Maejo International Journal of Science and Technology

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1 Maejo Int. J. Sci. Technol. 018, 1(01), 11-7 Full Paper Maejo International Journal of Science and Technology ISSN Available online at Parameter estimation of Pareto distribution: some modified moment estimators Shahzad Hussain, Sajjad Haider Bhatti *, Tanvir Ahmad, Muhammad Aftab and Muhammad Tahir Department of Statistics, Government College University, Faisalabad, Pakistan *Corresponding author, Received: 8 November 016 / Accepted: 31 December 017 / Published: 10 January 018 Abstract: This article focuses on modified moment estimators for the parameter estimation of Pareto distribution. Four modified moment estimators are proposed and compared with the traditional method of moments estimation. The comparison is carried out through a simulation study using total relative deviation (TRD) and mean square error (MSE) as performance indicators. The simulation study was carried out using R-Language. Modified moment estimators based on mean of first-order statistics and expectation of empirical cumulative distribution function of first-order statistics are demonstrated to perform better than the traditional estimator and other modified moment estimators. The modified moment estimators perform much better when applied on real-life data where bootstrap MSE and TRD are used as performance measures. Keywords: bootstrapping, method of moments, mean square error, modified moment estimators, Pareto distribution, total relative deviation INTRODUCTION Pareto distribution was developed by Pareto [1] and it is widely used as an income model. The distribution was originally used to define the allocation of wealth among individual units since it seems to show rather well the way that a large proportion of wealth in any society is owned by a small percentage of people in that society []. According to Burroughs and Tebbens [3], the main applications of Pareto distribution are in the modelling of earthquakes, forestry fire areas, and oil and gas explorations in different field sizes. The importance of the distribution in real-life problems is evident in many studies [4-9].

2 Maejo Int. J. Sci. Technol. 018, 1(01), Different estimation methods have been implemented to estimate the parameters of Pareto distribution. Quandt [10] derived algebraic expressions of different estimation methods, namely method of moments (MM), method of maximum likelihood, quantiles method and method of least squares. Afify [11] employed three distinct estimation procedures to estimate the parameters of Pareto distribution. He compared the maximum product spacing method of estimation with the ridge regression and least square methods. Lu and Tao [1] considered the weighted least-square method to estimate the parameters of Pareto distribution. In the recent literature different modifications have been proposed in different estimators, which performed better than the traditional estimators in most cases. Such modifications have been applied for the parameter estimation of different probability distributions. The concept of a modified estimator was coined by Cohen and Whitten [13], who derived the modified moment estimators and modified maximum likelihood estimators for Weibull distribution. Mostly their modifications were based on order statistics. Their numerical evaluation showed that the modified moment and modified maximum likelihood estimators gave superior accuracy as compared to the traditional moment and maximum likelihood estimators respectively. Rashid and Akhter [14] have suggested some modifications in the MM and maximum likelihood for Exponential distribution. Their results are in favour of the modified estimators in comparison to the traditional ones, particularly in the case of modified moment estimators. Zaka and Akhter [15] have also proposed some modified moment and modified maximum likelihood estimators for Power Function distribution. Their results suggest that the modified estimators outperform the basic or traditional estimators. However, some of their proposed modifications derived through the MM have no practical meanings because two modified moment estimators have the same expressions when variance or coefficient of variation is replaced in the nd moment of Power Function distribution. This makes the two modifications identical. These are the major drawbacks in their proposed modifications in the MM. In this article we propose some modifications in the moment estimators for parameter estimation of Pareto distribution. These newly proposed modifications are derived using variance, harmonic mean, mean of first-order statistics and expectation of empirical cumulative distribution function (CDF) of first-order statistics. The core motivation behind all modifications is to estimate parameters of the widely applied Pareto distribution as efficiently and accurately as possible. The performance of the proposed modified estimators is compared through a simulation study and a real-life data example. METHODS We propose four modifications to the MM based on variance, harmonic mean, mean of firstorder statistics and expectation of empirical CDF of first-order statistics of Pareto distribution. The following section briefly explains the derivation of moment estimators and proposed modified moment estimators for Pareto distribution. Method of Moments (MM) The MM proposed by Pearson [16] is one of the oldest methods for parameter estimation. Due to its conceptual and methodological simplicity, it is extensively used in the literature for parameter estimation of almost all probability distributions. This method consists in equating

3 Maejo Int. J. Sci. Technol. 018, 1(01), sample moments to corresponding moments of the parent population and then solving a system of equations for unknown parameters. Consider t1, t, t3,..., t n as a random sample from Pareto distribution with its probability density function given as 1 f t t and, 0 t r ti th i 1 The r sample moment is defined as m r n th The r population moment of Pareto distribution is defined as r r E t t. f t dt, n, where mstands r for th r sample moment. r r E t r Putting r 1, we get the first two population moments about the origin denoted by 1 and : 1 1 and Now equating the corresponding sample and population moments, we get m and, m 1 1 t, (1) 1 s t () Solving equations (1) and (), we get the estimates of α and β as 1 1 t, s (3) s t s s t t (4) Equations (3) and (4) are the estimators derived through MM for Pareto distribution. Modified Moment Estimator (I) For the first modification in the MM, we followed Cohen and Whitten [13], Rashid and Akhter [14] and Zaka and Akhter [15], who derived the modified moment estimators for Weibull, Exponential and Power Function distributions respectively. In this modification which is based on variance, equation () is replaced by the variance of Pareto distribution. The detailed derivation is given as s (5) 1.

4 Maejo Int. J. Sci. Technol. 018, 1(01), From equation (5) we get s 1. 1 (6) Solving equations (1) and (6) for and, we get the estimator for unknown parameters as t 1 1 s (7) Putting the value of from equation (7) in equation (6), we get t s , s t 1 1 s s t s s t s s s t s t s s t s s t, Equations (7) and (8) provide algebraic expressions for the 1 st modified moment (MM-I) estimators of Pareto distribution. Modified Moment Estimator (II) Our second modification is based on the harmonic mean of Pareto distribution. For this we also followed the pattern used by Cohen and Whitten [13], Rashid and Akhter [14] and Zaka and Akhter [15], for modified estimators of Weibull, Exponential and Power Function distributions respectively. In this modification equation () is replaced by the harmonic mean (HM) of Pareto distribution: From equation (9), 1 HM 1 (9) HM (10) 1 1 Solving equations (3) and (10) simultaneously for unknown parameters, we get the required estimators of and as HM 1, (11) t t H. M HM t t HM (8) (1)

5 Maejo Int. J. Sci. Technol. 018, 1(01), Thus, equations (11) and (1) provide algebraic expressions of the nd modified moment (MM-II) estimators of Pareto distribution. Modified Moment Estimator (III) For the third modification in the MM, we followed the procedure given by Rashid and Akhter [14], who derived the modified moment estimator for Exponential distribution. In this modification the second moment about origin (i.e. equation ) is replaced by the mean of first-order statistics of Pareto distribution. The detailed derivation is given below. The first moment about origin of Pareto distribution is t (13) 1 In general, for any probability distribution the mean of first order-statistics is 1 1 E t t For the particular case of Pareto distribution, the mean of fist-order statistics from Afify [17] is n t n 1 1 Solving equations (13) and (14), we get t 1 1, t n t1 nt n t t 1 Equations (15) and (16) are the 3 rd modified moment (MM-III) estimators of Pareto distribution. Modified Moment Estimator (IV) For our fourth modification in the MM, we followed Rashid and Akhter [14], who derived the modified moment estimator for Exponential distribution. In this modification the nd moment about origin (i.e. equation ) is replaced by expectation of the empirical CDF of first-order statistics. From equation (1), we have t 1 From the literature [13-15], the expectation of empirical CDF of first-order statistics is F t E F t, n 1 1 1, n 1 1 t (14) (15) (16) (17) (18) t 1 1 n n 1 (19)

6 Maejo Int. J. Sci. Technol. 018, 1(01), Putting the value of from equation (19) in equation (17), we get 1 t t n Applying Binomial series on the first term of the right hand side and ignoring the higher-order terms, t , t n n 1 t n t t 1 Equations (19) and (0) are algebraic expressions for the 4 th modified moment (MM-IV) estimators of Pareto distribution. Simulation Study A simulation study was performed to compare the performance of the proposed modified moment estimators with one another and with traditional moment estimators. The Simulation study was carried out for eight different combinations of parameter values (, 3; 3, 4; 0.5, 1; 1, 1;, 1; 4, 1; 1, ; and 4, 5 ) with various sample sizes (n = 0, 50, 100, 00, 500 and 1000). We generated random samples of different sizes by observing that if R is uniform (0, 1), then (0) t i 1 1 R is the Pareto random variable with parameters α and β. All the simulation results were based on 10,000 replications using R-Language [18]. Total relative deviation (TRD) and mean square error (MSE) were used as performance indicators, which are defined as TRD, MSE E, MSE E These performance indicators are commonly used for this purpose in such type of comparisons as in Al-Fawzan [19], Rashid and Akhter [14] and Zaka and Akhter [15]. RESULTS AND DISCUSSION Results from Simulation Study Results from the simulation study are given in Tables 1-8 for different combinations of parameter values. These results were obtained for different sample sizes and used for analysing the accuracy of estimators derived through the MM and the proposed modified estimators. From the results in Table 1 (for 3, ), it is observed that for scale parameter ( ), MM and MM-I overestimate while MM-II underestimates the true parameter values. The estimates

7 Maejo Int. J. Sci. Technol. 018, 1(01), from MM-III and MM-IV are much closer to the true parameter values. Similarly, with respect to shape parameter ( ), MM and MM-I overestimate while MM-II underestimates the true parameter values. However, estimates from MM-III and MM-IV are much closer to the true values. In general, MM-III and MM-IV outperform the traditional moment estimator and other two modified moment estimators throughout the sample sizes based on both of the performance indices used (i.e. MSE and TRD). For example, for first parameter combination ( 3, ), the values of TRD for MM- IV (TRD = , , , , and for sample size 0, 50, 100, 00, 500 and 1000 respectively) are much lower than those for MM, MM-I and MM-II, and closer (but lower) than that for MM-III. Similarly, values of MSE for different sample sizes are lower in the case of MM-IV compared to other estimators considered. Similar patterns of results are evident in Tables -8 for other parameter combinations considered. Although MM-III and MM-IV have almost identical results, we can make a choice in favour of MM-IV based on the values of both performance indices, while MM and MM-I are found to be poorest performers, particularly for small samples. Application to Real-Life Data After comparing the modified moment estimators by means of simulation study, we compared the performance of these estimators using a set of real-life data taken from Beirliant et al. [0]. The data consisted of fire damage claims (in thousands of Norwegian Krones) in Norway during The same data have also been used in different studies on Pareto distribution like those by Munir et al. [], Brazauskas and Serfling [1], Kaiser and Brazauskas [], and Obradović [3]. First, it was determined that the data followed Pareto distribution by applying Kolmogorov-Smirnov test (p-value>0.75) [4-5]. In comparison based on real-life data, bootstrap estimates of bias, mean square error and total relative deviation were used as performance indicators. Bootstrapping [6-8] was performed using boot package in R- language [9] with 1000 bootstrap replicates. Specifically, the following formulas were used for calculations of estimated bias, estimated MSE and estimated TRD with bootstrap samples: Estimated Bias( ) ( ), Bootstrapp s Original Estimated Bias ( ) ( ), MSE Bootstrapp s ( ), Bootstrapp Original s MSE Original ( ), Bootstrapp Original s ( ) ( ) Bootstrapp Original Bootstrapp Original s s TRD Original Results from the real-life data are given in Table 9. These results show that MM-III and MM-IV perform much better than other competing estimators. However, the results from these Original

8 Maejo Int. J. Sci. Technol. 018, 1(01), two estimators are almost identical in terms of bias, MSE and TRD computed through bootstrapping. Table 1. Comparison of MM, MM-I, MM-II, MM-III and MM-IV for, 3 Method Size True β True α E E MSE ( ) MSE ( ) MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E TRD

9 Maejo Int. J. Sci. Technol. 018, 1(01), Table. Comparison of MM, MM-I, MM-II, MM-III and MM-IV for 3, 4 Method E MSE ( ) MSE ( ) TRD Size MM True β True α E MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E

10 Maejo Int. J. Sci. Technol. 018, 1(01), Table 3. Comparison of MM, MM-I, MM-II, MM-III and MM-IV for 0.5, 1 Method Size True β True α E E MSE ( ) MSE ( ) MM E MM-I E MM-II MM-III MM-IV MM E MM-I E MM-II MM-III MM-IV MM E MM-I E MM-II MM-III MM-IV MM E MM-I E MM-II MM-III MM-IV MM E MM-I E MM-II MM-III E MM-IV E MM E MM-I E MM-II MM-III E MM-IV E TRD

11 Maejo Int. J. Sci. Technol. 018, 1(01), Table 4. Comparison of MM, MM-I, MM-II, MM-III and MM-IV for 1, 1 Method Size True β True α E E MSE ( ) MSE ( ) MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E TRD

12 Maejo Int. J. Sci. Technol. 018, 1(01), 11-7 Table 5. Comparison of MM, MM-I, MM-II, MM-III and MM-IV for, 1 Method Size True β True α E E MSE ( ) MSE ( ) MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E TRD

13 Maejo Int. J. Sci. Technol. 018, 1(01), Table 6. Comparison of MM, MM-I, MM-II, MM-III and MM-IV for 4, 1 Method Size True β True α E E MSE ( ) MSE ( ) MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E TRD

14 Maejo Int. J. Sci. Technol. 018, 1(01), Table 7. Comparison of MM, MM-I, MM-II, MM-III and MM-IV for 1, Method Size True β True α E E MSE ( ) MSE ( ) MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E TRD

15 Maejo Int. J. Sci. Technol. 018, 1(01), Table 8. Comparison of MM, MM-I, MM-II, MM-III and MM-IV for 4, 5 Method Size True β True α E E MSE ( ) MSE ( ) MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III MM-IV MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E MM MM-I MM-II MM-III E MM-IV E TRD

16 Maejo Int. J. Sci. Technol. 018, 1(01), Table 9. Application of proposed estimators to real-life data Estimated Estimated Method Parameter Estimate Bias MSE MM MM-I MM-II MM-III MM-IV Estimated TRD CONCLUSIONS Four modifications in the method of moments for parameter estimation of Pareto distribution have been presented. Through a simulation study it has been established that the modified estimators based on mean of first-order statistics and expectation of empirical cumulative distribution function of first-order statistics are found superior to the traditional and modified moment estimators derived using variance and harmonic mean. These two modified estimators may be recommended for parameter estimation as they minimise the magnitude of underestimation or overestimation of parameters. REFERENCES 1. V. Pareto, "The New Theories of Economics", J. Polit. Econ., 1897, 5, R. Munir, M. Saleem, M. Aslam and S. Ali, "Comparison of different methods of parameters estimation for pareto model", Casp. J. Appl. Sci. Res., 013,, S. M. Burroughs and S. F. Tebbens, "Upper-truncated power law distributions", Fractals, 001, 9, N. H. Abdel-All, M. A. W. Mahmoud and H. N. Abd-Ellah, "Geometrical properties of pareto distribution", Appl. Math. Comput., 003, 145, P. G. Sankaran and M. T. Nair, "On finite mixture of pareto distributions", Calcutta Stat. Assoc. Bull., 005, 57, M. E. J. Newman, "Power laws, Pareto distributions and Zipf s law", Contemp. Phys., 005, 46, I. B. Aban, M. M. Meerschaert and A. K. Panorska, "Parameter estimation for the truncated Pareto distribution", J. Am. Stat. Assoc., 006, 101, B. C. Arnold, "Pareto distribution", in "Encyclopedia of Statistical Sciences" (Ed. S. Kotz, C. B. Read, N. Balakrishnan and B. Vidakovic), John Wiley and Sons, Hoboken, D. R. Clark, "A note on the upper-truncated Pareto distribution", Casualty Actuar. Soc. E- Forum Winter, 013, 1, R. E. Quandt, "Old and new methods of estimation and the Pareto distribution", Metrika, 1966, 10, 55-8.

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