Comparison of some one sample confidence intervals for estimating the mean of the weibull distribution INTRODUCTION.

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Intenationl Reseach Jounal of Agicultual Economics and Statistics Volume 3 Issue 1 Mach, 1 9-34 Reseach Pape Compaison of some one sample confidence intevals fo estimating the mean of the weiull

8.11; Revised : 5.11.11; Accepted: 1.1.11 ABSTRACT : In this aticle, an attempt has een made to eview eisting inteval estimatos fo Weiull mean and compae them unde the same simulation condition.

1 Intenationl Reseach Jounal of Agicultual Economics and Statistics Volume 3 Issue 1 Mach, Reseach Pape Compaison of some one sample confidence intevals fo estimating the mean of the weiull distiution H. V. KULKARNI AND S. K. POWAR See end of the pape fo authos affiliations Coespondence to : H.V. KULKARNI, Depatment of Statistics, Shivaji Univesity, KOLHAPUR (M.S.) INDIA Pape Histoy : Received : ; Revised : ; Accepted: ABSTRACT : In this aticle, an attempt has een made to eview eisting inteval estimatos fo Weiull mean and compae them unde the same simulation condition. A compaison of the pefomance of the CI estimatos has een made with espect to simulated aveage widths and aveage coveage poailities. A compaative study evealed that the pefomances of the estimatos diffe significantly when the sample sizes ae small and shape paamete is small i e. data ae fom a highly skewed Weiull distiution. Results ae illustated with a eal data set fo pactitiones. Based on the simulation study, good inteval estimatos have een ecommended fo use in pactice. KEY WORDS : Confidence inteval, Weiull mean, Compaative study HOW TO CITE THIS PAPER : Kulkani, H.V. and Powa, S.K. (1). Compaison of some one sample confidence intevals fo estimating the mean of the weiull distiution, Intenat. Res. J. agic. Eco. & Stat., 3 (1) : INTRODUCTION Weiull distiution is widely used in eliaility and suvival analysis due to its fleile shape and aility to model a wide ange of failue ates. It can e deived theoetically as a fom of eteme value distiution, govening the time to occuence of the weakest link of many competing failue pocesses. Its special case with shape paamete = is the Rayleigh distiution which is commonly used fo modeling the magnitude of adial eo when and y coodinate eos ae independent nomal vaiales with zeo mean and the same standad deviation, while the case = 1 coesponds to the widely used eponential distiution. Let X follows a weiull distiution with scale paamete a and shape paamete. The poaility density function (pdf) of X is given y: f X ( ; a, ) = a a 1 ep a ; >, a >, > ; ; othewise The mean of the Weiull distiution is given y µ = a Γ ( 1 + (1/ )) whee, Γ( k 1 dm k ) = m ep( m) is the Gamma function. The polem of inteval estimation fo the mean m when oth paametes a and ae unknown has een less attended in the liteatue. Colosimo and Ho (1999) otained CI fo m using asymptotic-nomal theoy and it is usually called Wald CI. Yang et al. (7) poposed CI fo Weiull mean fom type-ii censoed data ased on a Chi-squae distiuted pivotal quantity given y Lawless (198), involving modified maimum likelihood estimato (MMLE) of in the esulting CI and is efeed to as naïve CI. An adjustment is suggested to the Chisquae quantile fo otaining moe efined esults. Kishnamoothy et al. (9) utilized genealized vaiale (GV) appoach fo developing confidence limits fo Weiull mean µ and studied coveages of the coesponding one sided confidence limits and compaed with coveages of the Wald confidence limits ased on log tansfomed vaiale. In this pape, we eviewed and compaed the CIs poposed

2 H. V. KULKARNI AND S. K. POWAR y diffeent eseaches at diffeent times and found good intevals fo estimating the mean of the Weiull distiution. Since a theoetical compaison is not tactale, a simulation study has een made to compae the pefomance of the estimatos. A compaison was made with espect to oth simulated aveage widths and aveage coveage poailities. The aticle is oganized as follows : Section povides a ief eview of the eisting methods in the liteatue fo CI estimation of the Weiull mean. Section 3 pesents the compaison of the CIs ased on the simulated aveage widths and the aveage coveage poailities. Section 4 illustates the methods using a eal data set. Section 5 povides oveall conclusions and ecommendations. MATERIALS AND METHODS Eisting methods : In this section, we eviewed eisting methods fo CI estimation of the Weiull mean. Fo convenience, methods discussed in this section ae aeviated and the aeviations ae given in the paentheses following the name of each method. Wald CI ased on log tansfomed vaiale () : Based on the log-tansfomed vaiale, the test statistic W = log( ) µ log( µ ) / va log( µ ) can e used to otain the CI fo the mean of the Weiull ( a, ) distiution, whee, µ = a Γ (1 + (1/ )) â is maimum likelihood estimato (MLE) of ( a, is MLE of and va[µ ] is estimated using delta method as, va[ µ ] µ a va[ ] µ = a, µ,, va[ µ + a] a a a, + côv[ a, ] a...(1) The fomulae fo v a [a ], va[ ] and côv[ a, ] given y Cohen (1965) ae used with appopiate modifications to adopt fo the fom of density function used in this wok. Employing the appoimation va[log( )] µ va[ ]/ µ µ and asymptotic nomality of the MLEs, the esulting CI is given y : ep log( µ ) ± Ζ 1 α / va[log( µ )] whee, Ζα is the α th quantile of standad nomal distiution. Kishnamoothy et al. (9) studied coveages of the one sided confidence limits of aove CI at 5 pe cent level of significance. Wald CI poposed y Colosimo and Ho (WALD-CH) : Colosimo and Ho (1999) poposed the following CI fo the mean µ of the Weiull ( a, ) distiution, µ ± Ζ1 α / va[ ] µ whee, v a [µ ] is otained using equation (1). Naive CI ased on modified MLE (NMMLE) : Yang et al. (7) consideed type II censoed life testing epeiment and used a pivotal quantity T ( ) = S ( ) / a to otain CI fo µ whee, S ( ) = n i + ( ) and = 1,,..., n is the level of censoing. Based on the fact that T() follows a Chi-squae distiution with degees pf feedom (df) (see Lowless (198)), fo known value of an eact CI fo µ is given y: Γ(1+ 1/ ) 1/ 1/ S( ) S( ), Γ(1 + 1/ χ,1 / χ () α, α / whee, χ v,α is the α th quantile of Chi-squae distiution with v df. When is unkown, it is eplaced y its appopiate estimato in equation (). Authos efeed to this CI as a naive CI and ased on compaative study etween MLE and MMLE, suggested using MMLE, ~, fo, which is a solution to the equation: i log( i ) i ) 1 + log( i ) = 3 Intenat. Res. J. agic. Eco.& Stat. 3(1)Mach, 1: 9-34

3 COMPARISON OF SOME ONE SAMPLE CONFIDENCE INTERVALS FOR ESTIMATING THE MEAN OF THE WEIBULL DISTRIBUTION whee, = + ( n ) i i. In this case, fo otaining moe efined esults, authos suggested an adjustment to Chi-squaed quantile as : χ v, α = cχv, α ( c 1) whee, c = 1+ τ k p log( log(1 α)) / p is the adjustment facto and χ v, α = cχv, α ( c 1) whee, c = 1+ τ k p log( log(1 α )) / p is the adjustment facto and p / n. Hee, τ and k ae functions of p and authos have taulated thei values fo p =.1,.,...,1 ased on a simulation study. Fo uncensoed data, p = 1 fo which τ = and k = CI ased on GV appoach () : Kishnamoothy et al. (9) have used the GV appoach developed y Tsui and Weeahandi (1989) and Weeahandi (1993) fo constuction of CI fo Weiull mean. Hee, authos otained the genealized pivotal quantity (GPQ) sepaately fo the paametes a and as: ( / ) G 1/ a a = a and G = / espectively. Hee â and ae MLEs of a and espectively ased on the oseved sample and â and ae MLEs ased on a andom sample of size n fom Weiull (1,1) distiution. Then GPQ fo the mean µ = a Γ ( 1 + (1 / )) is given y : ( / ) G G ( 1 1/ G ) 1/ a a (1 / a Γ + µ = Γ + = ). Authos suggested to geneate a desied nume say N=1 of independent copies of G µ y plugging in the simulated values of â and ased on N independent andom samples each of size n fom Weiull (1,1) distiution fo given â and. Then α /th and ( 1 α /) th pecentiles of the esultant sample of G µ can e used as CI fo µ. RESULTS AND DATA ANALYSIS The esults ae summaized elow accoding to ojectives of the study: Compaison : Since a theoetical compaison of the estimatos is not possile, a simulation study was conducted to compae the pefomance of the inteval estimatos. The CIs ae compaed with espect to thei simulated aveage coveage poailities and aveage widths ased on the following simulation pocedue. Simulation pocedue : Fo vaious fied sample sizes n =1,5, 45, level of significance α =. 5, shape paamete =.5,1,, 3 fo which skewness of the undelying distiution vaies fom high to modeate and scale paamete a =.5,1, 5; 5 simulated samples ae geneated fom Weiull ( a, )distiution. Fo each of these 5 samples, lowe and uppe confidence limits L i and U i ;,,..., 5 ae otained fo the population mean µ fo each of the aove fou CIs. It is noted that fo method, the computation of the CI ased on each of the single simulated 5 samples needs a futhe set of N (hee we have taken N=1) samples of the same size fom Weiull (1,1) distiution to geneate independent copies of Gµ as descied in section.4. The simulated aveage width of each CI is otained y aveaging the 5 quantities U i L i ;,,..., 5. The simulated aveage coveage poaility is the popotion of these 5 intevals that coveed the value of the mean. Simulated aveage coveage poailities and aveage widths (enclosed in paentheses) fo all fou CIs descied in pevious section fo aove mentioned sample sizes ae epoted in Tale 1, fo unknown scale and shape paametes a,. To enale a visual display and compaison among the methods, o plots of the aveage coveage poailities and aveage widths ove the set A =.5,1,, 3 fo sample sizes ( a, ) : a =.5,1,5, = n =1,5,45 ae plotted in Fig. 1. When the shape paamete is known, all CIs pefom ecellent with espect to coveage poailities. The coveages ae almost close to nominal level.95 unifomly fo all a and mentioned aove. Since all methods ehave almost unifomly, these esults ae not displayed hee. Intenat. Res. J. agic. Eco. & Stat. 3(1) Mach, 1:

4 H. V. KULKARNI AND S. K. POWAR Results of the simulation study : The following pominent facts ae clealy visile fom the Tale 1 and Fig. 1: Coveage poailities of each CI inceased as sample size and shape paamete inceased and ae invaiant of scale paamete. Aveage widths of each CI deceased as oth sample size and shape paamete inceased and inceased as scale paamete inceased. When the shape paamete is unknown, ased CI outpefoms the othe thee CIs almost unifomly, with espect to coveage poailities, and moe pominently fo small shape paamete. The coveage poailities ae close to the nominal size.95. The aveage widths of NMMLE ased CI wee makedly smalle than its good competito, fo small shape paamete and small sample size. Fo lage shape paamete whee the Weiull distiution is nealy symmetic and fo lage sample size, the pefomance of NMMLE and ased CIs wee almost equivalent with espect to oth aveage coveage poaility and aveage width. The computational pocedue fo NMMLE ased CI is simple than that fo ased CI. Aveage widths of Wald CIs wee smalle, howeve the coesponding coveage poailities wee vey poo and wee elow the nominal size.95. Based on aove osevations, we ecommend ased Tale 1 : Pecentage simulated coveage poailities and aveage widths (enclosed in paentheses) of 95% CI fo Weiull mean fo vaious sample sizes using, WALD-CH, NMMLE, and methods* n=1 n=5 n=45 a (4.58) WALD-CH (.88) NMMLE 93.5 (16.56) (59.1) 89.4 (.61) (.58) 93.9 (.87) (1.4) 9.5 (.7) 89.8 (.6) 94.1 (.33) 94.6 (.36) 9.6 (.19) (.18) 94. (.3) (.3) 9.4 (1.96) 86.4 (1.73) 94.8 (.61) (3.4) 9.86 (.39) 91.4 (.38) (.44) 95.4 (.47) (.18) (.18) (.19) 94.4 (.19) (.1) 9.9 (.1) 94.1 (.13) (.13) (1.37) 89.8 (1.8) (1.57) (1.7) 93.8 (.9) 9.76 (.9) 95. (.31) 95.1 (.3) (.13) (.13) (.14) (.14) (.9) 94.8 (.9) (.1) (.1) (9.15) (1.3) (.54) (.38) (3.9) (.78) (.35) (.5) (.74) (.58) (.7) (.19) WALD-CH (5.73) NMMLE (1.15) 93.8 (.53) 93.7 (.37) (3.48) 94.8 (.76) (.35) (.5) 95.4 (.54) 94.4 (.58) (.7) (.19) 94.8 (38.5) (1.74) (.66) (.45) (5.1) (.88) (.38) (.7) (3.16) (.6) (.8) (.19) (151.54) (.41) (.71) (.46) (6.56) (.94) (.38) (.6) (3.47) (.63) (.8) (.19) (44.4) (6.) (.66) (1.86) (19.8) (3.93) (1.77) (1.3) (13.6) (.9) (1.33) (.93) WALD-CH (9.11) NMMLE (5.84) (.65) (1.85) 94.3 (17.5) (3.83) (1.75) 94.6 (1.3) 94.6 (1.73) (.88) 94.7 (1.33) 95. (.93) (1.54) (8.57) (3.7) (.6) (5.81) (4.36) (1.9) (1.3) (15.64) (3.9) (1.39) (.97) (591.4) (1.66) (3.6) (.7) (3.89) (4.7) (1.9) (1.3) (17.3) (3.18) (1.39) (.96) 3 *Results ae otained fo unknown scale and shape paametes a,. Intenat. Res. J. agic. Eco.& Stat. 3(1)Mach, 1: 9-34

5 COMPARISON OF SOME ONE SAMPLE CONFIDENCE INTERVALS FOR ESTIMATING THE MEAN OF THE WEIBULL DISTRIBUTION Aveage Coveage 9 Aveage Width Aveage Coveage 9 Aveage Width Aveage Coveage 9 Aveage Width Fig. 1: Bo plots of simulated aveage coveage and aveage width fo the methods, WALD-CH, NMMLE, and fo vaious sample sizes (n) ove the ange of a=.5,1,5 and =.5,1,,3. Intenat. Res. J. agic. Eco. & Stat. 3(1) Mach, 1:

6 H. V. KULKARNI AND S. K. POWAR CI fo all sample sizes even fo small shape paametes. Fo lage shape paametes, NMMLE ased CI also can e used in pactice. Illustative eample : In this section we illustate the methods discussed in section with a eal data set. The data used hee ae the esults of an epeiment etacted fom Nelson (197). In this epeiment, 11 specimens of a paticula type of electical insulating fluid ae sujected to a constant voltage stess of 3 kilovats. The data set epesenting the failue time of each specimen is : 17.5,.66, 1., , 139.7, 144.1,.46, 43.4, 194.9, 47.3, The Kolmogoov-Sminov test to aove data set fo fitting Weiull distiution esulted p- value(-tail).61 indicating that Weiull is a good model fo ~ = aove data set. Hee a = = , sample mean is and sample standad deviation is The MLE of shape paamete,, indicates that the aove data set is faily skewed. The 95 pe cent CIs fo the mean time to failue (MTTF) and the coesponding CI widths (enclosed in paentheses) using, WALD-CH, NMMLE, and methods ae, espectively [43.39,13.6] (89.3), [33.48,118.3] (84.75), [43.93, 159.9] (115.97) and [45.11, ] (153.48). It seems that WALD-CH ased CI has the naowest width and the ased CI has the widest width. Oveall conclusion : The genealized vaiale appoach ased CI ehiits makedly well pefomance even fo small sample sizes fo almost all paamete cominations consideed in the simulation study fo inteval estimation of Weiull mean. Fo lage shape paametes and lage sample sizes, NMMLE and ased CIs pefom almost simila. The Wald ased CIs ae not ecommended due to thei poo coveage poailities. Authos affiliations: S.K. POWAR, Depatment of Statistics, Shivaji Univesity, KOLHAPUR (M.S.) INDIA sajeao.powa@ ediffmail.com LITERATURE CITED Cohen, A.C. (1965). Maimum likelihood estimation in the Weiull distiution ased on complete and on censoed samples. Technometics, 7: Colosimo, E.A. and Ho, L.L. (1999). Pactical appoach to inteval estimation fo the Weiull mean lifetime. Quality Engineeing, 1: Kishnamoothy,K., Lin, Y. and Xia,Y. (9). Confidence limits and pediction limits fo a Weiull distiution ased on the genealized vaiale appoach. J. Statistical Planning & Infeence, 139: Lawless,J.F.(198).Statistical models and methods fo lifetime data. John Wiley and Sons, New Yok. Nelson,W.B. (197). Gaphical analysis of acceleated life test data with the invese powe law model. IEEE Tansactions on Reliaility, R1: -11. Tsui, K.W. and Weeahandi, S. (1989). Genealized p-values in significance testing of hypotheses in the pesence of nuisance paametes. J. Ameican Statistical Assoc., 84: Weeahandi,S. (1993). Genealized confidence intevals. J. Ameican Statistical Assoc., 88: Yang,Z.L., Xie, M. and Wong, A.C.M. (7). A unified confidence inteval fo eliaility elated quantities of two-paamete Weiull distiution. J. Statistical Computation & Simulation, 77: * * * * * * * * 34 Intenat. Res. J. agic. Eco.& Stat. 3(1)Mach, 1: 9-34

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