International Journal of Mathematics Trends and Technology IJMTT - Volume 35 Number 3- July 20 Single Stage Shrinkage Estimator for the Shape Parameter of the Pareto Distribution Abbas Najim Salman*, Adel Abdulkadhim Hussein** Department of Mathematics-Ibn-Al-Haitham College of Education - University of Baghdad Abbasnajim@yahoocom* adilabed57@yahoocom** Abstract: In this paper, we employee single stage shrinkage estimator for estimate the shape parameter of the Pareto distribution when the scale parameter is known The proposed estimator is shown to have a smaller mean squared error in a region around when comparison with the usual and existing estimators Keywords: Pareto Distribution, Shrinkage Estimation, Prior Guess, Pre Test Region, Bias, Mean Squared Error and Relative Efficiency Introduction The Model "Pareto distribution is a power law probability distribution that coincides with social, scientific, geophysical and many other types of observable phenomena It is a continuous distribution bounded on the lower side and it has two parameters, shape and mode and it is a highly skewed distribution It is a decreasing function and it has a finite value at the minimum value It is a heavy tailed distribution meaning that a random variable following a Pareto distribution can have extreme values The mode parameter for Pareto distribution sets the position of the left edge of the probability density function The only outcomes that can be observed from this distribution are greater than or equal to the value of the mode parameter Changes in the mode simply shift the boundary to the left or right Pareto distribution named after the Italian economist and sociologist Vilfredo Pareto -923 It was proposed first by Pareto in 97 at the university of Lausanne as a model for the distribution of income The pareto model is very often used as a basis of Excess of loss quotations as it gives a pretty good description of the random behavior of large losses, and often used to model the distribution of income ; [], [7],[],[2] and [] Consider a random sample t, t2,, tn from the Pareto distribution with the following pdf ISSN: 223-5373 Page 5
International Journal of Mathematics Trends and Technology IJMTT - Volume 35 Number 3- July 20 <0 =0 >0 0 ft 0 0 02 0 0 2 0 t Figure Shown pdf of Pareto distribution In conventional notation, we write t par,, where and are the shape and scale parameter respectively Sometimes we may have a prior guess value due past experiences or from an acquaintance with similar situation of the parameter to be estimated, if this value is very close to the true value, the shrinkage technique is useful to obtain an improved estimator According to Thompson [3], such prior estimate may arise for any one of a number of reasons, eg, we are estimating and i we believe is close to the true value of ; or ii We fear that may be near the true value of ; ie; A bad thing can happen when you do not we use it if = For such cases, this prior guess value may be utilized to improve the estimation procedure with usual estimator MLE via shrinkage estimator which including shrinkage weight factor as follows, see[5] = 2 The authors in [9],[3],[5],[] and others suggested shrinkage estimator 2 for estimating the parameters of different distribution when a guess prior value is available They showed that these estimators perform better than usual estimator in the term of mean square error when the guess value is close to the true value Consequently, to make sure whether the prior guess value is approximately or close to the true value or not, we may test H0: = vs HA: and a preliminary test shrunken estimator of significance was employed as follows: ISSN: 223-5373 / / 2 3 Page 57
International Journal of Mathematics Trends and Technology IJMTT - Volume 35 Number 3- July 20 Using pre-assigned level of significance, the critical region R for such test is pretest region using specific test statistic, where, i =, 2 is shrinkage weight factors such that 0 and is the usual estimator MLE for example Preliminary test shrinkage estimators 3 have been considered in different contexts by [],[2],[3], [0], [], [5] The aim of this paper is to estimate the shape parameter of pareto distribution using mentioned preliminary test shrinkage estimator 3 and study its behavior when we derived the expressions of its Bias, Mean Square Error and Relative Efficiency and study their performance Numerical results and conclusions for these expressions were made to show the effective of the proposed estimator as well as some comparisons with the usual and existing estimators were made 2 Usual Estimation of the Shape Parameter Let t, t2,, tn be identically independent pareto distributed random variables when the scale parameter is known, then the maximum likelihood estimator of as below: See[2] It follows easily that Ln t will be exponentially distributed with mean / y= will be Gamma distributed with parameters n and ie; y Gn, Therefore, when E Then, we get and var and the mean square error is 2 Single Stage Shrunken Estimator Recall the estimator 3 when Where R is the pre test region for test H0: ISSN: 223-5373 = vs HA: 3 using test statistic Page 5
International Journal of Mathematics Trends and Technology IJMTT - Volume 35 Number 3- July 20, ie; - For simplest we denote to R as below R=[a,b], where and are the lower and upper 00 /2 percentile of chi square distribution with degree of freedom 2n The Expression for Bias of the proposed estimator is defined as = [ ] where R is the complement region of R in real space and f is a pdf of which has the following form: [ ] 5 Then the Bias will be *, - +, we denote to the Bias Ratio as B which is defined as: 7 The expression of the Mean Squared Error MSE of, *,, The Efficiency of ISSN: 223-5373 is, - relative to the, - denoted by REff + R is defined as below: Page 59
International Journal of Mathematics Trends and Technology IJMTT - Volume 35 Number 3- July 20 9 see [],[2],[3] 3 Numerical Results and Discussion The computations of the statistical indicators Relative Efficiency [REff ] and Bias Ratio [B ] expressions were used for the considered testimators These computations were performed for the constants = 00,005,0, n =,,,0,2 and = 0250252 and k=exp-n Some of these computations are displayed in attached table for some samples of these constants The observation mentioned in the table leads to the following results: i The Relative Efficiency [REff ] of are adversely proportional with small value of especially when =, ie = 00 yield highest efficiency iithe Relative Efficiency [REff ] of has maximum value when = =, for each n,, and decreasing otherwise This feature shown the important usefulness of prior knowledge which given higher effects of proposed estimator as well as the important role of shrinkage technique iiibias ratio [B ] of increases when increases ivbias ratio [B ] of are reasonably small when = for each n,, and increases otherwise This property shown that the proposed estimator is very close to unbiased property especially when = vthe Relative efficiency [REff ] of increases function with increases value of n for =, and vice versa otherwise for each vithe Effective Interval [the value of that makes REff greater than one] using proposed estimator is [075,25] Here the pretest criterion is very important for guarantee that prior information is very closely to the actual value and prevent it far away from it, which get optimal effect of the considered estimator to obtain high efficiency viithe considered estimator is better than the classical estimator especially when, which is given the effective of and important weight of prior knowledge as well as the increment of efficiency may be reach to tens time viiithe proposed estimator has smaller MSE than some existing estimators introduced by authors, see for examples [] Conclusions From the above discussions it is obvious that by using guess point value one can improve the usual estimator It can be noted that if the guess point is very close to the true value of the parameter ie; =, the proposed estimators perform better than the usual estimator If one has no confidence in the guessed value then proposed preliminary test shrinkage estimators can be suggested We can safely use the proposed estimators for small sample size at usual level of significance and moderate value of shrinkage weight factors ISSN: 223-5373 5 Page 0
International Journal of Mathematics Trends and Technology IJMTT - Volume 35 Number 3- July 20 References Al-Hemyari,ZA, Khurshid, A and Al-Joboori, A N,2009, On Thompson Type 2 3 5 7 9 0 2 3 5 Estimators for the Mean of Normal Distribution, REVISTA INVESTIGACION OPERACIONAL J Vol 30, No2,09- Al-Joboori, AN, et al 20, Single and Double Stage Shrinkage Estimators for the normal Mean with the Variance Cases, International Journal of statistic, Vol3,2,PP, 27-3 Al-Joboori, AN, Khalef, BA and Hayder SK, 20, Improved of Stein-Type Estimator of the Variance of Normal Distribution via Shrinkage Estimation Technique, International Journal of Inventive Engineering and Science IJIES, Vol2,, PP-7 Al-Joboori, AN,AA Hussein and MAMohammd, 20,On Significance Testimator in Pareto Distribution Via Shrinkage Technique, Journal of College of E Casella, G, 2002, Shrinkage, In: Encyclopedia of Envirnmetrics,, 992-99 EdsAHELShaarawi and wwpiegorsch, John Wiley :New York Cirillo R, "The Economics of Vilfredo Pareto", Frank Case and Company Limited, London, 979 Malik, HJ, 970, Estimation of the Parameter of the Pareto Distribution, Metrika, Vol5, pp 2-32 Ord, JK, 975, Statistical Models for Personal income Distributions in Statistical 5, Dordrecht, Reidel Mehta,JS and Srinivasan,R, 97, "Estimation of the Mean by Shrinkage to a Point", J Amer Statist Assoc,, p-90 Saleh, AKE, 200, Theory of Preliminary Test and Stein-Type Estimators with Application, Wiley and Sons, NewYork Rytgaard, M, 990, Estimation in the Pareto Distribution, Astin Bulletin, Vol20, No2, pp20-2 Rashid, H A and Al-Gazi, N A A, 20, Bayes Estimators for the Shape Parameter of Pareto Distribution under Generalized Square Errors Loss Function, Mathematical Theory and Modeling, Vol,No pp 20-32 Thompson, JR, 9, Some Shrinkage Techniques for Estimating the Mean, J Amer Statist Assoc, Vol3, pp3-22 Prakash, G, Singh, DC and Singh, RD, 200, Some Test Estimator for the Scale Parameter of Classical Pareto Distribution, Journal of Statistical Research, Vol02, pp-5 Singh, DC, Prakash, G and Singh, P, 2007, Shrinkage Testimators of the Shape Parameter of Pareto Distribution Using LINEX Loss Function, Communication in Statistics-Theory and Methods, Vol3, pp7-753 Singh, DC, Singh, P and Singh, PR, 99, Shrunken Estimators for the Scale Parameter of Classical Pareto Distribution, Microelectron Reliability, Vol33, pp3539 ISSN: 223-5373 Page
International Journal of Mathematics Trends and Technology IJMTT - Volume 35 Number 3- July 20 Table : Shown Bais Ratio B and Relative Efficiency REff of estimator Δ n 00 0 2 005 0 2 0 0 2 025 REff 77777 B - REff 07 B - REff 0232 B - REff 02929 B -075000 REff 0222 B 0-07500 REff 77777 B - REff 07 B - REff 02320 B - REff 02929 B - REff 0222 B -075000 REff 77777 B - REff 07 B - REff 0232 B - REff 02929 B - REff 0222 B -075000 ISSN: 223-5373 05 0000 0 - - 0000 0-0000 0-075 00000 0000 - - - 00000 0000 - - 0000 0000-2779e+09 57e-0 0030e+02 5309e-0 599e+029 50e-0 5235e+03 03772e-09 379057e+039 33e-02 77e+05 75e-009 97e+020 5e-0 52e+025 3057e-0 955e+029 09e-0 5750e+03 7220e-09 29993e+03 395e-00 597e+0 735e-00 7037e+022 755e-03 955e+027 22e-05 399e+03 2775e-07 25 599999 9 0000 599999 00000 9 599999 0000 7 5 0000 0 0 0000 0 0000 0 75 77777 07 02320 02929 075000 0222 075000 77777 07 0232 02929 0222 075000 77777 07 0232 02929 0222 075000 2 00000 00000 02309 0 02727 00000 02309 0 027272 00000 02309 0 027272 Page 2