Dd31720725

161 views

Published on

IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
161
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
2
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Dd31720725

  1. 1. Amal S. Hassan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.720-725 Maximum Likelihood and Bayes Estimators of the Unknown Parameters For Exponentiated Exponential Distribution Using Ranked Set Sampling Amal S. Hassan**(Department of Mathematical Statistics in Institute of Statistical Studies and Research (ISSR).Cairo University, Egypt)ABSTRACT Estimation of the shape and scale efficiency of the RSS estimator without increasing theparameters of exponentiated exponential set size m. It was shown that the double ranked setdistribution is considered based on simple random sample estimator of the mean is more efficient thansample and ranked set sample. Bayesian method of that using RSS. Al-Saleh and Muttlak [7] used RSS inestimation under squared error loss function and Bayesian estimation for exponential and normalmaximum likelihood method will be used. distributions to reduce Bayes risk. Al-Hadhrami [8]Comparison between estimators is made through studied the estimation problem of the unknownsimulation via their absolute relative biases, mean parameters for the modified Weibull distribution.square errors, and efficiencies. Comparison study Maximum likelihood estimators for the parametersrevealed that the Bayes estimator is better than were also investigated mathematically andmaximum likelihood estimator under both numerically. The numerical results show that thesampling schemes. The results show that the estimators based on RSS are more efficient than thatestimators based on ranked set sample are more from SRS. Helui et al [9] studied the estimation of theefficient than that from simple random sample at unknown parameter for Weibull distribution underthe same sample size. different sampling. Methods of estimation used areKeywords - Bayes; Estimation; Ranked Set ML, moments and Bayes. They concluded thatSampling; Simple Random Sample; Exponentiated estimators based on RSS and modified RSS haveexponential distribution. many advantages over those that are based on simple random sample (SRS). This article is concerned with the maximum I. Introduction likelihood (ML) and Bayes estimators of the shape A method of sampling based on ranked sets and scale parameters of exponentiated exponentialis an efficient alternative to simple random sampling based on SRS and RSS. Bayes estimators underthat has been shown to outperform simple random squared error loss function are discussed assumingsampling in many situations by reducing the variance gamma prior distribution for both parameters. Theof an estimator, thereby providing the same accuracy performance of the obtained estimators is investigatedwith a smaller sample size than is needed in simple in terms of their absolute relative biases (ARBs) andrandom sampling. Ranked set sample (RSS) can be mean square errors (MSEs). Relative efficiency of theapplied in many studies where the exact measurement estimators is also calculated.of an element is very difficult (in terms of money, The rest of the article is organized as follows. Intime, labour and organization) but the variable of Section 2, ML and Bayesian methods of estimation ofinterest, although not easily measurable, can be unknown parameters are discussed under SRS. Inrelatively easily ranked (order) at no cost or very little Section 3, the same methods of estimation areadditional cost. The ranking can be done on the basis discussed based on RSS.of visual inspection, prior information, earlier Numerical illustration is carried out to illustratesampling episodes or other rough methods not theoretical results in Section 4. Simulation results arerequiring actual measurement. displayed in Section 5. Finally, conclusions are RSS was first suggested by McIntyre [1] and was presented in Section 6.supported by Takahasi and Wakimoto [2] bymathematical theory. Dell and Clutter [3] showed thatRSS is more efficient than simple random sampling II. Estimation of Parameters Using SRSeven with an error in ranking. Samawi et al. [4] This Section discusses the ML and Bayessuggested using extreme ranked set samples for estimators under squared error loss function for theestimating a population mean. Muttlak [5] introduced unknown parameters of EE distribution under SRS.median ranked set sampling to estimate the populationmean. Al-Saleh and Al-Kadiri [6] considered double 2.1 Maximum Likelihood Estimationranked set sample, as a procedure that increases the Gupta and Kundu [10] proposed an exponentiated exponential (EE) distribution as an alternative to the 720 | P a g e
  2. 2. gamma and Weibull distributions and studied its  ( x ) different properties. EE has the following probability ndensity function (PDF)  ( a  c   xi ) n (4)   n d 1 nb1 e i 1  (1  e xi ) 1 d f ( x, ,  )    (1- e-  x )  1 e  x , , , x  0 0 i 1 ,Let X 1 , X 2 ,..., X n be independent and identically H where,distributed random variables from EE, then the log- nlikelihood function, L( ,  ) , is   ( a   c    xi ) n n H     n  d 1 n  b 1 e i 1  (1  e  i 1  x i  1 ) d d  L( ,  )  n ln( )  n ln(  )    xi 0 0 i 1 Similarly integrating the joint posterior with respect to n  , the marginal posterior  can be obtained as  (   1) ln(1  e  xi ).  ( x)  i 1 nThe normal equations become:   ( a  c   xi ) n (5) L n n x n   (  1)  xi i   xi  0, (1)   n  d 1 n  b 1 e i 1 i 1  (1  e xi ) 1 d  i 1 e  1 i 1 0 . HL n n    ln(1  e  xi )  0. (2) The Bayes estimators for parameters  and  of EE  i 1 distribution under squared error loss function may beObviously, it is not easy to obtain a closed form defined, respectively, assolution for the two non-linear equations (1) and (2). ~ So, iterative procedure must be applied to solve these bays  E ( x)     ( x) d ,equations numerically. Newton Raphson method is 0 used to obtain ML estimates of  and  , say,  MLE ~ ~  bays  E ( x)     ( x) d . (6) ~and  MLE . 0 There are no explicit forms for obtaining estimators for the EE, therefore, numerical solution and2.2 Bayesian Estimation computer facilities are needed.Assume that  and  are independent randomvariables. Following Kundu and Gupta [11], it is III. Estimation of Parameters Using RSSassumed that  and  have the following gamma The aim of this Section is to find the ML andprior distributions; Bayes estimators for the unknown parameters of EE  1 ( )  b 1e  a ,   0; distribution under RSS.  2 ( )   d 1e  c ,   0; (3) 3.1Maximum Likelihood Estimationwhere, all the hyper parameters a; b; c; d are assumed RSS is recognized as a useful samplingto be known and non-negative. Suppose technique for improving the precision and increasing x  ( x1 , x 2 ,..., x n ) is a random sample from EE, the efficiency of estimation when the variable underthen based on the likelihood function of the observed consideration is expensive to measure or difficult todata; obtain but cheap and easy to rank. n The procedure of using RSS is as follows:   xi nL( x  ,  )   n n e i 1  (1  e  i 1  xi )  1 . Step1: Randomly draw m random sets with m elements in each sample and ranked them (withoutUsing the priors given in equations (3), then the joint actual measuring) with respect to the variable ofposterior density function of  and  can be written interest. Step 2: From the first set, the element with theas; L( x  ,  ) 1 ( ) 2 ( )  ( ,  x)    . smallest rank was chosen. From the second set, the element with the second smallest rank was chosen.   L( x  ,  ) 1 ( ) 2 ( ) d d 0 0 The procedure was continued until the element with the largest rank from the mth sample was chosen. ThisMarginal posterior of a parameter is obtained by procedure yielded a total number of m elementsintegrating the joint posterior distribution with respect chosen to be measured, one from each sample.to the other parameter and hence the marginal Step 3: Repeat the above steps k times ( k cycles)posterior of  can be written as until the desired sample size ( n  mk ) is obtained for analysis. Suppose that X ( i ) ic ., i  1... m, c  1... k is a ranked set sample from EE distribution, with sample size n  mk , m is the set size and k is the number of 721 | P a g e
  3. 3. cycles. For simplicity, let Yic  X ( i ) ic , then for fixed k m   ( a  c   yic ) H 2   (m!)  mk  mk  d 1 mk  b 1 e c 1 i 1 c, Yic are independent with PDF equal to PDF of ith 0order statistics and given by k (1  e  yic )i 1[1  (1  e yic ) ]m i m g ( yic )  m! (i  1)!(m  i)! [ F ( yic )]i1 f ( yic )[1  F ( yic )]mi  c 1 i 1 (i  1)!(m  i )! d , yi c  0, and, k mThe likelihood function of the sample  ( a  c   yic )y1c , y 2c ,..., y mc  (m!) ( )mk mk  d 1 ( ) mk  b 1 e c 1 i 1  H1    .L( y  ,  )  k m (1  e yic ) i 1[1  (1  e yic ) ]m i k m m!  e  yic (1  e  yic )  i 1[1  (1  e  yic )  ]mi 0 0  (i  1)!(m  i)! d d  c 1 i 1 (7) c 1 i 1 (i  1)!(m  i )!The log-likelihood function, is given by Similarly integrating the joint posterior with respect toLog L  C  mk (ln   ln  )   (i  1) ln(1  e k m  yic )  , the marginal posterior  can be obtained as c 1 i 1 H3 k m k m    yic   (m  i ) ln[1  (1  e  yic )  ],  ( y )  , (11) c 1 i 1 c 1 i 1 H1where, c is a constant.  ( a c  k m  yic )Differentiate the log likelihood with respect to  and H 3   (m!)  mk  mk  d 1 mk b1 e c 1 i 1  and equating to zero 0 k m (1  e  yic ) i1[1  (1  e yic )  ]miLog L mk k m (i  1) y  d k m     yic    yic ic (i  1)!(m  i )!   c1 i1 c 1 i 1 e 1 c 1 i 1  yic  1  yic The Bayes estimators for parameters  and  of EE k m (m  i )(1  e ) e yic    , (8) distribution under squared error loss function may be c 1 i 1 1  (1  e  yic ) defined, respectively, as Log L mk (m  i ) ln(1  e k m )  y i c bays  E ( y)     ( y) d ,     y i c   (9)   c 1 i 1 (1  e ) 1 0  k m  bays  E ( y)     ( y) d . (12)   i ln(1  e  yic ) . 0 c 1 i 1 These integrals cannot be obtained in a simple closedThe ML estimators of  and  , say  MLE ,  MLE , form; therefore, numerical solution is applied.are the solution of the two nonlinear equations (8) IV. Simulation Procedureand (9). Since it is difficult to find a closed form It is very difficult to compare the theoreticalsolution for the parameters, numerical technique is performances of the different estimators proposed inneeded to solve them. the previous Sections. Therefore, this Section presents3.2 Bayesian Estimation the numerical solutions to obtain the ML and BayesUsing the priors defined in equations (3) and the estimators of the unknown parameters  and  forlikelihood function given in equation (7), then the the EE distribution based on RSS and SRS. Ajoint posterior density function of  and  under comparison studies between these estimators will beRSS can be written as; carried out through MSEs, ARBs and relative L( y  ,  ) 1 ( ) 2 ( ) efficiency. Monte Carlo simulation is applied for ( ,  y )    . different set sizes, different number of cycles and   L( y  ,  ) 0 0 1 ( ) 2 ( ) d d different parameter values. The simulation procedures are described through the followingMarginal posterior of a parameter is obtained by algorithmintegrating the joint posterior distribution with respect Step 1: A random sample of size n = 10, 15, 16, 20,to the other parameter and hence the marginal 24 and 30 with set size m=(2,3), number of cycles k =posterior of  can be written, as (5,8,10), where n  m  k are generated from EE distribution. H2 ( y )  , (10) Step 2: The parameters values are selected as H1   0.5(0.3)1.1 for   1 in estimation procedure.where, Step 3: For the chosen set of parameters and each ~ ~ sample of size n, eight estimators (  MLE ,  MLE , 722 | P a g e
  4. 4. ~ ~ Bayes  Bayes ,  MLE ,  MLE ,  Bayes and  Bayes ) are  Bayes is better than  MLE based on SRScomputed under SRS and RSS. Bayes is better than MLE based on SRSStep 4: Repeat the pervious steps from 1 to 3 N times ~ ~representing different samples, where N=1000. Then,  Bayes is better than  MLE based on RSSthe ARBs and MSEs of the estimates are computed. ~ ~Step 5: Compute the efficiency of estimators, that  Bayes is better than  MLE based on RSSdefined as, Efficiency = MSE(SRS)/MSE(RSS). 5- Comparing the biases of estimators, it is noted that the Bayes estimates have the minimum biases in V. Simulation Results almost all of the cases expect few cases. All simulated studies presented here are 6. It is clear from Table 2 that the efficiency ofobtained via MathCAD (14). The results are reported estimators increases as the sample sizes increase. Thein Tables 1 and 2. Table 1 contains the estimates of estimators based on RSS have smaller MSE than theparameters for EE under SRS and RSS for different corresponding ones based on SRS. The efficiency ofvalue of sample sizes and different value of RSS estimators with respect to SRS estimator isparameters. Table 2 contains the efficiency of greater than one and increases when the sample sizeestimators for SRS relative to RSS. From Tables 1 increases.and 2 many conclusions can be made on theperformance of both method of estimation based on VI. ConclusionsRSS and SRS. These conclusions are summarized as This article deals with the estimationfollows: problem of unknown parameters of EE distribution1. Based on SRS, ARBs and MSEs for the estimates based on RSS. ML and Bayesian methods ofof  and  are greater than the corresponding in estimation are used. Bayes estimates are obtainedRSS. under squared error loss function. Comparing the2. For both method of estimation, it is clear that ARBs performance of estimators, it is observed that theand MSEs decrease as set sizes increase for fixed Bayes performs better than ML relative to their ARBsvalue of  . and MSE’s. Furthermore, ARBs and MSEs of the3. As the value of  increases, the ARBs and MSEs estimates for both shape and scale parameters relativeincrease in almost all of the cases. to RSS are smaller than the corresponding SRS. This4- The MSEs for the Bayes estimates of both study revealed that the estimators based on RSS are more efficient than the corresponding SRS.parameters  and  is smaller than the MSEs for theMLEs of  and  in almost all the cases under bothSRS and RSS, in sense that,Table 1: Results of simulation study of ARBs and MSEs of estimates for different values of parameters ( ,  ) for EE distribution under SRS and RSS. RSS SRS (  0.5,  1) (  0.5,  1) ~ ~ ~ ~ (m,k) MLE Bayes  MLE  Bayes  MLE  Bayes  MLE  Bayes (2,5) 0.288 0.123 0.370 0.439 0.650 0.332 0.656 0.461 0.043 0.039 0.163 0.217 0.054 0.052 0.172 0.280 (2,8) 0.173 0.096 0.173 0.268 0.557 0.197 0.650 0.216 0.033 0.031 0.158 0.165 0.044 0.044 0.169 0.215 (2,10) 0.135 0.081 0.173 0.083 0.392 0.109 0.649 0.192 0.020 0.028 0.154 0.129 0.038 0.040 0.165 0.184 (3,5) 0.195 0.107 0.272 0.382 0.574 0.297 0.656 0.358 0.036 0.037 0.162 0.168 0.046 0.050 0.171 0.215 (3,8) 0.115 0.080 0.147 0.082 0.301 0.082 0.574 0.250 0.015 0.028 0.116 0.116 0.029 0.039 0.130 0.171 (3,10) 0.104 0.062 0.137 0.062 0.240 0.073 0.572 0.096 0.011 0.010 0.029 0.019 0.025 0.022 0.059 0.054 723 | P a g e
  5. 5. Continued Table 1 RSS SRS (  0.8,  1) (  0.8,  1) ~ ~ ~ ~ (m,k) MLE Bayes  MLE  Bayes  MLE  Bayes  MLE  Bayes (2,5) 0.343 0.276 0.196 0.301 0.451 0.276 0.540 0.336 0.339 0.089 0.233 0.255 0.345 0.125 0.345 0.266 (2,8) 0.116 0.127 0.105 0.219 0.293 0.230 0.532 0.239 0.118 0.065 0.118 0.185 0.121 0.101 0.200 0.237 (2,10) 0.088 0.068 0.093 0.123 0.193 0.130 0.425 0.179 0.079 0.045 0.079 0.114 0.095 0.075 0.135 0.219 (3,5) 0.254 0.157 0.194 0.236 0.337 0.234 0.535 0.252 0.138 0.085 0.131 0.225 0.146 0.120 0.217 0.256 (3,8) 0.079 0.048 0.054 0.096 0.169 0.118 0.404 0.135 0.031 0.036 0.032 0.088 0.068 0.063 0.080 0.196 (3,10) 0.065 0.047 0.036 0.070 0.168 0.011 0.388 0.107 0.023 0.024 0.020 0.066 0.062 0.044 0.060 0.185 RSS SRS (  1.1,  1) (  1.1,  1) ~ ~ ~ ~ (m,k) MLE Bayes  MLE  Bayes  MLE  Bayes  MLE  Bayes (2,5) 0.405 0.124 0.341 0.227 0.593 0.365 0.452 0.338 0.311 0.188 0.266 0.280 0.388 0.280 0.310 0.466 (2,8) 0.230 0.065 0.128 0.113 0.558 0.273 0.445 0.202 0.176 0.111 0.174 0.104 0.228 0.235 0.278 0.198 (2,10) 0.156 0.058 0.109 0.105 0.478 0.171 0.300 0.155 0.173 0.085 0.102 0.085 0.225 0.199 0.206 0.197 (3,5) 0.318 0.076 0.220 0.190 0.582 0.316 0.451 0.260 0.183 0.169 0.244 0.167 0.238 0.345 0.294 0.310 (3,8) 0.144 0.051 0.089 0.070 0.470 0.148 0.288 0.086 0.090 0.064 0.097 0.083 0.180 0.159 0.197 0.195 (3,10) 0.116 0.049 0.081 0.036 0.456 0.065 0.086 0.083 0.085 0.043 0.038 0.077 0.171 0.150 0.102 0.189 Note: The first entry is the simulated about ARBs. The second entry is the simulated about MSEs.Table 2: Efficiency of the estimators under SRS with respect to RSS in both estimation methods.  Efficiency of Efficiency of  (  0.5,  1) (  0.5,  1) (m,k) ML Bayesian ML Bayesian (2,5) 1.256 1.333 1.055 1.290 (2,8) 1.333 1.419 1.070 1.303 (2,10) 1.900 1.429 1.071 1.426 (3,5) 1.278 1.351 1.056 1.297 (3,8) 1.933 1.393 1.121 1.474 (3,10) 2.273 2.200 2.034 2.842  Efficiency of Efficiency of  (  0.8,  1) (  0.8,  1) (m,k) ML Bayesian ML Bayesian (2,5) 1.018 1.404 1.481 1.043 (2,8) 1.025 1.554 1.695 1.281 (2,10) 1.203 1.666 1.709 1.921 (3,5) 1.057 1.412 1.656 1.138 724 | P a g e
  6. 6. Continued Table 2 (3,8) 2.194 1.750 2.500 2.227 (3,10) 2.696 1.833 3.000 2.803  Efficiency of Efficiency of  (  1.1,  1) (  1.1,  1) (m,k) ML Bayesian ML Bayesian (2,5) 1.248 1.489 1.165 1.664 (2,8) 1.295 2.117 1.597 1.904 (2,10) 1.306 2.341 2.019 2.318 (3,5) 1.301 2.041 1.205 1.856 (3,8) 2.000 2.484 2.031 2.349 (3,10) 2.012 3.488 2.684 2.455References[1] G. A. McIntyre, A method of unbiased selective 11] D. Kundu and R. D.Gupta, Generalized sampling, using ranked sets, Australian Journal exponential distribution; Bayesian Inference, of Agricultural Research, 3, 1952, pp 385-390. Computational Statistics and Data Analysis, 52,[2] K. Takahasi and K. Wakimoto, On unbiased 2008, pp 1873-1883. estimates of the population mean based on the sample stratified by means of ordering, Annals of the Institute of Statistical Mathematics 20, 1968, pp 1-31.[3] T. R. Dell and J. L. Clutter, Ranked set sampling theory with order statistics background, Biometrics 28, 1972, pp 545–555.[4] H. M. Samawi, M. S. Ahmed, and W. A. Abu- Dayyeh, Estimating the population mean using extreme raked set sampling, The Biometrical Journal, 38(5), 1996, pp. 577-86.[5] H. A. Muttlak, Median ranked set sampling. Journal of Applied Statistical Science, 6(4), 1997, pp 245-255.[6] M. F. Al-Saleh and M. Al-Kadiri, Double ranked set sampling, Statistics and Probability Letters. 48, 2000, pp 205–212.[7] M. F. Al-Saleh and H. A. Muttlak, A note in Bayesian estimation using ranked set sampling, Pakistan Journal of Statistics 14, 1998, pp 49– 56.[8] S. A. Al-Hadhram, Parametric estimation on modified Weibull distribution based on ranked set sampling, European Journal of Scientific Research 44(1), 2010, pp.73-78. [9] A. Helu, M. Abu-Salih, and O. Alkami, Bayes estimation of Weibull distribution parameters using ranked set sampling, Communications in Statistics—Theory and Methods, 39, 2010, pp 2533–2551. [10] R. D. Gupta and D Kundu, Generalized exponential distributions, Australian and New Zealand Journal of Statistics, 41(2), 1999, pp 173 - 188.. 725 | P a g e

×