc Dstrbuto (GHLD-II). Its pdf, cdf ad hazard fucto are gve by: x ( e ) f ( x), x 0, 0 (4) x 1 (1 e ) x e F( x) 1, x 0, 0 x 1 e (5) h( x), x 0, 0. (6) x 1 e 68

4 KANTAM ET AL Balakrsha ad Sadhu (1995) suggested a ew probablty model wth a stadard pdf ad cdf gve by: (1/ k ) 1 (1 kx) 1 f ( x),0 x, k 0 1/ k [1 (1 kx) ] k (7) 1/ k 1 (1 kx) 1 F( x),0 x, k 0. (8) 1/ k 1 (1 kx) k The lmts of (7) ad (8) as k are respectvely (1) ad () the pdf ad cdf of HLD. Balakrsha ad Sadhu (1995) called the dstrbuto (7) ad (8) Geeralzed HLD. Olapade (008) cosdered two dstrbutos ad dscussed ther dstrbutoal propertes, order statstcs samples from these dstrbutos: He amed these dstrbutos type-i ad type-iii GHLD, respectvely. The types of geeralzed HLD of Olapade (008) are through trucato of the type-i ad type- III geeralzed logstc dstrbutos from Balakrsha ad Leug (1988) at the org. Thus, ths type-ii GHLD s coceptually dfferet from the GHLDs of Balakrsha ad Sadhu (1995) ad Olapade (008). Hece, the proposed models motvated a separate research study. Estmato Type-II Geeralzed Half Logstc Dstrbuto (GHLD-II) The probablty desty fucto ad dstrbuto fucto of GHLD-II wth scale parameter ad power parameter are gve by: x/ ( e ) f( x), 0 x, 0, q 0 (9) x/ 1 ( 1 e ) x/ e F( x) 1, 0 x, 0, x/ 1 e q 0. (10) 69

5 ESTIMATION AND TESTING IN TYPE II GHLD Let x1 < x < < x be a ordered sample of sze from GHLD-II. The log lkelhood fucto of the sample s by x x/ x/ log L (log log ) log log(1 e ) log(1 e ) 1 The log lkelhood equatos to estmate the parameters ad are gve log L log L 0, 0, x / log L x e 0 x / 1 1e (11) log L 0 x / log(1 e ) x log (1) It ca be see that these two equatos must be solved teratvely for ad for a gve sample. The asymptotc varaces ad covaraces of MLEs of ad ca be obtaed by vertg the formato matrx whose elemets are the mathematcal expectato of the followg expressos: x/ x/ log L x (1 e ) e 4 x / 1 (1 e ) (13) log L (14) log L 1 x x / 1 (1 e ) (15) 70

6 KANTAM ET AL These equatos, evaluated at estmates of ad, provde am estmated dsperso matrx. I order to obta a aalytcal estmator for, ts estmatg equato s approxmated by some admssble expresso. Equato (11) to get MLE of, after smplfcato would become z z( e ) x 0 where z z (16) 1 e 1 To obta the aalytcal expresso for, approxmate the followg expresso (16) by some lear fucto the correspodg populato quartle. Let, approxmate Gz ( ) z z ( e ) z (1 e ) (17) G( z ) z (18) where, are to be sutably foud. After usg ths approxmato (16) the soluto for s x 1 ˆ 1 (19) Ths estmator s amed the MMLE of, whch s a lear estmator x s * To obta,, let p ; 1,,..., ad let t, t be the solutos of 1 equatos: pq ' F( t ) p j p (for example) (0) * pq " F( t ) p p (for example), where q 1 p (1) 71

7 ESTIMATION AND TESTING IN TYPE II GHLD where F(.) s cdf of GHLD-II. The tercept ad slope of lear approxmato the Equato (18) are respectvely gve by G t * ( ) G( t) * t t () G(t ) t. (3) * * gve by Usg dstrbuto fucto F(.) of GHLD-II, the expressos for t, t are * t ' 1/ (1 p ) t log, ' 1/ t (1 pt ) * " 1/ (1 pt ) log " 1/. (1 pt ) Table 1 shows the values of α, β for varous ө ad. The MMLE of σ ca be show to be equvalet to the exact MLE wth respect to the asymptotc varace. Ther performace small samples s also studed through smulato because the exact MLE s a teratve soluto. The emprcal sample characterstcs are gve Table, whch dcates the followg: 1. The emprcal sample characterstcs bas, varace ad MSE decrease as sample sze creases.. MMLE s geerally more based tha MLE; wth referece to varace as well as MSE, MMLE s better tha MLE for small samples. 7

8 KANTAM ET AL Table 1. Itercept ad Slope of the Approxmato G(Z ) = α + β z (GHLD II) θ = θ = 3 θ = 4 α β α β α β

9 ESTIMATION AND TESTING IN TYPE II GHLD Table 1, cotued θ = θ = 3 θ = 4 α β α β α β Table. Emprcal Sample Characterstcs (Type-II GHLD) Bas Varace MSE θ MLE MMLE MLE MMLE MLE MMLE

10 KANTAM ET AL GHLD-II vs. Expoetal Model The dscrmato betwee GHLD-II ad the expoetal model s made usg the lkelhood rato (LR) crtero. Specfy GHLD-II as ull populato (P0) ad the expoetal model as alteratve populato (P1). A ull hypothess s proposed as H0: a gve sample belogs to GHLD-II (P0) versus a alteratve hypothess H1: the sample belogs to the populato Expoetal model (P1). Let L1, L0, respectvely, stad for the lkelhood fucto of a sample wth populato P1 ad P0. The percetles of the LR crtero L1/L0 are obtaed by smulato as: 10,000 radom samples of szes = 5, 10, 15, 0 are geerated from the ull populato P0 ad ts parameters are estmated usg each sample. The value of the lkelhood fucto of the ull populato s computed at the geerated sample observatos ad the correspodg parameter estmates; ths value s deoted by L0. Usg the same sample, geerated from P0, the parameters ad lkelhood fucto value of the alteratve populato are calculated, for example, L1. The values of L1/L0 over 10,000 rus are sorted ad selected percetles are detfed for a gve, θ (see Table 3). Table 3. Percetles of L1/L0 (P0 : GHLD-II, P1: Expoetal) θ \ p The etres uder the colum headgs 0.95 Table 3 may be take as 5% level of sgfcace crtcal values for dscrmatg betwee the GHLD-II ad expoetal models. The powers of the test statstc L1/L0 are also evaluated through smulato by calculatg L1/L0 wth samples geerated from expoetal 75

11 ESTIMATION AND TESTING IN TYPE II GHLD populato (P1) ad estmatg, the parameters calculatg the values of the lkelhood fuctos L1, L0 wth sample from P1. The proporto of L1/L0 values fallg above 95 th percetle of L1/L0 would become the power of the LR test crtero (see Table 4). It s observed that the dscrmato betwee GHLD-II ad expoetal models falls wth creased sample sze, dcatg less dstgushablty betwee the expoetal model ad GHLD-II. Table 4. Powers of LR Test Crtero at α = 0.05 θ \ Dstrbutos GHLD-II vs. Expoetal Refereces Balakrsha, N. (1985). Order statstcs from the half logstc dstrbuto. Joural of Statstcal Computato ad Smulato, 0: Balakrsha, N., & Leug, M. Y. (1988). Order statstcs from type-i geeralzed logstc dstrbuto. Commucato Statstcs Smulato ad Computg, 17(1): Balakrsha, N., & Sadhu, R. A. (1995). Recurrece relatos for sgle ad product momets of order Statstcs from a geeralzed half logstc dstrbuto wth applcatos to ferece. Joural of Statstcal Computato ad Smulato, 5(4): Gupta, R. D., & Kudu, D. (1999). Geeralzed expoetal dstrbutos. Australa ad New Zealad Joural of Statstcs, 41:

12 KANTAM ET AL Katam, R. R. L., & Srvasa Rao, G. (1993). Relablty estmato Raylegh dstrbuto wth cesorg some approxmatos to ML Method. Proceedgs of II Aual Coferece of Socety for Developmet of Statstcs, Acharya Nagarjua Uversty: Katam, R. R. L., & Srvasa Rao, G. (00). Log-logstc dstrbuto: Modfed Maxmum lkelhood estmato. Gujarat Statstcal Revew, 9(1-): Katam, R. R. L., ad Srram, B. (003). Maxmum lkelhood estmato from cesored samples: Some modfcatos legth based verso of expoetal model. Statstcal methods, 5(1): Mehrotra, K. G., & Nada, P. (1974). Ubased estmato of parameters by order statstcs the case of cesored samples. Bometrka, 61: Mudholkar, G. S., Srvastava, D., & Fremer, M. (1995). Expoetated Webull famly: A reaalyss of the bus-motor falure data. Techometrcs, 37(4): Olapade, A. K. (008). O Type III Geeralzed Half Logstc Dstrbuto. arxv: v1 [math.st] 10 Ju 008. Pearso, T., & Rootze, H. (1977). Smple hghly effcet estmators for a type-i Cesored Normal sample, Bometrka, 64: Rajarsh, S., & Rajarsh, M. B. (1988). Bathtub dstrbutos: A revew. Commucato Statstcs Theory & Methods, 17: Rosaah, K., Katam, R. R. L., & Narasmham, V. L. (1993a). ML ad Modfed ML Estmato gamma dstrbuto wth kow pror relato amog the Parameters. Paksta Joural of Statstcs, 9(3)B: Rosaah, K., Katam, R. R L., & Narasmham, V. L. (1993b). O modfed maxmum lkelhood estmato of gamma parameters. Joural of Statstcal Research, 7(1-): Tku, M. L. (1967). Estmatg the mea ad stadard devato from a cesored Normal sample. Bometrka, 54: Tku, M. L., & Suresh, R. P. (199). A ew method of estmato for locato ad scale parameters. Joural of Statstcal Plag ad Iferece, 30:

Comparing Different Estimators of three Parameters for Transmuted Weibull Distribution

Comparing Different Estimators of three Parameters for Transmuted Weibull Distribution Global Joural of Pure ad Appled Mathematcs. ISSN 0973-768 Volume 3, Number 9 (207), pp. 55-528 Research Ida Publcatos http://www.rpublcato.com Comparg Dfferet Estmators of three Parameters for Trasmuted