Available online at http://ijim.srbiau.ac.ir/ Int. J. Industrial Mathematics ISSN 008-561 Vol. 9, No. 1, 017 Article ID IJIM-00675, 7 pages Research Article Distribution of Ratios of Generalized Order Statistics From Pareto Distribution Inference M. Rajaei Salmasi, G. Yari Received Date: 015-08-09 Revised Date: 016-06-06 Accepted Date: 016-08-06 Abstract The aim of this paper is to study distribution of ratios of generalized order statistics from pareto distribution. parameter estimation of Pareto distribution based on generalized order statistics ratios of them have been obtained. Inferences using method of moments unbiased estimator have been obtained to develop point estimations. Consistency of unbiased estimator has been illustrated. To compare the performances of the employed methods, numerical results have been computed. Illustrative example using real data is also given. Keywords : Generalized Order Statisitcs; Pareto Distribution; Parameter Estimation. 1 Introduction as: amps [5, 6] discussed generalized order statistics GOS as a unified form of differ- K ent models of ordered rom variablesr.v. s. Let F be a comulative distribution function CDF with probability density function PDF of f, with constants parameters, m = m 1, m,..., m n 1, k 1, n N, n is given for all 1 r n 1, m 1, m,..., m n 1 R, where = k + n r + M r 1, M r = n 1 j=r m j. Suppose X=1,..., n denote n GOS, then their joint PDF, f X 1,n, m,k,...,x n,n, m,k x 1,..., x n, can be written f X 1,n, m,k,...,x n,n, m,k x 1,..., x n = n 1 k on the cone of j=1 n 1 γ j [ i=1 F mi x i fx i ] F k 1 x n fx n 1.1 F 1 0 < x 1... x n < F 1 1, where c r 1 = r i=1 γ i, r = 1,..., n, F = 1 F. is the survival function. Marginal PDF of r is Department of Statistics, Science Research Branch, Islamic Azad University, Tehran, Iran. Corresponding author. yari@iust.ac.ir Department of Mathematics, Iran University of Science Technology, Tehran, Iran. f r x r = c r 1 m + 1 r Γr F γr 1 x r [1 F x r ] r 1 fx r 1. 91
9 M. Rajaei Salmasi et al. /IJIM Vol. 9, No. 1 017 91-97 also the joint PDF of X r X s, where r < s, is f X n,m,k,x s,n,m,k x r, x s = c s 1 m + 1 s ΓrΓs r F m x r [1 F x r ] r 1 [F x r F x s ] F γ s 1 xs fx r fx s. 1.3 [9] had studied a model of personal income exceeded given level he showed that it can be approximated by Pareto law. The PDF CDF of Pareto law or exactly Pareto distribution PD are as follows: fx = α α x α+1, 1.4 F x = 1 F x = 1 x α, x >. 1.5 For detailed information one can refer to [4]. Several methods have been proposed in the literature for estimating the PD parameters. [11] introduced uniformly minimum variance unbiased estimatior for unknown parameters of Pareto PD. [10] studied consistency of some estimators, he suggested different popular techniues for inference. [4] discussed best linear unbiased estimates, [3] estimated PD unknown parameters using generalized median estimator. Estimation of parameters from PD through order statistics some other ordered rom variables illustrated by [1] [13]. Distribution of sum, product or ratios of rom variables is an important issues of reliability distribution theory which considered by many authors. [] studied distribution of ratios of generalized life distribution, [7] derived distribution of ratio of two normal rom variables. Also, Kotz type distributions was studied by [8] about ratios. The present paper has focused on point estimations based on GOS from PD. Ratios of GOS has been studied related distributions has obtained. Moment estimator based on ratios Pareto rom variables Unbiased estimator have been employed in the present paper. To investigate consistency of the unbiased estimato single product moments for GOS of PD have been calculated. Numerical results for comparison of the estimators have been computed. Illustrative example based on real data from iranian rural household income is also given. The rest of paper is organized as follows: Section includes ratio distribution of GOS from PD. inferences about parameters of PD have been discussed in Section 3. Sections 4 5 include numerical results conclusions, respectively. Distribution of ratios of GOS from PD Theorem.1 Let r s denote r th s th GOS having Pareto as underlying distribution. Then r a s distributed as beta with parameters s where r < s. γ s Proof. For simplicity, define x r = r x s = s. Considering R = x r x s Q = x s, it has been obtained, X r = QR X s = Q, so Jacobian has been computed J = 1 0 R Q = Q. Using 1.3 joint distribution of Q = R = R can be written as f Q,R, R = f Xr,X s R, = C s 1 m + 1 s ΓrΓs r F m R [ ] 1 F r 1 R [ F R F ] s r 1 F γs 1 frf..6 Using 1.4 1.5, above relation can be rewritten as f Q,R, R = 1 s r 1 α α C s 1 m + 1 s ΓrΓs r αγr α 1 R α 1 [ 1 R α] s r 1 [ ] α r 1 1 R α..7 In order to compute the distribution of R, integration from.7 with respect to is needed. Integral limits can be calculated, since X r < X s
M. Rajaei Salmasi et al. /IJIM Vol. 9, No. 1 017 91-97 93 we have R < 1 so it can be obtaied b R > b, furthermore, it is known that b < Q, so it is been concluded b < b R < Q, so taking integral as follows, f R R = where, [ gr 1 R α 1 αγr α R α ] r 1,.8 gr = 1 s r 1 α α C s 1 m + 1 s ΓrΓs r R α 1 [ 1 R α] s r 1. To get the solution of.8, following change of variable has been considered z = 1 it can be written f R R = grhr where hr = α R α, 1 0 z r 1 1 z γr 1 z, Rα, so we have α α f R R = grhr, m + 1.9 where, a, b = ΓaΓb Γa+b is beta function. According to the fact, relation 1. is a density function, integration based on it over the support of the rom variable X euals to 1, therefore, = 1, so we have C r 1 Γr r C r 1 Γrm + 1 r =..10 m + 1 Using recent eulity relations +1 = α+1 1 αs r 1 = αγ s 1, DF of R could be written as γ αc s 1 r f R R = m + 1 s 1 ΓrΓs r R αγs 1 [ 1 R α] s r 1..11 By change of variable W = R α, using euality Γ s r + γs j = = r + 1s γ j Γ γs m + 1 s r, we get to γ C s 1 r m + 1 s ΓrΓs r = 1, γs, s r which completes the proof. Theorem. Let r denotes r th GOS based on Pareto distribution with parameters α. r Then r 1 has Pareto distribution with parameters α. Proof. Using 1.3, 1.4 1.5 transformations U = z = r 1, it can be r written f s u, z = r 1 zc r 1 m + 1 r Γr 1 [ αm ] α r 1 z z αγr 1 α α z α+1 γr uz α+1. uz Following transformation has been assumed h = α, z after some calculation it been obtained fu = α αm+α C r 1 m + 1 r Γr 1 αγr 1 u α+1 u αγr 1 γ α+1 r αm + 1 α mα alpha+1 1 which gives the proof of theorem, 0 h γr+ 1 1 h r h, f U u = α α u α+1, u >..1
94 M. Rajaei Salmasi et al. /IJIM Vol. 9, No. 1 017 91-97 Table 1: Moment Estimations. Estimations MSE n ˆα ˆ ˆα ˆ 10 4.137 3.4718.1811 0.843 50 3.1574 3.5103 0.8615 0.4187 100.9899 3.5143 0.645 0.3333 Table : Estimation based on GOS. Method of Moments Unbiased n k m ˆα MSE for ˆα ˆ MSE for ˆ V ar ˆ 90.5cm10 30.5cm1 0.3403 0.161 3.777 0.77 0.3316.948 0.053 3.476 1.093 0.0301 5.83 0.17 3.3536 0.678 0.0071 30.5cm 0.3773 0.145 3.359 1.0643 0.1865.3089 0.191 3.3469 0.1484 0.055 5.905 0.095 3.3766 0.8139 0.0068 30.5cm6 0.4957 0.011 3.511 0.1104 0.0761.3645 0.1356 3.55 0.144 0.0185 5.305 0.1795 3.4371 0.376 0.0056 90.5cm50 30.5cm1 0.509 0.006 3.464 0.1716 0.0491.484 0.0177 3.5455 1.964 0.0051 5.477 0.09 3.4539 1.0198 0.001 30.5cm 0.54 0.045 3.579 1.654 0.049.4895 0.0106 3.376 0.6766 0.0045 5.4894 0.0107 3.476 0.0314 0.0047 30.5cm6 0.6048 0.105 3.3868 0.0855 0.031.5177 0.0178 3.4417 0.955 0.004 5.495 0.0051 3.559 0.003 0.001 90.5cm100 30.5cm1 0.6353 0.1353 3.436 0.9898 0.0.607 0.107 3.4738 0.6066 0.00 5.6171 0.1117 3.5148 0.6436 0.00053 30.5cm 0.6501 0.1501 3.7037 06445 0.044.658 0.158 3.399 0.473 0.00 5.6196 0.1196 3.5093 0.0806 0.0005 30.5cm6 0.7099 0.1 3.694 0.9148 0.003.6459 0.1459 3.4613 0.479 0.00 5.698 0.198 3.4964 0.604 0.0005 Table 3: Real data. 41187 5796 1133 167 1 3403 1666 467 3606 61 1591 488 5537 3701 185.1 Real DATA been estimated. Real data survey are also included. 15 data from census results of 014 rural household income in islamic republic of Iran are taken illustrative example has constructed. Based on proposed methods, unknown parameters of population has
M. Rajaei Salmasi et al. /IJIM Vol. 9, No. 1 017 91-97 95 Table 4: Estimation results. Method of Moments GOS Moment Unbiased ˆα ˆ ˆα ˆ.44 1114.3157 116 k = 1, m = 0 3 Inference 3.1 GOS moments of Pareto distribution It is well known that the expectation value variance of PD 1.4 are respectively, V arx = EX = α α 1 α α 1 α α > 1 3.13 α > 1, 3.14 second moment of PD can be derived by 3.13 3.14 as follows: EX = α α α > 3.15 In order to compute GOS moments of PD, it is necessary to obtain distribution of GOS from PD. Using 1., 1.4 1.5, PDF of rth GOS from PD can be presented as: f r x = ααγr C r 1 Γrm + 1 r 1 x >. 3.16 Based on 3.16, t th moment is given by E r t = x t f r xx, taking integral leads to the following relation, C r 1 E r t = t Γrm + 1 r Setting t=0 leads to.10. 3.17 can be rewritten as: E r t = t α t. 3.17 m + 1α Using.10, relation α t α. 3.18 First second moments can be obtained by setting t = 1, in the relation 3.18, therefore we get to: E r = α 1 α E r = αγr α, 3.19. 3.0 Variance of GOS from PD can be calculated as V ar r = [ α α αγr 1 α 3. Method of moments ]. 3.1 In order to get the moment estimators of unknown parameters of PD, we have to consider following relations, EX = X, EX = X, 3. where X = 1 n n i=1 x i X = 1 n n i=1 x i. 3..1 Method of moments using PD Using 3.13, 3.15 3., it can be obtained = α 1 α x, = α α x. 3.3 Using 3.3 simple calculation moment estimators of α could be achieved respectively: x ˆα m = 1 + x x, 3.4 ˆ m = ˆα m 1 ˆα m x. 3.5
96 M. Rajaei Salmasi et al. /IJIM Vol. 9, No. 1 017 91-97 3.. Method of moments using GOS from PD Using Theorem. similar to the 3..1, moment estimator of α could be obtained based on GOS from PD. According to relations 3., 3.13 3.15, clearly it can be written E where E X r X r 1 X r X r 1 = αγ r α 1 = Y, 3.6 = αγ3 r α = Y, α >, ny = γ 1 Y 1 + ny = γ 1 Y 1 + n i= n i= X i γ i X i 1 X i γ i X i 1,, 3.7 furthermore, Y i, i = 1,,, n are GOS from PD. Therefore, moment estimator for α based on GOS can be obtained as: Y ˆα gm = 1 + Y Y. 3.3 Unbiased estimation At this subsection has been estimated through unbiased estimator when α has been assumed to be known. Considering 3.19, it can be written where, Gα, r = r E =, 3.8 Gα, r αγr 1 α γr r Gα,r, therefore, it can be concluded ˆ Unb = is the unbiased estimator of. In order to evaluate consistency of the estimato variance can be computed. Using 3.1, variance of unbiased estimator can be obtained, V ar ˆUnb = 1 G α, r varr. 4 Numerical study 4.1 Simulated DATA For comparison performances of methods evaluation of estimators in different circumstances numerical study are considered. different samples with different sample sizes n = 10, 50, 100, values k=1,,6 m=0,,5 also with parameter values of α =.5, = 3.5 are derived based on algorithm discussed in [1]. 5 Conclusion In this paper distribution of ratios of GOS from PD were obtained. Single moments of were derived based on them moment estimators for PD unknown parameters were constructed. Using ratio distribution moment inference was done. Unbiased estimator based on moments of PD through GOS was derived, consistency of estimator was studied. To compare of methods different parameter values the numerical studies were presented. Based on numerical results following conclusions are obtained: Table 1 shows that when sample size increases, the MSE of both unknown parameters of PD estimated based on method of moments is low, so the estimators give better performances. Table 1 showed that the estimators based on GOS are better than the others. Based on GOS method of moments when n m increase simultaneously estimator gives better performances. Table shows that when n increases k decreases, estimator slightly performs good. Unbiased estimator gives better results when n k increase. Table shows that unbiased estimator of gets close to real parameter value, while n is growing, variance of estimator is decreasing, so unbiased estimator is asymptotically consistent.
M. Rajaei Salmasi et al. /IJIM Vol. 9, No. 1 017 91-97 97 References [1] Z. Aboeleneen, Inference for weibull distribution under generalized order statistics, Mathematics Computers in Simulation 81 010 6-36. [] A. P. Basu, R. H. Lochne On the distribution of the ratio of two rom variables having generalized life distributions, Technometrics 13 1971 81-87. [3] V. Brazauskas, R. Serfling, Robust estimation of tail parameters for two-parameter Pareto exponential models via generalized uantile statistics, Extremes 3 000 31-49. [4] N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions, Wiley, NewYork, ckem Col. 1, nd Ed. [5] U. Kamps, A concept of generalized order statistics, Teubner Skripten zur mathematischen Stochastik. B.G. Teubner 1995a [6] U. Kamps, A concept of generalized order statistics, Journal of Statistical Planning Inference 481995b 1-3. [13] S. J. Wu, Estimation for the two-parameter Pareto distribution under progressive censoring with uniform removals, J. Statist. Comput. Simul. 73 003 15-134. Mehdi Rajaei Salmasi was born in 1983 in Salmas, Iran. He is Ph.D of statistics received from science research branch of Islamic Azad University. He is member of department of mathematics at the Islamic Azad University, Urmia Branch, Iran. His research area is distribution estimation theory. Gholamhossein Yari was born in 1954 in Iran. He is associated professor of statistics at Iran University of Science Technology, Tehran. His interests are information theory stochastics processes. [7] G. Marsaglia, Ratios of normal variables ratios of sums of uniform variables, Journal of American Statistical Asoociation 60 1965 193-04. [8] S. Nadarajah, The Kotz type ratio distribution, Statistics 46 01 167-174. [9] V. Pareto, Cours conomie Politiue Profess a l Universit de Lausanne, Vol. I, 1896; Vol. II, 1897. [10] R. E. Qut, Old new methods of estimation the Pareto distribution, Metrika 10 1966 55-8. [11] S. K. Saksena, Estimation of parameters in a Pareto distribution simultaneous comparison of estimation, PH.D. Thesis, Louisiana Tech. University, Ruston, Lousiana. [1] K. Vnnman, Estimators Based on Order Statistics from a Pareto Distributio, Journal of the American Statistical Association 711976 704-708.