Journal of Modern Applied Statistical Metods Volue 14 Issue 1 Article 19 5-1-015 Estiation for te Paraeters of te Exponentiated Exponential Distribution Using a Median Ranked Set Sapling Monjed H. Sau Palestine Polytecnic University, onjedsau@ppu.edu Areen Qtait Palestine Polytecnic University, areen_cutegirl@otail.co Follow tis and additional works at: ttp://digitalcoons.wayne.edu/jas Part of te Applied Statistics Coons, Social and Beavioral Sciences Coons, and te Statistical Teory Coons Recoended Citation Sau, Monjed H. and Qtait, Areen (015) "Estiation for te Paraeters of te Exponentiated Exponential Distribution Using a Median Ranked Set Sapling," Journal of Modern Applied Statistical Metods: Vol. 14 : Iss. 1, Article 19. DOI:.37/jas/1430453880 Available at: ttp://digitalcoons.wayne.edu/jas/vol14/iss1/19 Tis Regular Article is brougt to you for free and open access by te Open Access Journals at DigitalCoons@WayneState. It as been accepted for inclusion in Journal of Modern Applied Statistical Metods by an autorized editor of DigitalCoons@WayneState.
Journal of Modern Applied Statistical Metods May 015, Vol. 14, No. 1, 15-37. Copyrigt 015 JMASM, Inc. ISSN 1538 947 Estiation for te Paraeters of te Exponentiated Exponential Distribution Using a Median Ranked Set Sapling Monjed H. Sau Palestine Polytecnic University Palestinian Territories Areen Qtait Palestine Polytecnic University Palestinian Territories Te etod of axiu likeliood estiation based on Median Ranked Set Sapling (MRSS) was used to estiate te sape and scale paraeters of te Exponentiated Exponential Distribution (EED). Tey were copared wit te conventional estiators. Te relative efficiency was used for coparison. Te aount of inforation (in Fiser's sense) available fro te MRSS about te paraeters of te EED were be evaluated. Confidence intervals for te paraeters were constructed using MRSS. Keywords: Exponentiated exponential distribution; Fiser inforation; Maxiu likeliood estiation; Median ranked set sapling; Ranked set sapling Introduction One of te ost coon approaces of data collection is tat of a siple rando saple (). Oter ore structured sapling designs, suc as stratified sapling or probability sapling, are also available to elp ake sure tat te obtained data collection provides a good representation of te population of interest. Any suc additional structure of tis type revolves around ow te saple data teselves sould be collected in order to provide an inforative iage of te larger population. Wit any of tese approaces, once te saple ites ave been cosen, te desired easureents are collected fro eac of te selected ites. Many efforts are ade to develop statistical tecniques for data collection tat generally leads to ore representative saples (saples wose caracteristics accurately reflect tose of te underlying population). To tis end, ranked set sapling and soe of its variations were developed. Dr. Sau is Assistant Professor of Statistics in te College of Applied Sciences. Eail i at onjedsau@ppu.edu. 15
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS In tis section, te ranked set sapling (RSS) and edian ranked set sapling (MRSS) are presented. Te exponentiated exponential distribution (EED) and its properties are also discussed. Ranked set sapling RSS was proposed by McIntyre (195) to estiate a pasture yield in Australia. Tis etod was not used for a long tie, but in te last 30 years a lot of researc work was done using tis etod, wic as becoe very iportant in different aspects. and RSS are bot independent, but tey differ in several ways, like: 1. RSS is ore efficient tan wit te sae nuber of easured eleents.. Developent of RSS procedure is ore difficult tan tat of. 3. In, just eleents are needed but in RSS eleents are cosen out of to acieve te desired saple. Also stratified rando sapling and RSS are different in soe tings like: 1. In stratified sapling we liited wit no ore six strata but in RSS we are not restricted ourselves wit te nuber of sets.. In bot of te is used but in RSS ordering te eleents in eac set is needed before selecting te saple. RSS as a etod used basically for infinite population were te set of sapling units drawn fro a population can ranked in a ceap way wic is not costly and/or tie consuing. Te steps of coosing RSS are as follows: 1. Randoly select sets eac of size eleents fro te population under study.. Te eleents for eac set in Step (1) are ranked visually or by any negligible cost etod tat does not require actual easureents. 3. Select and quantify te i t iniu fro te i t set, i = 1,,, to get a new set of size, wic is called te ranked set saple. 4. Repeat Steps (1) (3) ties (cycles) until obtaining a saple of size n =. 16
SAMUH & QTAIT Figure 1 illustrates te procedure of RSS in ters of atrices. Let Y i = { (ii) ; i = 1,, }; tat is, te obtained RSS, { (11), (),, () }, is denoted by {Y 1, Y,, Y }. If te process is repeated cycles, ten te RSS can be represented as a atrix of size n = as it is sown in Step 4 of Figure 1. Step 1: Step : 11 1 1 11 1 1 1 1 1 1 Step 3:, 11,, Step 4: Y11 Y1 Y1 Y1 Y Y 1 Figure 1. Ranked set sapling procedure RSS as a etod is applicable were ranking and sapling units is uc ceaper tan te easureent of te variable of interest. In particular RSS can be used in te following situation: 1. Ranking units in a set can be done easily by judgent in te variable of interest troug visual inspection or wit te elp of certain auxiliary eans.. If tere is a concoitant variable can be obtained easily (concoitant is a variable wic is not of ajor concern but are correlated wit te variable of interest). Median ranked set sapling MRSS was suggested by Muttlak (1997) as a etod to estiate te population ean instead of RSS to reduce te errors, and increase te efficiency over RSS and. It is described by te following steps: 17
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS 1. Randoly select saple units fro te target population.. Allocate te units into sets eac of size, and rank te units witin eac set. 3. Fro eac set in Step (), if te saple size is odd select fro eac set te 1 t sallest rank unit i.e te edian of eac set. Wile if te saple size is even select fro te first sets te sallest rank unit and fro te second sets take te sallest rank unit. Tis step yields saple eleents wic is te edian RSS. 4. Repeat Steps (1) (3) ties (cycles) until obtaining a saple of size n =. Figure illustrates te procedure of MRSS wen = 4 in ters of atrices. Let us denote te MRSS,, 1 33, 43 by Y, Y, Y, Y 1 3 4 t t. If te process is repeated cycles, ten te RSS can be represented as a atrix of size n = 4 as it is sown in Step 4 of Figure. Step 1: Step : 11 1 13 14 11 1 13 14 1 3 4 1 3 4 31 3 33 34 31 3 33 34 41 4 43 44 41 4 43 44 Step 3:,,, Step 4: 1 33 44 Y Y Y Y Y Y Y Y Y Y Y Y 11 1 13 14 1 3 4 1 3 4 Figure. Median ranked set sapling procedure 18
SAMUH & QTAIT Te exponentiated exponential distribution Te exponentiated exponential distribution (EED) introduced by Gupta and Kundu (1999) as a generalization of te exponential distribution. It is of great interest and is popularly used in analyzing lifetie or survival data. Consider te rando variable tat is exponentiated exponential-distributed wit scale paraeter λ > 0 and sape paraeter α > 0. Te probability density function of is given by x x 1 f x;, e 1 e ; x 0. Te corresponding cuulative distribution function is given by x ;, 1 e F x. It is clear tat te EED is siply te α t power of te exponential cuulative distribution. So, te case were α = 1 is called te exponential distribution. Te ean, variance, skewness, kurtosis and te pdf's curves of te EED for different values of te scale and sape paraeters are sown in Table 1. Te properties of te EED ave been studied by any autors, see for exaple Gupta and Kundu (001), Nadaraja (011), Gitany et al. (013), and Ristić and Nadaraja (014). Literature Review Stokes (1976) used RSS for estiating te paraeters in a location-scale faily of distributions. Te RSS estiators of te location and scale paraeters are sown to be ore efficient tan te estiators. Se also used RSS to estiate te correlation coefficient of a bivariate noral distribution. La et al. (1994) used RSS for estiating two-paraeter exponential distribution. f y 1 y exp (1) 19
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS Table 1. Te ean, variance, skewness, kurtosis and te pdf's curves of te EED for different values of α and λ. (α,λ) (1,1) Properties of te EED Mean: 1 Skewness: Variance: 1 Kurtosis: 9 PDFs Curve: (0.5,1.5) Mean: 0.845 Skewness: 3.8514 Variance: 0.790 Kurtosis: 19.6675 PDFs Curve: (1.5,.5) Mean: 0.7364 Skewness: -0.5053 Variance: 1.0640 Kurtosis: 0.513 PDFs Curve: (0.5,0.5) Mean: 0.8536 Skewness: 3.8514 Variance:.59 Kurtosis: 19.6675 PDFs Curve: An unbiased estiators of and based on RSS wit teir variances are derived. Tey ade a coparison between tese estiators wit teir counterpart in. x Stokes (1995) considered te location-scale distribution, F, and estiated μ and σ using te etods of axiu likeliood estiation and best linear unbiased estiation witin te fraework of RSS. Sina et al. (1996) used RSS to estiate te paraeters of te noral and exponential distributions. Teir work assued partial knowledge of te underlying distribution witout any knowledge of te paraeters. For eac paraeter, tey proposed best linear unbiased estiators for full and partial RSS. 0
SAMUH & QTAIT Saawi and Al-Sageer (001) studied te use of Extree RSS (ERSS) and MRSS for distribution function estiation. For a rando variable, it is sown tat te distribution function estiator wen using ERSS and MRSS are ore efficient tan wen using and RSS for soe values of a given x. Abu-Dayye and Sawi (009) considered te axiu likeliood estiator and te likeliood ratio test for aking inference about te scale paraeter of te exponential distribution in case of oving extree ranked set sapling (MERSS). Te estiators and test cannot be written in closed for. Terefore, a odification of te axiu likeliood estiator using te tecnique suggested by Maarota and Nanda (1974) was considered. It was used to odify te likeliood ratio test to get a test in closed for for testing a siple ypotesis against one-sided alternatives. Al-Oari and Al-Hadrai (011) used ERSS to estiate te paraeters and population ean of te odified Weibull distribution. Te axiu likeliood estiators are investigated and copared to te corresponding one based on. It was found tat te estiators based on ERSS are ore efficient tan estiators using. Te ERSS estiator of te population ean was also found to be ore efficient tan te based on te sae nuber of easured units. Haq et al. (013) proposed a partial ranked set sapling (PRSS) etod for estiation of population ean, edian and variance. On te basis of perfect and iperfect rankings, Monte Carlo siulations fro syetric and asyetric distributions are used to evaluate te effectiveness of te proposed estiators. It was found tat te estiators under PRSS are ore efficient tan te estiators based on siple rando sapling. Abu-Dayye et al. (013) used RSS for studying te estiation of te sape and location paraeters of te Pareto distribution. Te estiators were copared wit teir counterpart in in ters of teir biases and ean square errors. It was sown tat te estiators based on RSS can be real copetitors against tose based on. Sarikavanij et al. (014) considered siultaneous coparison of te location and scale estiators of a two-paraeter exponential distribution based on and RSS by using generalized variance (GV). Tey suggested various RSS strategies to estiate te scale paraeter. Teir perforances in ters of GV were copared wit strategy. It was sown tat te iniu values of set size,, based on RSS, wic would result in saller GV tan tat based on. 1
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS Maxiu likeliood estiation and fiser inforation based on Consider a rando saple coing fro te EED f(x; α; λ) were te values of α and λ are unknown. Te likeliood function is given by n n xi n n xi 1 i1 (, ) (1 ) ; 0, 0. i1 L e e Tus, te log likeliood function is xi log L (, ) nlog nlog ( 1) log(1 e ) x. n n i () i1 i1 Te noral equations becoe log L (, ) n x e ( 1) xi 0. n xi n i xi i11e i1 (3) n log L (, ) n xi log(1 e ) 0, (4) i1 Fro Equation 4, te axiu likeliood estiator (MLE) of α as a function of λ, say ˆ, is ˆ( ) n n. xi log 1 e i1 Substituting ˆ in Equation, we obtain te profile log-likeliood of λ as xi log L ( ˆ ( ), ) nlog n nlog log(1 e ) n n i1 xi nlog n log(1 e ) x. i1 i1 Te MLE of λ, can be obtained by axiizing (5) w.r.t λ as n i (5)
SAMUH & QTAIT n xi n n log L ( ˆ ( ), ) n xie n xie x x i i n i1 1 e i1 x 1 1 i i e log(1 e ) i1 xi xi. (6) However, te solutions are not in closed fors, in order to obtain estiates for α and λ, te noral equations can be solved nuerically. Fiser inforation (FI) nuber is used to easure te aount of inforation tat an observable saple carries about te paraeter(s). Te FI nuber for te paraeter θ is defined as log L( ) FI( ). Based on te rando saple 1,,, n te FI nubers of α and λ are, respectively, given by FI FI log L (, ) n ( ), log L (, ) n x e ( ) ( 1). n xi i xi i1 (1 e ) Maxiu likeliood estiation and fiser inforation based on MRSS Consider te axiu likeliood estiation of te paraeters α and λ of EED under MRSS paying attention to te odd and even set sizes. Odd set sizes Suppose {Y ; j = 1,,, } is a MRSS fro an EED, were is te nuber of cycles and is te set size. Since te set size is assued to be odd, te Y are independent and identically distributed as te distribution of te 1 t order statistics of te rando saple 1,,, ; tat is 3
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS 1 1 1! y y y fy y f y e 1 e 1 1 e. 1 1! Te likeliood function of MRSS for odd set size is given by j1 i1 L f y c e, MRSSO Y j1 i1 1 1 y y 1 e 1 1 e j1 i1 j1 i1 1 y 1 (7) were c 1 is a constant. Tus, te log likeliood function is log L, d log log y MRSSO 1 j1 i1 1 y 1 1 e j1 i1 j1 i1 1 y log1 1 e j1 i1 y (8) were d 1 is a constant. Te noral equations becoe log L, MRSSO j1 i1 y 1 ye 1 j1 i1 1 e y y y 1 y e 1 e j1 i1 1 1e y y 1 0 (9) 4
SAMUH & QTAIT log LMRSSO (, ) 1 log 1 e j1 i1 1 j1 i1 1 1 y y y 1e log 1e y e 0, () Te MLEs of te paraeters α and λ are te solutions of te Equations (9) and (). However, te solutions are not in closed fors, in order to obtain estiates for α and λ, te noral equations can be solved nuerically. Based on te MRSS {Y ; j = 1,,, ; i = 1,,,}, for odd set size, te FI nubers of α and λ are, respectively, given by FI FI MRSSO MRSSO log LMRSSO (, ) ( ), log LMRSSO (, ) ( ). Te observed FI nubers are evaluated at te axiu likeliood estiates. Even set sizes Because te set size is assued to be even, for eac j 1,,, ; Y ~ f y, Y i f Yi f y for i 1,, ( y) f y for i,, t were and are te and te order statistics of te rando saple 1,,, ; terefore, for i1,, ; Y are independent and identically distributed as t 5
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS! f y e e e 1! y y 1 ( ) y Y 1 1 (1 ), i! y y 11 1 ( ) y f Y y e 1 e 1 (1 e ). i 1! (11) (1) and for i,, ; Y are independent and identically distributed as Note {Y ; j = 1,,, ; i = 1,,,} are independent. Tus, te likeliood function of MRSS for even set size is given by LMRSSE (, ) fy ( y ) ( ) i fy y i j1 i1 j1 i y y j 1 i 1 j 1 i 1 y c e e 1 e j1 i1 y y 1 e 1 1 e j1 i j1 i1 1 1 j1 i y 1 1 e, 1 (13) were c is a constant. Tus, te log likeliood function is 6
SAMUH & QTAIT log L (, ) d log log MRSSE j1 i1 j1 i y j1 i1 1log 1 e j1 i1 11 log 1 e j1 i j1 i log 1 1 e y y y y e y 1 log 1 1, (14) were d is a constant. Te noral equations becoe log L (, ) d log log MRSSE j1 i1 j1 i y j1 i1 1log 1 e j1 i1 11 log 1 e j1 i j1 i log 1 1 e y y y y e y 1 log 1 1, (15) 7
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS log LMRSSE (, ) log 1 e j1 i1 1log 1 e j1 i y y 1e log 1e y e y y 1e log 1e 1 y j i e j1 i1 1 1 1 1 1 0 y y (16) Te MLEs of te paraeters α and λ are te solutions of te Equations (15) and (16). However, te solutions are not in closed fors, in order to obtain estiates for α and λ, te noral equations can be solved nuerically. Based on te MRSS {Y ; j = 1,,, ; i = 1,,,}, for even set size, te FI nubers of α and λ are, respectively, given by log LMRSSE (, ) FIMRSSE ( ), log LMRSSE (, ) FIMRSSE ( ). Te observed FI nubers are evaluated at te axiu likeliood estiates. Te coparison between te resulting estiators under MRSS and can be done using te asyptotic efficiency (see Stokes, 1995). Te asyptotic efficiency of MRSS w.r.t for estiating θ is defined by ˆ ˆ ˆ ˆ FIMRSS ( ) Aeff ( MRSS ; ) li eff ( MRSS ; ). n FI ( ) 8
SAMUH & QTAIT Interval Estiates Let 1,, n be a rando saple fro f (x;θ), were θ is an unknown quantity. A confidence interval for te paraeter θ, wit confidence level or confidence coefficient 1 γ, is an interval wit rando endpoints [S L ( 1,, n ), S u ( 1,, n )]. It is given by 1 1 P S,, S,, 1. L n U n Te interval [S L ( 1,, n ), S u ( 1,, n )] is called a 0(1 γ)% confidence interval for θ. For large saple size, te axiu likeliood estiator, under appropriate regularity conditions (see Davison, 008, p.118), as any useful properties, including reparaetrization-invariance, consistency, efficiency, and te sapling distribution of a axiu likeliood estiator ˆMLE is asyptotically unbiased and also asyptotically noral wit its variance obtained fro te inverse Fiser inforation nuber of saple size 1 at te unknown paraeter θ; tat is, ˆ 1 MLE N, FI ( ) as n. Terefore, te approxiate 0(1 γ)% confidence liits for te ˆMLE of θ can be constructed as ˆ P z z 1, 1 FI ( ) were z γ is te γ t upper percentile of te standard noral distribution. Terefore, te approxiate 0(1 γ)% confidence liits for te scale and location paraeters of te EED are given, respectively, by 1 1 P ˆ z FI ( ) ˆ z FI ( ) 1, (17) ˆ 1 ˆ 1 P z FI ( ) z FI ( ) 1. (18) Ten, te approxiate confidence liits for α and λ will be constructed using Equation (17) and (18), respectively. 9
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS Siulation Study To investigate te properties of te axiu likeliood estiators of te scale and locations paraeters of te EED a siulation study is conducted. Monte Carlo siulation is applied for different saple sizes, = {,3,4,5} and = {,50,0}, and for different paraeter values, (α, λ) = {(1,1),(0.5,1.5),(1.5,.5)}. Te estiates of α and λ, te bias estiates, te MSEs, and te efficiency values are coputed over 000 replications for different cases. Te results are reported in Tables -4. Moreover, te observed Fiser inforation atrices and te asyptotic efficiency in estiating α and λ under and MRSS are calculated and te results reported in Table 5. Te observed Fiser inforation nubers of α and λ based on are denoted by FI ( ˆ ) and FI ( ˆ ), respectively, and te observed inforation nubers of α and λ based on MRSS are denoted by FI MRSS ( ˆ ) and FI MRSS ( ˆ ), respectively. Te asyptotic efficiency, Aeff, for estiating α is found as te ratio and for estiating λ is found as te ratio FIMRSS ( ) Aeff ( ˆ ; ˆ MRSS ), FI ( ) ˆ ˆ FIMRSS ( ) Aeff ( MRSS ; ). FI ( ) Confidence intervals based on and MRSS for (α, λ) = (1.5,.5) for different saple sizes are constructed at 1 γ = 0.95 level of confidence using Equation (17) and (18), respectively, and te results are sown in Table 5. 30
SAMUH & QTAIT Table. Te Bias, MSE, and Efficiency values of estiating te paraeters (α = 1, λ = 1) under and MRSS. α = 1 λ = 1 Sapling ˆ Bias( ˆ ) MSE( ˆ ) eff ˆ ˆ ˆ Bias( ) MSE( ) eff 50 00 1.1731 0.1731 0.063 1.1346 0.1346 0.1490 MRSS 0.7835-0.165 0.170 1.647 0.750-0.480 0.1446 1.0300 1.098 0.098 0.013 1.031-0.3187 0.1147 MRSS 0.7075-0.95 0.0954 0.8 0.6813-0.3187 0.1147 0.1791 1.0066 0.0066 0.0044 1.0090 0.0090 0.0047 MRSS 0.6915-0.3085 0.0973 0.0451 0.666-0.3374 0.1168 0.0401 3 4 5 50 00 50 00 50 00 1.15 0.15 0.38 1.078 0.078 0.0650 MRSS 1.0949 0.0949 0.0880 1.1795 1.0730 0.0730 0.0650 1.1968 1.09 0.09 0.015 1.0160 0.0160 0.0117 MRSS 1.0165 0.0165 0.0116 1.0770 1.0114 0.0114 0.03 1.1317 1.0056 0.0056 0.0030 1.0037 0.0037 0.008 MRSS 1.005 0.005 0.007 1.1180 1.0036 0.0036 0.006 1.19 1.061 0.061 0.0599 1.0550 0.0550 0.0561 MRSS 0.8719-0.181 0.049 1.174 0.8553-0.1447 0.0561 0.990 1.0154 0.0154 0.0091 1.0141 0.0141 0.0091 MRSS 0.898-0.170 0.0347 0.61 0.8189-0.1811 0.0396 0.91 1.007 0.007 0.00 1.0034 0.0034 0.00 MRSS 0.830-0.1770 0.036 0.0669 0.81-0.1878 0.0369 0.0600 1.0514 0.0514 0.0474 1.0460 0.0460 0.0415 MRSS 1.0571 0.0571 0.0396 1.1966 1.0486 0.0486 0.0351 1.181 1.0099 0.0099 0.0076 1.0076 0.0076 0.0074 MRSS 1.0099 0.0099 0.0064 1.181 1.0075 0.0075 0.0060 1.46 1.0044 0.0044 0.0019 1.008 0.008 0.0018 MRSS 1.005 0.005 0.0014 1.3314 1.0014 0.0014 0.0014 1.649 31
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS Table 3. Te Bias, MSE, and Efficiency values of estiating te paraeters (α = 0.5, λ = 1.5) under and MRSS. α = 0.5 λ = 1.5 Sapling ˆ Bias( ˆ ) MSE( ˆ ) eff ˆ Bias( ˆ ) MSE( ˆ ) eff 50 00 0.5687 0.0687 0.0349 1.7938 0.938 0.638 MRSS 0.406-0.0938 0.040 1.4557 1.083-0.4177 0.457 1.384 0.514 0.014 0.0041 1.5499 0.0499 0.0730 MRSS 0.3748-0.15 0.0178 0.317 0.9367-0.5633 0.3566 0.048 0.504 0.004 0.0009 1.5181 0.0181 0.0160 MRSS 0.3680-0.130 0.0179 0.0493 0.9006-0.5994 0.3676 0.0445 3 4 5 50 00 50 00 50 00 0.54 0.04 0.0188 1.6688 0.1688 0.3043 MRSS 0.5366 0.0366 0.0146 1.87 1.6565 0.1565 0.594 1.173 0.50 0.00 0.005 1.5341 0.0346 0.0418 MRSS 0.5067 0.0067 0.001 1.1871 1.546 0.046 0.038 1.0930 0.504 0.004 0.0006 1.5080 0.0080 0.0099 MRSS 0.501 0.001 0.0005 1.38 1.5077 0.0077 0.0094 1.053 0.538 0.038 0.01 1.6165 0.1165 0.066 MRSS 0.4463-0.0537 0.0090 1.301 1.44-0.576 0.1895 1.0905 0.506 0.006 0.0018 1.584 0.084 0.0318 MRSS 0.490-0.07 0.006 0.99 1.1656-0.3344 0.1345 0.365 0.50 0.00 0.0004 1.5070 0.0070 0.0077 MRSS 0.463-0.0737 0.0057 0.0775 1.150-0.3480 0.163 0.0608 0.505 0.005 0.0091 1.5984 0.0984 0.1544 MRSS 0.519 0.019 0.0065 1.3956 1.609 0.9 0.1371 1.165 0.5041 0.0041 0.0015 1.5166 0.0166 0.060 MRSS 0.5039 0.0039 0.0011 1.3306 1.5160 0.0160 0.05 1.1578 0.5019 0.0019 0.0004 1.5055 0.0055 0.006 MRSS 0.50 0.00 0.0003 1.514 1.5033 0.0033 0.0053 1.1736 3
SAMUH & QTAIT Table 4. Te Bias, MSE, and Efficiency values of estiating te paraeters (α = 1.5, λ =.5) under and MRSS. α = 1.5 λ =.5 Sapling ˆ Bias( ˆ ) MSE( ˆ ) eff ˆ Bias( ˆ ) MSE( ˆ ) eff 50 00 1.8041 0.3041 0.6164.7886 0.886 0.7157 MRSS 1.1505-0.3495 0.3413 1.8061 1.9-0.5790 0.769 0.9304 1.5507 0.0507 0.0575.5493 0.0493 0.40 MRSS 1.01-0.4788 0.538 0.64 1.7673-0.737 0.608 0.17 1.5119 0.0119 0.0116.5198 0.0198 0.038 MRSS 0.9948-0.505 0.605 0.0445 1.756-0.7744 0.6157 0.0387 3 4 5 50 00 50 00 50 00 1.676 0.176 0.947.6681 0.1681 0.3835 MRSS 1.6680 0.1680 0.600 1.1336.6581 0.1581 0.30 1.1976 1.5380 0.0380 0.039.5341 0.0341 0.0589 MRSS 1.585 0.085 0.030 1.070.545 0.045 0.050 1.1341 1.509 0.009 0.0078.5080 0.0080 0.0145 MRSS 1.5088 0.0088 0.0074 1.068.5078 0.0078 0.0130 1.13 1.6086 0.86 0.1688.6194 0.1194 0.785 MRSS 1.888-0.11 0.1343 1.568.1679-0.331 0.911 0.9568 1.567 0.067 0.041.5311 0.0311 0.0461 MRSS 1.173-0.87 0.0951 0.531.0889-0.4111 0.045 0.55 1.5049 0.0049 0.0057.5074 0.0074 0.0113 MRSS 1.054-0.946 0.0901 0.0636.074-0.458 0.1897 0.0594 1.5897 0.0897 0.186.5990 0.0990 0.071 MRSS 1.601 0.1 0.1160 1.95.6059 0.59 0.1740 1.1900 1.5169 0.0169 0.001.5163 0.0163 0.0375 MRSS 1.517 0.017 0.0181 1.81.5160 0.0160 0.0307 1. 1.5073 0.0073 0.0050.5063 0.0063 0.0093 MRSS 1.5041 0.0041 0.0040 1.501.5030 0.0030 0.007 1.845 33
ESTIMATION FOR PARAMETERS OF THE EED USING MRSS Table 5. Te observed Fiser inforation atrix, te variance-covariance atrix, a 95% confidence interval of te paraeters (α = 1.5, λ =.5), and te asyptotic efficiency under and MRSS. Fiser Inforation Sapling ˆ Aeff 00 MRSS MRSS Variance- Covariance 95% CI ˆ ˆ ˆ Lower Upper Widt ˆ 8.67-4.3 3.60 a 0.067 0.1870 0.9130.695 1.78 ˆ -4.3 4.68.17 b 0.1870 0.3830 1.5756 4.0016.460 ˆ 31.6-1.88 0.0670 0.0851 0.643 1.6578 1.0147 ˆ -1.88.15 0.0851 0.065 1.0303.8117 1.7813 ˆ 177.59-89.5 3.37 0.0111 0.01 1.3054 1.7184 0.4130 ˆ -89.5 90.67.16 0.01 0.018.304.8094 0.5788 ˆ 597.86-57.35 0.0039 0.0051 0.874 1.117 0.448 ˆ -57.35 195.78 0.0051 0.0118 1.517 1.9385 0.458 3 00 MRSS MRSS ˆ 13.31-6.53 1.91 0.1389 0.199 0.9457.4067 1.46 ˆ -6.53 6.98.00 0.199 0.649 1.6593 3.6769.0176 ˆ 5.45-15.57 0.135 0.1376 0.9793.3568 1.3776 ˆ -15.57 13.98 0.1376 0.49 1.786 3.5876 1.8590 ˆ 66. -134.3 1.9 0.0075 0.0073 1.3395 1.6789 0.3395 ˆ -134.3 136.78.04 0.0073 0.0145.70.7440 0.470 ˆ 51.14-30.54 0.0070 0.0080 1.3448 1.678 0.380 ˆ -30.54 78.70 0.0080 0.018.861.796 0.4435 4 00 MRSS MRSS ˆ 17.86-8.77 3.73 0.5 0.0 0.979.443 1.714 ˆ -8.77 9..98 0.0 0.039 1.7344 3.5044 1.7701 ˆ 66.59-36.3 0.0540 0.0714 0.8333 1.7443 0.99 ˆ -36.3 7.45 0.0714 0.1309 1.4588.8770 1.4183 ˆ 355.8-179.45 3.74 0.0056 0.0055 1.358 1.6516 0.933 ˆ -179.45 18.44.99 0.0055 0.09.308.7 0.4093 ˆ ˆ 139.79-734.7 0.009 0.0040 1.0999 1.39 0.111-734.7 545.1 0.0040 0.007 1.9079.405 0.336 (continued) 34
SAMUH & QTAIT 5 00 MRSS MRSS ˆ.49-11.06.79 0.0843 0.0811 1.006.1588 1.138 ˆ -11.06 11.49 3.06 0.0811 0.1651 1.806 3.3954 1.598 ˆ 6.70-41.6 0.0748 0.0886 1.0651.1373 1.071 ˆ -41.6 35.1 0.0886 0.1335 1.8898 3.30 1.433 ˆ 443.0-3.87.88 0.0045 0.0044 1.3758 1.6388 0.630 ˆ -3.87 7.76 3.11 0.0044 0.0087.335.6891 0.3656 ˆ 177.40-854.0 0.0041 0.0049 1.3786 1.696 0.5 ˆ -854.0 707.46 0.0049 0.0073.3355.6705 0.3349 *Note: a) Aeff ( ˆ MRSS, ˆ ˆ ˆ ), b) Aeff ( MRSS, ) Conclusion Te etod of axiu likeliood estiation for estiating te sape and scale paraeters of te EED is studied in te MRSS fraework. Te new obtained estiators are co-pared wit te conventional estiators obtained by. Te relative efficiency are calculated for coparing te estiators. Te aount of inforation available fro te MRSS about te paraeters of te EED is evaluated. Confidence intervals for te paraeters are constructed using and MRSS. More specifically, we ave te following conclusions. 1. Fro Tables -4 it can be concluded tat: a. For eac sapling etod, te MSEs of te estiators decrease as te set size increases and as te nuber of cycle increases. b. It is clear, fro te biases, tat MRSS overestiate and wen te set size is odd and underestiate and wen te set size is even. c. Te biases of te estiators based on MRSS wen te set size is odd decrease as te nuber of cycle increases. Wen te set size is even te biases of te estiators based on MRSS increase as te nuber of cycle increases. d. Te efficiency is always greater tan 1 wen te set size is odd; tat is, MRSS is ore efficient tan in estiating te paraeters of te EED.. Fro Table 5 it can be concluded tat: 35
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